Applications of Stochastic Calculus of Variations to ...duedahl.org/1665_Duedahl_materie.pdf · Applications of Stochastic Calculus of Variations to Sensitivity Analysis and Related

Applications of Stochastic Calculus of Variations toSensitivity Analysis and Related Problems in Finance and Insurance

Sindre Duedahl

Dissertation presented for the Degree of Philosophiæ Doctor

Department of Mathematics

University of Oslo

2015

© Sindre Duedahl, 2015 Series of dissertations submitted to the Faculty of Mathematics and Natural Sciences, University of Oslo No. 1665 ISSN 1501-7710 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. Cover: Hanne Baadsgaard Utigard. Print production: John Grieg AS, Bergen. Produced in co-operation with Akademika Publishing. The thesis is produced by Akademika Publishing merely in connection with the thesis defence. Kindly direct all inquiries regarding the thesis to the copyright holder or the unit which grants the doctorate.

Acknowledgements

I wish to thank my main advisor, Frank Proske, for his suggestion thatI should begin in February of 2012 the work toward a Ph. D. degree instochastic analysis, mathematical finance and insurance, and for his generoushelp and encouragement throughout the process.

I would also like to thank our coauthors David Ruiz Banos, Erik Bølviken,and Thilo Meyer-Brandis, for a very interesting collaboration.

Thanks to Erik Bedos for agreeing to be my second advisor. He has beenvery helpful and supportive whenever I have talked to him.

It should also be mentioned here that my work on the thesis has bene-fited concretely from discussions with Fred Benth, Lars Lødøen Halsteinslid,Henrik Paulov Hammer, and Olav Skutlaberg.

Moreover I want to thank Professor Reinhard Klein for giving me theopportunity since February of 2015 to be employed in his research group atthe University of Bonn, Germany, where I have enjoyed great flexibility tofinish the thesis alongside other highly interesting projects.

I also want to thank my family and friends for support and encourage-ment.

Bonn, October 23, 2015Sindre Duedahl

1

Contents

• An introductory note

• Paper I Banos, D.R, Duedahl, S., Meyer-Brandis, T., Proske, F. N. Construction ofMalliavin differentiable strong solutions of SDEs under an integrability condition onthe drift without the Yamada-Watanabe principle.

• Paper II Banos, D.R, Duedahl, S., Meyer-Brandis, T., Proske, F., Computing deltaswithout derivatives.

• Paper III Duedahl, S., Implementation of stochastic yield curve duration and port-folio immunization strategies.

• Paper IV Bølviken, E., Duedahl, S., Proske, F., Modeling and estimation of sto-chastic transition rates in life insurance with regime switching based on generalizedCox processes.

1. An introductory note

This thesis deals with the application of several different kinds of variational or infinite-dimensional methods in stochastic analysis to problems of mathematical finance and insur-ance. The first three papers involve applications to the problem of measuring the sensitivityof portfolio values to changes in different market conditions, such as the term structure ofinterest rates, the spot price of a security underlying a derivative contract, or other quantitiesof interest for investors and risk managers. The last paper deals with estimation of the pa-rameters of a life insurance model with stochastic transition rates based on generalized Coxprocesses and nonlinear filtering theory.

1.1. Paper I: Strong solutions and the Bismut-Elworthy-Li formula. This paperaims at the construction of strong solutions of stochastic differential equations where thedrift condition is allowed to be highly irregular, with the sole restriction being that of globalintegrability. The solutions are shown to be Malliavin differentiable, and a Bismut-Elworthy-Li formula is derived for the solutions of the associated Kolmogorov equation. In the nextpaper, a similar formula is applied to the computation of sensitivity parameters in an optionpricing model.

1.2. Paper II: Sensitivity analysis of derivative prices with irregular drift coeffi-cients. An important problem in mathematical finance is the computation of the parametersknown as Greeks, which measure the sensitivity of the price of a derivative contract to changesin different market conditions, such as the price of an underlying security or the term struc-ture of interest rates. In practice, the payoff functions involved are often not smooth or evencontinuous, e.g. binary options, European call options, etc., and it is of interest to constructrepresentations of these quantities which do not assume smoothness of the payoff function.The Malliavin calculus has been successfully applied to this problem. It leads to numericallytractable computations even in the case of discontinuous payoff functions. However, in theexisting literature it is still assumed that the SDE which governs the dynamics of the under-lying diffusion process has smooth coefficients. This excludes certain scenarios which couldarise in practice, such as regime-switching; an example of this could be an abrupt changein central bank policy in the case of, for instance, interest rates passing through a certainthreshold. In this paper a representation for the Delta, one of the most important Greeks, isderived, only assuming that the drift coefficient is decomposable into the sum of a bounded

and measurable part and a Lipschitz continuous part. The method is applied to lookback andAsian options, and several simulation examples are given which involve a regime-switchingfeature of the dynamics.

1.3. Paper III: Numerical implementation of duration in stochastic interest ratemodels, and applications to the immunization of portfolios. In this paper we buildon earlier work by Kettler, Proske and Rubtsov (2012), which introduced a new notion ofstochastic duration, i.e. sensitivity of a portfolio with respect to the stochastic evolution ofthe term structure of interest rates. This notion of duration generalizes the much earlierdeterministic concept introduced by Macaulay (1938), which is widely used by practitioners,despite its severe limitation of assuming a deterministic and piecewise-constant interest ratestructure. The new notion of duration leads to an analytically intractable forward-backwardstochastic partial differential equation, which makes it desirable to develop a method of di-rectly numerically estimating the parameters which determine the stochastic duration, andPaper III makes an attempt at this, using a Monte Carlo simulation method based on non-linear filtering to estimate the parameters. An implementation of the algorithm in MatLabis included in the paper.

1.4. Paper IV: Stochastic transition rates, life insurance, and nonlinear filtering.The last paper concerns a life insurance model featuring stochastic transition rates of the stateprocesses. The model is non-Gaussian, and is constructed using generalized Cox processes,which makes it useful for capturing ”regime switching” effects similar to the ones mentionedin connection with Paper II. A method based on nonlinear filtering for Levy processes is usedto estimate the unknown parameters of the model.

I

CONSTRUCTION OF MALLIAVIN DIFFERENTIABLE STRONG SOLUTIONS

OF SDES UNDER AN INTEGRABILITY CONDITION ON THE DRIFT

WITHOUT THE YAMADA-WATANABE PRINCIPLE

DAVID R. BANOS, SINDRE DUEDAHL, THILO MEYER-BRANDIS, AND FRANK PROSKE

Abstract. In this paper we aim at employing a compactness criterion of Da Prato, Malliavin,Nualart [7] for square integrable Brownian functionals to construct unique strong solutions ofSDE’s under an integrability condition on the drift coe�cient. The obtained solutions turn outto be Malliavin di�erentiable and are used to derive a Bismut-Elworthy-Li formula for solutionsof the Kolmogorov equation.

Key words and phrases: Strong solutions of SDEs, Malliavin regularity, Kolmogorov equa-tion, Bismut-Elworthy-Li formula, singular drift coe�cient.

MSC2010: 60H10, 60H07, 60H40, 60J60.

1. Introduction

The object of study of this paper is the stochastic di�erential equation (SDE)

Xt = x+

Z t

0

b(s,Xxs )ds+Bt, 0 � t � T, x 2 R

d, (1.1)

where B· is a d-dimensional Brownian motion on some complete probability space (�,F , μ) withrespect to a μ-completed Brownian �ltration {Ft}0�t�T and where b : [0, T ] � R

d ! Rd is a

Borel-measurable function.In this article we are interested in the analysis of strong solutions X· of the SDE (1.1), that is

an {Ft}0�t�T -adapted solution processes on (�,F , μ) when the drift coe�cient is irregular, e.g.non-Lipschitzian or discontinuous.A widely used construction method for strong solutions in this case in the literature is based on

the so-called Yamada-Watanabe principle. Using this principle, a once constructed weak solution,that is a solution which is not necessarily a functional of the driving noise, combined with pathwiseuniqueness gives a unique strong solution. So

Weak solution + Pathwise uniqueness ) Unique strong solution . (1.2)

Here, pathwise uniqueness means the following: If X(1)· and X

(2)· are {F

(1)t }0�t�T - and re-

spectively {F(2)t }0�t�T -adapted weak solutions on a probability space, then these solutions must

coincide a.s. See [63]. In the milestone paper from 1974 [64], A.K. Zvonkin used the Yamada-Watanabe principle in the one-dimensional case in connection with PDE techniques to constructa unique strong solution to (1.1), when b is merely bounded and measurable. Subsequently, thelatter result was generalised by A.Y. Veretennikov [60] to the multidimensional case.Important other and more recent results in this direction are e.g. [30], [22] and [29]. See also the

striking work [8] in the Hilbert space setting, where the authors use solutions of in�nite-dimensionalKolmogorov equations to obtain unique strong solutions of stochastic evolution equations withbounded and measurable drift for a.e. initial values.In this article we want to employ a construction principle for strong solutions developed in

[41]. This method which relies on a compactness criterion from Malliavin Calculus for square1

2 D. R. BANOS, S. DUEDAHL, T. MEYER-BRANDIS, AND F. PROSKE

integrable functionals of the Brownian motion [7] is in diametrical opposition to the Yamada-Watanabe principle (1.2) in the sense that

Strong existence + Uniqueness in law ) Strong uniqueness ,

that is the existence of a strong solution to (1.1) and uniqueness in law of solutions imply theexistence of a unique strong solution. A crucial consequence of this approach is the additionalinsight that the constructed solutions are regular in the sense of Malliavin di�erentiability.We mention that this method has been recently applied in a series of other papers. See e.g.

[38], where the authors obtain Malliavin di�erentiable solutions when the drift coe�cient in Rd

is bounded and measurable. Other applications pertain to the stochastic transport equation withsingular coe�cients [42], [44] or stochastic evolution equations in Hilbert spaces with boundedHolder-continuous drift [16]. See also [25] in the case of truncated �-stable processes as drivingnoise and [1] in the case of fractional Brownian motion for Hurst parameter H < 1/2, which is anon-Markovian driving noise.Using the above mentioned new approach, one of the objectives of this paper is to construct

Malliavin di�erentiable unique strong solutions to (1.1) under the integrability condition

b 2 Lq([0, T ], Lp(Rd,Rd)) (1.3)

for p � 2, q > 2 such thatd

p+2

q< 1.

The idea for the proof rests on a mixture of techniques in [38] and [19]. More precisely, weapproximate in the �rst step the drift coe�cient b by smooth functions bn with compact supportand apply the Ito-Tanaka-Zvonkin ”trick” by transforming the solutions Xn,x

t of (1.1) associatedwith the coe�cients bn to processes

Y n,xt := Xn,x

t + Un(t,Xn,xt ),

where the processes Y n,xt satisfy an equation with more regular coe�cients than (1.1) given by

dY n,xt = �Un(t,X

n,xt )dt+ (Id +rUn(t,X

n,xt )) dBt

for solutions Un to the backward PDE’s

@Un

@t+1

2�Un + bnrUn = �Un � bn, Un(T, x) = 0. (1.4)

In the second step we use the compactness criterion for L2(�) in [7] applied to the sequenceY n,xt , n � 1 in connection with Schauder-type of estimates of solutions of (1.4) and techniquesfrom white noise analysis to show that

Y n,xt

n!1��! Y x

t

in L2(�) for all t and thatXx

t = '(t, Y xt ),

where '(t, ·) is the inverse of the function x 7! x+ U(t, x) for all t and U a solution of (1.4), is aMalliavin di�erentiable unique strong solution of (1.1).Our paper is organised as follows: In Section 2 we present our main results on the construction

of strong solutions (Theorem 2.1 and Theorem 2.15). As an application of the results obtainedin Section 2 we establish in Section 3 a Bismut-Elworthy-Li formula for the representation of �rstorder derivatives of solutions of Kolmogorov equations.

2. Main results

In this section, we want to further develop the ideas introduced in [19] and [41] to derive Malli-avin di�erentiable strong solutions of stochastic di�erential equations with irregular coe�cients.More precisely, we aim at analyzing the SDE’s of the form

dXt = b(t,Xt)dt+ dBt, 0 � t � 1, X0 = x 2 Rd , (2.1)

CONSTRUCTION OF MALLIAVIN DIFFERENTIABLE STRONG SOLUTIONS 3

where the drift coe�cient b : [0, T ] � Rd �! R

d is a Borel measurable function satisfying someintegrability condition and Bt is a d-dimensional Brownian motion with respect to the stochasticbasis

(�,F , μ) , {Ft}0�t�T (2.2)

for the μ�augmented �ltration {Ft}0�t�T generated by Bt. At the end of this section we shall

also apply our technique to equations with more general di�usion coe�cients (Theorem 2.15).Consider the space

Lqp := Lq

�[0, T ], Lp(Rd,Rd)

�for p, q 2 R satisfying the following condition

p > 2, q > 2 andd

p+2

q< 1 (2.3)

and denote by | · | the Euclidean norm in Rd. The Banach space Lq

p is endowed with the norm

kfkLqp=

Z T

0

�ZRd

|f(t, x)|pdx

�q/p

dt

!1/q

<1 (2.4)

for f 2 Lqp.

The main goal of the paper is to show that SDE’s of the type (2.1) with drift coe�cient bsatisfying the integrability condition given in (2.4) admit strong solutions that are unique and inaddition, Malliavin di�erentiable.So, our main result is the following theorem:

Theorem 2.1. Suppose that the drift coe�cient b : [0, T ] � Rd ! R

d in (2.1) belongs to Lqp.

Then there exists a unique global strong solution X to equation (2.1) such that Xt is Malliavindi�erentiable for all 0 � t � T .

An important step of the proof of Theorem 2.1 is directly based on the study of the regularityof solutions to the following associated PDE to equation SDE (2.1).

@tU(t, x) + b(t, x) · rU(t, x) +1

2�U(t, x)� �U(t, x) + b = 0, t 2 [0, T ], U(T, x) = 0, (2.5)

where U : [0, T ]� Rd ! R

d, � > 0 and b 2 Lqp.

The following result is due to [18] and stablishes the well-posedness of the above PDE problemin a certain space.First, recall the de�nition of the following functional spaces

Hq�,p = Lq([0, T ],W�,p(Rd)), H

�,qp =W �,q([0, T ], Lp(Rd))

andHq

�,p = Hq�,p \H

1,qp .

The norm in Hq�,p can be taken to be

kukHq�,p

� kukHq�,p+ k@tukLq

p.

Theorem 2.2. Let p, q be such that p � 2, q > 2 and dp +

2q < 1 and � > 0. Consider two vector

�elds b,� 2 Lqp. Then there exists a unique solution of the backward parabolic system

@tu+1

2�u+ b · ru� �u+ � = 0, t 2 [0, T ], u(T, x) = 0 (2.6)

belonging to the space

Hq2,p := Lq([0, T ],W 2,p(Rd)) \W 1,q([0, T ], Lp(Rd)),

i.e. there exists a constant C > 0 depending only on d, p, q, T,� and kbkLqpsuch that

kukHq2,p

� Ck�kLqp. (2.7)

The following result is a part of [30, Lemma 10.2] that gives us some properties on the regularityof u 2 Hq

2,p that we will need for the proof of Theorem 2.1.


Lemma 2.3. Let p, q 2 (1,1) such that dp +

2q < 1 and u 2 Hq

2,p, then ru is Holder continuous

in (t, x) 2 [0, T ]� Rd, namely for any " 2 (0, 1) satisfying

"+d

p+2

q< 1

there exists a constant C > 0 depending only on p, q and " such that for all s, t 2 [0, T ] andx, y 2 R

d, x 6= y

kru(t, x)�ru(s, x)k � C|t� s|"/2kruk1�1/q�"/2

Hq2,p

k@tuk1/q+"/2

Lqp

, (2.8)

kru(t, x)k+kru(t, x)�ru(t, y)k

|x� y|"� CT�1/q

�kukHq

2,p+ Tk@tukLq

p

�, (2.9)

where k · k denotes any norm in Rd�d

Our method to construct strong solutions is actually motivated by the following observation in[34] and [39] (see also [40]).

Proposition 2.4. Suppose that the drift coe�cient b : [0, T ]� Rd�! R

d in (2.1) is bounded andLipschitz continuous. Then the unique strong solution Xt = (X

1t , ..., X

dt ) of (2.1) has the explicit

representation

'�t,Xi

t(!)�= Eeμ

h'�t, eBi

t(e!)� E�T (b)

i(2.10)

for all ' : [0, T ]�R �! R such that '�t, Bi

t

�2 L2(�) for all 0 � t � T, i = 1, . . . , d,. The random

element E�T (b) is given by

E�T (b)(!, e!) := exp� � dX

j=1

Z T

0

�W j

s (!) + bj(s, eBs(e!))� d eBj

s(e!)�1

2

Z T

0

�W j

s (!) + bj(s, eBs(e!))��2

ds�. (2.11)

Here�e�, eF , eμ� ,� eBt

�t�0

is a copy of the quadruple (�,F , μ) , (Bt)t�0 in (2.2). Further Eeμ denotes

a Pettis integral of random elements � : e� �! (S)�with respect to the measure eμ. The Wick

product � in the Wick exponential of (2.11) (see A.15) is taken with respect to μ and W jt is the

white noise of Bjt in the Hida space (S)

�(see (A.12)). The stochastic integrals

R T

0�(t, e!)d eBj

s(e!)in (2.11) are de�ned for predictable integrands � with values in the conuclear space (S)

�. See e.g.

[26] for de�nitions. The other integral type in (2.11) is to be understood in the sense of Pettis.

Remark 2.5. Let 0 = tn1 < tn2 < . . . < tnmn= T be a sequence of partitions of the interval

[0, T ] with maxmn�1i=1

��tni+1 � tni�� ! 0 . Then the stochastic integral of the white noise W j can be

approximated as follows:Z T

0

W js (!)d

eBjs(e!) = lim

n�!1

mnXi=1

( eBjtni+1(e!)� eBj

tni(e!))W j

tni(!)

in L2(� � eμ; (S)�). For more information about stochastic integration on conuclear spaces thereader is referred to [26].

In the sequel we shall use the notation Y i,bt for the expectation on the right hand side of (2.10)

for '(t, x) = x, that is

Y i,bt := Eeμ

h eB(i)t E�

T (b)i

for i = 1, . . . , d. We set

Y bt =

�Y 1,bt , . . . , Y d,b

t

�. (2.12)

The form of Formula (2.10) in Proposition 2.4 actually gives rise to the conjecture that theexpectation on the right hand side of Y b

t in (2.12) may also de�ne solutions of (2.1) for driftcoe�cients b lying in Lq

p.


Our method to construct strong solutions to SDE (2.1) which are Malliavin di�erentiable isessentially based on three steps.

• First, we consider a sequence of compactly supported smooth functions bn : [0, T ]�Rd !

Rd, n � 0 such that b0 := b and supn�0 kbnkLq

p< 1 approximating b 2 Lq

p a.e. withrespect to the Lebesgue measure and then we prove that the sequence of strong solutionsXn

t = Y bnt , n � 1, is relatively compact in L2(�;Rd) (Corollary 2.9) for every t 2 [0, T ].

The main tool to verify compactness is the bound in Lemma 2.6 in connection with acompactness criterion in terms of Malliavin derivatives obtained in [7] (see Appendix B).This step is one of the main contributions of this paper.

• Secondly, given a merely measurable drift coe�cient b in the space Lqp, we show that Y

bt , t 2

[0, T ] is a generalized process in the Hida distribution space and we invoke the S-transform(A.13) to prove that for a given sequence of a.e. approximating, smooth coe�cients bnwith compact support such that supn�0 kbnkLq

p, a subsequence of the corresponding strong

solutions Xnj

t = Ybnj

t ful�ls

Ybnj

t ! Y bt

in L2(�;Rd) for 0 � t � T (Lemma 2.12).• Finally, using a certain transformation property for Y b

t (Lemma 2.14) we directly showthat Y b

t is a Maaliavin di�erentiable solution to (2.1).

We turn now to the �rst step of our procedure. The successful completion of the �rst step relieson the following essential lemma:

Lemma 2.6. Let bn : [0, T ] � Rd ! R

d, n � 1 be a sequence of functions in C10 (R

d) (space ofin�nitely often di�erentiable functions with compact support) approximating b 2 Lq

p a.e. such that

b0 := b and supn�0 kbnkLqp< 1. Denote by Xn,x

t the strong solution of SDE (2.1) with drift

coe�cient bn for each n � 0. Then for every t 2 [0, T ], 0 � r0 � r � t there exist a 0 < � < 1 anda function C : R ! [0,1) depending only on p, q, d, � and T such that

EkDr0X

n,xt �DrX

n,xt k2

� C(kbnkLp,q )|r0 � r|� (2.13)

withsupn�1

C(kbnkLqp) <1.

Here k · k denotes any norm in Rd�d.

Moreover,

supn�1

supr2[0,T ]

E [kDrXn,xt kp] <1 (2.14)

for all p � 2.

Proof. Throughout the proof we will denote by C� : R ! [0,1) any function depending on theparameters �. We will also use the symbol . to denote less or equal up to a positive real constantindependent of n.We will prove the above estimates by considering the solution of the associated PDE presented

in (2.5) with bn, n � 0 in place of b which we denote by Un, n � 0 and then using the resultsintroduced at the beginning of this section on the regularity of its solution.First, let us introduce a new process that will be useful for this purpose. Consider for each

n � 0 and t 2 [0, T ] the functions �t,n : Rd ! R

d de�ned as �t,n(x) = x+Un(t, x). It turns out, see[18, Lemma 3.5], that the functions �t,n, t 2 [0, T ], n � 0 de�ne a family of C1-di�eomorphisms

on Rd. Furthermore, consider the auxiliary process Xn,x

t := �t,n(Xn,xt ), t 2 [0, T ], n � 1. One

checks using Ito’s formula and (2.5) that Xn,xt satis�es the following SDE

dXn,xt = �Un(t, �

�1t,n(X

n,xt ))dt+

�Id +rUn(t, �

�1t,n(X

n,xt ))

�dBt, Xn,x

0 = x+ Un(0, x) (2.15)

which is equivalent to SDE (2.1) if we replace b by bn, n � 1. Using the chain rule for Malliavinderivatives (see e.g. [45]) we see that for 0 � r � t,

DrXn,xt = r�t,n(X

n,xt )DrX

n,xt .


Because of Lemma B.4 it su�ces to prove the estimates (2.13) and (2.14) for the process Xn,xt .

Since bn are now smooth we have that (2.15) admits a unique strong solution which takes theform

Xn,xt = x+ Un(0, x) + �

Z t

0

Un(s, ��1s,n(X

n,xs ))ds+

Z t

0

�Id +rUn(s, �

�1s,n(X

n,xs ))

�dBs.

Then the Malliavin derivative of Xn,xt for 0 � r � t, which exists (see e.g. [45]), is

DrXn,xt = Id +rUn(r, �

�1r,n(X

n,xr ))

+ �

Z t

r

rUn(s, ��1s,n(X

n,xs ))r��1

s,n(Xn,xs )DrX

n,xs ds

+

Z t

r

r2Un(s, ��1s,n(X

n,xs ))r��1

s,n(Xn,xs )DrX

n,xs dBs.

Denote for simplicity, Znr,t := DrX

n,xt . Then for r0 < r we can write

Znr0,t � Zn

r,t = rUn(r0, ��1

r0,n(Xn,xr0 ))�rUn(r, �

�1r,n(X

n,xr ))

+ �

Z r

r0rUn(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )Zn

r0,sds

+ �

Z t

r

rUn(s, ��1s,n(X

n,xs ))r��1

s,n(Xn,xs )

�Znr0,s � Zn

r,s

�ds

+

Z r

r0r2Un(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )Zn

r0,sdBs

+

Z t

r


n,xs ))r��1

s,n(Xn,xs )

�Znr0,s � Zn

r,s

�dBs

= Znr0,r � Zn

r,r

+ �

Z t

r

rUn(s, ��1s,n(X

n,xs ))r��1

s,n(Xn,xs )

�Znr0,s � Zn

r,s

�ds

+

Z t

r


n,xs ))r��1

s,n(Xn,xs )

�Znr0,s � Zn

r,s

�dBs.

By dint of Lemma B.3 we know that rUn is bounded uniformly in n and Lemma B.2 showsthat r2Un belongs, at least, to L

qp uniformly in n. This implies that the stochastic integral in

the expression for Znr0,t � Zn

r,t is a true martingale, which we here denote my Mnt . As a result,

since the initial condition Znr0,r �Zn

r,r is Fr-measurable for each n � 0, for a given � � 2, by Ito’sformula we have

kZnr0,t � Zn

r,tk� . kZn

r0,r � Znr,rk

� +

Z t

r

kZnr0,s � Zn

r,sk�ds+Mn

t

+

Z t

r

kZnr0,s � Zn

r,sk��2Tr

"�r2Un(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )(Zn

r0,s � Znr,s)

�

�

�r2Un(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )(Zn

r0,s � Znr,s)

��

#ds

(2.16)

where here Tr stands for the trace and � for the transposition of matrices.We proceed then using the fact that the trace of the matrix appearing in (2.16) can be bounded

by a constant Cp,d independent of n, times kZnr0,s � Zn

r,sk2kr2Un(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )k2.

Altogether,


kZnr0,t � Zn

r,tk� . kZn

r0,r � Znr,rk

� +

Z t

r

kZnr0,s � Zn

r,sk�ds+Mn

t

+

Z t

r

kZnr0,s � Zn

r,sk�kr2Un(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )k2ds

(2.17)

Consider thus the process

V nt :=

Z t

r

kr2Un(s, ��1s,n(X

n,xs ))r��1

s,n(Xn,xs )k2ds. (2.18)

The process V nt is a continuous non-decreasing and {Ft}t2[0,T ]-adapted process such that V

nr = 0.

Then Lemma B.2 in connection with Theorem 2.2 we have that supn�0

E[V nt ] <1.

Then Ito’s formula yields

e�V nt kZn

r0,t � Znr,tk

� . kZnr0,r � Zn

r,rk� +

Z t

r

e�V ns kZn

r0,s � Znr,sk

�ds+

Z t

r

e�V ns dMs. (2.19)

Then taking expectation

Ehe�V n

t kZnr0,t � Zn

r,tk�i. E

kZn

r0,r � Znr,rk

�+

Z t

r

Ehe�V n

s kZnr0,s � Zn

r,sk�ids. (2.20)

Then Gronwall’s inequality gives

Ehe�V n

t kZnr0,t � Zn

r,tk�i. E

kZn

r0,r � Znr,rk

�. (2.21)

At this point, it is easy to see, following similar steps, that for the process Znr,t one has

Ehe�V n

t kZnr,tk

�i. E

kZn

r,rk�,

where Znr,r = Id +rUn(r, �

�1r,n(X

n,xr )). So

supn�0

supr2[0,T ]

Ehe�V n

t kZnr,tk

�i. 1 + sup

n�0sup

r2[0,T ]

EhkrUn(r, �

�1r,n(X

n,xr ))k�

i<1 (2.22)

because of Lemma B.3 (ii) for a su�ciently large � 2 R.Then, the Cauchy-Schwarz inequality and Lemma B.5 give

supn�0

supr2[0,T ]

EkZn

r,tk�� sup

n�0sup

r2[0,T ]

Ehe�2V n

t kZnr,tk

2�i1/2

supn�0

Ehe2V

nT

i1/2<1.

We continue to prove the estimate (2.13). Recall that

Znr0,r � Zn

r,r = rUn(r0, ��1

r0,n(Xn,xr0 ))�rUn(r, �

�1r,n(X

n,xr ))

+ �

Z r

r0rUn(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )Zr0,sds (2.23)

+

Z r

r0r2Un(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )Zr0,sdBs.

Then taking norm and using Burkholder-Davis-Gundy inequality we get

EkZn

r0,r � Znr,rk

�. E

hkrUn(r

0, ��1r0,n(X

n,xr0 ))�rUn(r, �

�1r,n(X

n,xr ))k�

i(2.24)

+ ��E

��Z r

r0krUn(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )Zr0,skds

��

+ E

"�Z r

r0kr2Un(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )Zr0,sk

2ds

��/2#.

=: i)n + ii)n + iii)n


The aim now is to �nd Holder bounds in the sense of (2.13) for the expressions appearing in(2.24).For i)n we may write

i)n = EhkrUn(r

0, ��1r0,n(X


�1r,n(X

n,xr ))k�

i. E

hkrUn(r

0, ��1r0,n(X


�1r0,n(X

n,xr0 ))k

�i

+ EhkrUn(r, �

�1r0,n(X


�1r,n(X

n,xr ))k�

i.

Then by Lemma 2.3 there exists an " 2 (0, 1/�) and a constant Cp,q,d,� > 0 independent ofn � 0 such that

E

�krUn(r

0, ��1r0,n(X


�1r0,n(X

n,xr0 ))k

�

�

� Cp,q,d,�

�|r0 � r|"/2krUnk

1�1/q�"/2

Hq2,p

k@tUnk1/q+"/2

Lqp

��

and

E

�krUn(r, �

�1r0,n(X


�1r,n(X

n,xr ))k�

�

� Cp,q,d,�T��/qE

h|��1

r0,n(Xn,xr0 )� ��1

r,n(Xn,xr )|�"

i�kUnkHq

2,p+ Tk@tUnkLq

p

��

.

The above bounds in connection with inequality (2.7) in Theorem 2.2 give

i)n � Cp,q,d,�,T (kbnkLqp)�|r0 � r|�"/2 + E

h|��1

r0,n(Xn,xr0 )� ��1

r,n(Xn,xr )|�"

i�for some continuous function Cp,q,d,�,T (·) and hence

supn�0

Cp,q,d,�,T (kbnkLqp) <1.

Moreover, using Girsanov’s theorem, we obtain that

E

�|��1

r0,n(Xn,xr0 )� ��1

r,n(Xn,xr )|

�= E [|Xn,x

r0 �Xn,xr |]

. E

"��Z r

r0bn(s, x+Bs)ds

�� E Z T

0

bn(u, x+Bu)dBu

!#+ E [|Br0 �Br|]

. |r0 � r|1/2E

�Z r

r0|bn(s, x+Bs)|

2ds

�1/2+ |r0 � r|1/2

. |r0 � r|1/2

, where we used, Cauchy-Schwarz inequality and both that

supn�0

E

24E Z T

0

bn(u, x+Bu)dBu

!235 <1

and

supn�0

E

�Z r

r0|bn(s, x+Bs)|

2ds

�1/2<1,

see [30, Lemma 3.2] or Lemma B.1.By Jensen’s inequality for concave functions and the previous estimate we have

Eh|��1

r0,n(Xn,xr0 )� ��1

r,n(Xn,xr )|�"

i� E

h|��1

r0,n(Xn,xr0 )� ��1

r,n(Xn,xr )|

i�". |r0 � r|�"/2.

Altogether,

i)n � Cp,q,d,�,T (kbnkLqp)|r0 � r|�


for a � 2 (0, 1).For the second term, ii)n, we use Holder’s inequality, Lemma B.3 (ii) for a su�ciently large

� 2 R, Lemma B.4 and the estimate (2.14) to obtain

ii)n . ��|r0 � r|��1

sup

s2[0,t]

EhkrUn(s, �

�1s,n(X

n,xs ))r��1

s,n(Xn,xs )k2�

ids

!1/2 �Z r

r0EkZn

r0,sk2�ds

�1/2

� Cp,q,d,�,T |r0� r|�

for a � 2 (0, 1).Finally, for the third term, for � � 2, we use Holder’s inequality to obtain

iii)n . |r0 � r|��22 E

�Z r

r0kr2Un(s, �

�1s,n(X

ns ))k

�kr��1s,n(X

ns ))k

�kZnr0,sk

�ds

�.

Then choose � = 2(1 + �) with � 2 (0, 1/4) and use Lemma B.4 to get

iii)n . |r0 � r|�E

�Z r

r0kr2Un(s, �

�1s,n(X

ns ))k

2(1+�)kZnr0,sk

2(1+�)ds

�.

Then Fubini’s theorem, Holder’s inequality once more with respect to μ(d!), with exponent 1+�0,�0 2 (0, 1/4) and Cauchy-Schwarz yield

E

� Z r

r0kr2Un(s, �

�1s,n(X

ns ))k

2(1+�)kZnr0,sk

2(1+�)ds

�=

Z r

r0Ehkr2Un(s, �

�1s,n(X

ns ))k

2(1+�)kZnr0,sk

2(1+�)ids

.

Z r

r0Ehkr2Un(s, �

�1s,n(X

ns ))k

2(1+�)(1+�0)i1/(1+�0)

EhkZn

r0,sk2(1+�) 1+�

0

�0

i �0

1+�0

ds

. supn�0

sups2[r0,r]

EhkZn

r0,sk2(1+�) 1+�

0

�0

i �0

1+�0

Z r

r0Ehkr2Un(s, �

�1s,n(X

ns ))k

2(1+�)(1+�0)i1/(1+�0)

ds

.

Z T

0

Ehkr2Un(s, �

�1s,n(X

ns ))k

2(1+�)(1+�0)i1/(1+�0)

ds

where the last step follows from (2.14). For the last factor, since 0 < 1/(1 + �0) < 1, using theinverse Jensen’s inequality and the fact that 1 < (1 + �)(1 + �0) < 2 for suitable �, �0 2 (0, 1/4) inconnection with Lemma B.2 we haveZ T

0

E

�kr2Un(s, �

�1s,n(X

ns ))k

2(1+�)(1+�0)

�1/(1+�0)

ds

� T 1�1/(1+�0)

E

"Z T

0

kr2Un(s, ��1s,n(X

ns ))k

2(1+�)(1+�0)ds

#!1/(1+�0)

�M <1

for every n � 0, w.r.t. a constant M .As a summary, it follows from (2.21) that

Ehe�V n

t kZnr0,t � Zn

r,tk2(1+�)

i� Cp,q,d,�,T (kbnkLq

p)|r0 � r|�.

Then by Holder’s inequality with exponent 1+�, � 2 (0, 1) together with Lemma B.5 we obtain

EkZn

r0,t � Znr,tk

2= E

he

11+�

V nt e�

11+�

V nt kZn

r0,t � Znr,tk

2i

� Ehe

1�V nt

i �

1+�

Ehe�V n

t kZnr0,t � Zn

r,tk2(1+�)

i 11+�

� Cp,q,d,�,T (kbnkLqp)|r0 � r|�/(1+�)

withsupn�0

Cp,q,d,�,T (kbnkLqp) <1.


�

Remark 2.7. The bound given in (2.14) is in fact uniform in x 2 Rd. Indeed, by Lemma B.3 item

(ii) we have that the bound given in (2.22) is also uniform in x 2 Rd. Moreover, since �Un 2 Lq

p

for all n � 0, then by Lemma B.3 item (iii) in connection with Lemma B.1 we have that for anyk 2 R

supx2Rd

supn�0

E[ekVnT ] <1.

Hence, for any � � 1

supx2Rd

supr2[0,T ]

supn�0

E [kDrXn,xt k�] <1.

Remark 2.8. One also checks that the same holds for the spatial derivatives, that is for any � � 1

supx2Rd

supr2[0,T ]

supn�0

E

�k@

@xXn,x

t k��<1

by using the fact that @@xX

n,xt solves the same SDE as DrX

n,xt , starting at r = 0.

As a repercussion of Lemma 2.6 we have the following result which is central in the proof ofthe existence of strong solutions of (2.1).

Corollary 2.9. Let {bn}n�0 be a sequence of compactly supported smooth functions approximatingb in Lq

p. Denote, as before, Xx,nt the solution to equation (2.1) with drift coe�cient bn. Then for

each t 2 [0, T ] the sequence of random variables Xn,xt , n � 0 is relatively compact in L2(�).

Proof. This is a direct consequence of the compactness criterion that can be found in Appendix C,Lemma C.1 and C.2, which is due to [7], together with Lemma 2.6. One can check that the doubleintegral in Lemma C.2 is �nite. NamelyZ T

0

Z T

0

EkZn

r0,t � Zr,tk2

|r0 � r|1+2�dr0dr �

Z T

0

Z T

0

1

|r0 � r|2�+1��dr0dr <1

for any 0 < � < 1 and 2� + 1� � < 1. �

The following lemma gives a criterion under which the process Y bt belongs to the Hida distri-

bution space.

Lemma 2.10. Suppose that

Eμ

"exp

36

Z T

0

|b(s,Bs)|2ds

!#<1, (2.25)

where the drift b : [0, 1]�Rd�! R

d is measurable (in particular, (2.25) is valid for b 2 Lqp because

of Lemma (B.1)). Then the coordinates of the process Y bt , de�ned in (2.12), that is

Y i,bt = Eeμ

h eB(i)t E�

T (b)i, (2.26)

are elements of the Hida distribution space.

Proof. See [41] for a similar proof. �

Lemma 2.11. Let " 2 (0, 1) and de�ne p" := 1 + " and q" :=1+"" . Let bn : [0, T ]� R

d�! Rd be

a sequence of Borel measurable functions with b0 = b such that

supn�0

E

"exp

16q"(8q" � 1)

Z T

0

|bn(s, Bs)|2ds

!#<1 (2.27)

holds. Then ��S(Y i,bnt � Y i,b

t )(�)�� const · E[Jn]

1p" · exp

2(8q" � 1)

Z T

0

|�(s)|2ds

!


for all � 2 (SC([0, 1]))d, i = 1, . . . , d, where S denotes the S-transform (see Section A.1 in Appendix

A) and where the factor Jn is de�ned by

Jn =dX

j=1

2

��Z T

0

(b(j)n (s, B(j)s )� b(j)(s, B(j)

s ))2ds

��p"2

+

��Z T

0

(b(j)(s, B(j)s )2 � b(j)n (s, B

(j)s )2)ds

��p"

.

(2.28)

Here SC([0, 1]) is the complexi�cation of the Schwarz space S([0, 1]) on [0, 1], see Section A.1 inAppendix A.

In particular, if bn approximates b in the following sense

E[Jn]! 0 (2.29)

as n! 1, it follows that

Y bnt ! Y b

t in (S)�

as n! 1 for all 0 � t � 1, i = 1, . . . , d.

Proof. For i = 1, . . . , d we obtain by Proposition 2.4 and (A.14) that

|S(Y i,bnt � Y i,b

t )(�)| � Eμ

"|B

(i)t | exp

(dX

j=1

Re

� Z T

0

(b(j)(s, B(j)s ) + �(j)(s))dB(j)

s

�1

2

Z T

0

(b(j)(s, B(j)s ) + �(j)(s))2ds

�)

�

�� exp(

dXj=1

Z T

0

(b(j)n (s, B(j)s )� b(j)(s, B(j)

s ))dB(j)s

+1

2

Z T

0

(b(j)(s, B(j)s )2 � b(j)n (s, B

(j)s )2)ds

+

Z T

0

�(j)(s)(b(j)(s, B(j)s )� b(j)n (s, B

(j)s ))ds

)� 1

��#.

Since | exp{z}� 1| � |z| exp{|z|} it follows from Holder’s inequality with exponents p" = 1 + "and q" =

1+"" , for an appropriate " > 0, that

|S(Y i,bnt � Y i,b

t )(�)| � Eμ [|Qn|p" ]

1p" Eμ

"�|B

(i)t | exp

(dX

j=1

Re

� Z T

0

(b(j)(s, B(j)s ) + �(j)(s))dB(j)

s

�1

2

Z T

0

(b(j)(s, B(j)s ) + �(j)(s))2ds

�)�q"

exp {q"|Qn|}

# 1q"

,

where

Qn =dX

j=1

Z T

0

(b(j)n (s, B(j)s )� b(j)(s, B(j)

s ))dB(j)s +

1

2

Z T

0

(b(j)(s, B(j)s )2 � b(j)n (s, B

(j)s )2)ds

+

Z T

0

�(j)(s)(b(j)(s, B(j)s )� b(j)n (s, B

(j)s ))ds.


Then using the Cauchy-Schwarz inequality on the last integral and the fact that |x| � ex and1 � ex for x � 0 we may write

Eμ [|Qn|p" ] � C exp

8<: Z T

0

|�(s)|2ds

!p"/29=;Eμ

"dX

j=1

��Z T

0

(b(j)n (s, B(j)s )� b(j)(s, B(j)

s ))dB(j)s

��p"

+

��1

2

Z T

0

(b(j)(s, B(j)s )2 � b(j)n (s, B

(j)s )2)ds

��p"

+

��Z T

0

(b(j)(s, B(j)s )� b(j)n (s, B

(j)s ))2ds

��p"2#

= C exp

8<: Z T

0

|�(s)|2ds

!p"/29=;Eμ

"dX

j=1

2

��Z T

0

(b(j)n (s, B(j)s )� b(j)(s, B(j)

s ))2ds

��p"2

+

��Z T

0

(b(j)(s, B(j)s )2 � b(j)n (s, B

(j)s )2)ds

��p"#,

where in the last inequality we used the Burkholder-Davis-Gundy inequality for the stochasticintegral. Then

Eμ [|Qn|p" ]

1p" � C exp

8<: 1

p"

Z T

0

|�(s)|2ds

!p"/29=;Eμ [Jn]

1p" ,

where

Jn =dX

j=1

2

��Z T

0

(b(j)n (s, B(j)s )� b(j)(s, B(j)

s ))2ds

��p"2

+

��Z T

0

(b(j)(s, B(j)s )2 � b(j)n (s, B

(j)s )2)ds

��p"

.

Further we get that

Eμ

"�|B

(i)t | exp

(dX

j=1

Re

� Z T

0

(b(j)(s, B(j)s ) + �(j)(s))dB(j)

s

�1

2

Z T

0

(b(j)(s, B(j)s ) + �(j)(s))2ds

�)�q"

exp {q"|Qn|}

# 1q"

� Eμ

"�|B

(i)t | exp

(dX

j=1

Re

� Z T

0

(b(j)(s, B(j)s ) + �(j)(s))dB(j)

s

�1

2

Z T

0

(b(j)(s, B(j)s ) + �(j)(s))2ds

�)�2q"# 1

2q"

Eμ

"exp {2q"|Qn|}

# 12q"

.

Then for z 2 C one has exp{|z|} �12 (exp{2Re z}+ exp{�2Re z}+ exp{2Im z}+ exp{�2Im z}).

Thus

Eμ

"exp {2q"|Qn|}

# 12q"

�1

22q"

Eμ

"exp {4q"Re Qn}

# 12q"

+ Eμ

"exp {�4q"Re Qn}

# 12q"

+ Eμ

"exp {4q"Im Qn}

# 12q"

+ Eμ

"exp {�4q"Im Qn}

# 12q"!.


By the Cauchy-Schwarz inequality and the supermartingale property of Doleans-Dade expo-nentials we get

Eμ

"exp {4q"Re Qn}

#� Eμ

"exp

(dX

j=1

32q2"

Z T

0

(b(j)n (s, B(j)s )� b(j)(s, B(j)

s ))2ds

+ 4q"

Z T

0

(b(j)(s, B(j)s )2 � b(j)n (s, B

(j)s )2)ds

+ 8q"

Z T

0

Re �(j)(s)(b(j)(s, B(j)s )� b(j)n (s, B

(j)s ))ds

�)# 12

� Ln exp

(2q"

Z T

0

|�(s)|2ds

),

where the last step follows from the fact that hf, gi � 12 (kfk

2 + kgk2), f, g 2 L2([0, T ]) and where

Ln = Eμ

"exp

(dX

j=1

4q"(8q" + 1)

Z T

0

(b(j)n (s, B(j)s )� b(j)(s, B(j)

s ))2ds

+ 4q"

Z T

0

(b(j)(s, B(j)s )2 � b(j)n (s, B

(j)s )2)ds

�)# 12

.

Similarly, one also obtains

Eμ

"exp {�4q"Re Qn}

#� Ln exp

(2q"

Z T

0

|�(s)|2ds

).

In the same way, one also obtains the same bounds for Eμ [exp{4q"Im Qn}] andEμ [exp{�4q"Im Qn}].Finally, for the remaining factor we see that

Eμ

"�|B

(i)t | exp

(dX

j=1

Re

� Z T

0

(b(j)(s, B(j)s ) + �(j)(s))dB(j)

s

�1

2

Z T

0

(b(j)(s, B(j)s ) + �(j)(s))2ds

�)�2q"# 1

2q"

� Eμ

h|B

(i)t |4q"

i 14q"

Eμ

"exp

(4q"

dXj=1

Re

� Z T

0

(b(j)(s, B(j)s ) + �(j)(s))dB(j)

s

�1

2

Z T

0

(b(j)(s, B(j)s ) + �(j)(s))2ds

�)# 14q"

� Eμ

h|B

(i)t |4q"

i 14q"

Eμ

"exp

(dX

j=1

4q"(8q" � 1)

Z T

0

Re (b(j)(s, B(j)s ) + �(j)(s))2ds

)# 14q"

.

Now, since Re (z2) � (Re z)2, z 2 C we have that Re (b + �)2 � (b + Re �)2 then usingMinkowski’s inequality, i.e. kf + gkpp � 2p�1(kfkpp + kgkpp) for any p � 1 and Cauchy-Schwarzinequality w.r.t. μ one �nally obtains


Eμ

"�|B

(i)t | exp

(dX

j=1

Re

� Z T

0

(b(j)(s, B(j)s ) + �(j)(s))dB(j)

s

�1

2

Z T

0

(b(j)(s, B(j)s ) + �(j)(s))2ds

�)�2q"# 1

2q"

� CEμ

"exp

(16q"(8q" � 1)

Z T

0

|b(s, Bs)|2ds

)# 18q"

exp

(2(8q" � 1)

Z T

0

|�(s)|2ds

).

Altogether, we obtain

��S(Y i,bnt � Y i,b

t )(�)�� const · E[Jn]

11+" · exp

(2

�81 + "

"� 1

� Z T

0

|�(s)|2ds

).

�

Lemma 2.12. Let bn : [0, T ]�Rd�!R

d be a sequence of smooth functions with compact supportwith b0 := b which approximate the coe�cient b : [0, T ]�R

d�!Rd in Lq

p. Then for any 0 � t � T

there exists a subsequence of the corresponding strong solutions Xnj ,t = Ybnj

t , j = 1, 2..., such that

Ybnj

t �! Y bt

for j ! 1 in L2(�). In particular this implies Y bt 2 L2(�), 0 � t � T .

Proof. By Corollary 2.9 we know that there exists a subsequence Ybnj

t , j � 1, converging in L2(�).Further, we need to show that E[Jnj

] ! 0 as j ! 1 with Jnjas in (2.28). To this end, observe

that for a function f 2 Lqp one has

E

"Z T

0

f(s, Bs)ds

#=

Z T

0

(2�s)�d/2

ZRd

f(s, z)e�|z|2/(2s)dzds.

Then by using Holder’s inequality with respect to z and then to s we see that for anyp0, q0 2 [1,1] satisfying

d

p0+2

q0< 2,

we have

E

"Z T

0

f(s, Bs)ds

#� Ckfk

Lq0

p0

,

where C is a constant depending on T, d, p0, q0. Then from condition (2.3), since p, q > 2 we can�nd an � 2 [0, 1) small enough so that p, q > 2(1 + �). For these p, q de�ne p0 := p

2(1+�) � 1 and

q := q0

2(1+�) > 1 and apply the above estimate to |f |2(1+�) to obtain

E

"Z T

0

|f(s, Bs)|2(1+�)ds

#� CkfkLq

p. (2.30)

Now since b(j)n � b(j) 2 Lq

p for every j = 1, . . . , d and 0 <1+"2 < 1 we have

E

24 Z T

0

(b(j)n (s, B(j)s )� b(j)(s, B(j)

s ))2ds

! 1+"2

35 � E

"Z T

0

(b(j)n (s, B(j)s )� b(j)(s, B(j)

s ))2ds

# 1+"2

which goes to zero by the above estimate (2.30) by just taking the case where � = 0.Finally, for the the second term in E[Jnj

] we have


E

"�� Z T

0

(b(j)(s, B(j)s )2 � b(j)n (s, B

(j)s )2)ds

��1+"#

� T "E

"Z T

0

(b(j)(s, B(j)s ) + b(j)n (s, B

(j)s ))1+"(b(j)(s, B(j)

s )� b(j)n (s, B(j)s ))1+"ds

#

� T "

Z T

0

Eh(b(j)(s, B(j)

s ) + b(j)n (s, B(j)s ))2(1+")

i1/2Eh(b(j)(s, B(j)

s )� b(j)n (s, B(j)s ))2(1+")

i1/2ds

� T "E

"Z T

0

(b(j)(s, B(j)s ) + b(j)n (s, B

(j)s ))2(1+")ds

#1/2E

"Z T

0

(b(j)(s, B(j)s )� b(j)n (s, B

(j)s ))2(1+")ds

#1/2.

Then since b(j) + bn(j) 2 Lqp for every n � 0 we have

supn�0

E

"Z T

0

(b(j)(s, B(j)s ) + b(j)n (s, B

(j)s ))2(1+")ds

#1/2<1

for a su�ciently small " 2 (0, 1) by Lemma B.2 and

E

"Z T

0

(b(j)(s, B(j)s )� b(j)n (s, B

(j)s ))2(1+")ds

#1/2! 0

as n! 1 by estimate (2.30) for a su�ciently small " > 0.

Thus, by Lemma 2.11, Ybnj

t ! Y bt as j ! 1 in (S)�. But then, by uniqueness of the limit,

also Ybnj

t ! Y bt in L

2(�). �

Remark 2.13. It follows from the above proof that Y bnt ! Y b

t as n ! 1 in L2(�;Rd) for all tand x.

In fact, Lemma 2.12 enables us now to state the following ”transformation property” for Y bt .

Lemma 2.14. Assume that b : [0, T ]� Rd�! R

d is in Lqp. Then

'(i)�t, Y b

t

�= Eeμ

h'(i)

�t, eBt

�E�T (b)

i(2.31)

a.e. for all 0 � t � T, i = 1, . . . , d and ' = ('(1), . . . , '(d)) such that '(Bt) 2 L2(�;Rd).

Proof. See [49, Lemma 16] or [39]. �

Using the above auxiliary results we can �nally give the proof of Theorem 2.1.

Proof of Theorem 2.1. We want to use the transformation property (2.31) of Lemma 2.14 to show

that Y bt is a unique strong solution of the SDE (2.1). To shorten notation we set

R t

0'(s, !)dBs :=Pd

j=1

R t

0'(j)(s, !)dB

(j)s and x = 0. Also, let bn, n = 1, 2, ..., be a sequence of functions as required

in Lemma 2.12.We comment on that Y b

· has a continuous modi�cation. The latter can be seen as follows:

Since each Y bnt is a strong solution of the SDE (2.1) with respect to the drift bn we obtain from

Girsanov’s theorem and our assumptions that

Eμ

��Y i,bnt � Y i,bn

u

�4�= Eeμ

"� eB(i)t � eB(i)

u

�4

E

Z T

0

bn(s, eBs)d eBs

!#� const · |t� u|

2

for all 0 � u, t � T , n � 1, i = 1, ..., d. The above constant comes from the fact thatnE�R T

0bn(s, eBs)d eBs

�on�1

is bounded in L2(�;Rd) with respect to the measure μ, see Lemma

3.2. in [30] or Lemma B.1.


By Remark 2.13 we know that

Y bnt �! Y b

t in L2(�;Rd)

and hence we have almost sure convergence for a further subsequence, 0 � t � T . So we get thatby Fatou’s lemma

Eμ

��Y i,bt � Y i,b

u

�4�� const · |t� u|

2(2.32)

for all 0 � u, t � T , i = 1, ..., d. Then Kolmogorov’s lemma guarantees a continuous modi�cationof Y b

t .

Since eBt is a weak solution of (2.1) for the drift b(s, x) + �(s) with respect to the measure

dμ� = E�R T

0

�b(s, eBs) + �(s)

�d eBs

�dμ we get that

S(Y i,bt )(�) = Eeμ

" eB(i)t E

Z T

0

�b(s, eBs) + �(s)

�d eBs

!#= Eμ�

h eB(i)t

i= Eμ�

�Z t

0

�b(i)(s, eBs) + �(i)(s)

�ds

�

=

Z t

0

Eeμ

"b(i)(s, eBs)E

Z T

0

�b(u, eBu) + �(u)

�d eBu

!#ds+ S

�B

(i)t

�(�).

Thus the transformation property (2.31) applied to b yields

S(Y i,bt )(�) = S(

Z t

0

b(i)(u, Y i,bu )du)(�) + S(B

(i)t )(�).

Then it follows from the injectivity of the S-transform that

Y bt =

Z t

0

b(s, Y bs )ds+Bt .

See Section A in the Appendix.

The Malliavin di�erentiability of Y bt comes from the fact that Y i,bn

t ! Y i,bt in L2(�) and

supn�1

kY i,bnt kD1,2 �M <1

for all i = 1, . . . , d and 0 � t � 1. See e.g. [45].On the other hand, using uniqueness in law, which is a consequence of Lemma B.2 and Propo-

sition 3.10, Ch. 5 in [27] we may apply, under our conditions, Girsanov’s theorem to any othersolution. Then the proof of Proposition 2.4 (see e.g. [48, Proposition 1]) shows that any othersolution necessarily coincides with Y b

t . �

We conclude this section with a generalisation of Theorem 2.1 to a class of non-degenerated�dimensional Ito-di�usions.

Theorem 2.15. Assume the time-homogeneous Rd�valued SDE

dXt = b(Xt)dt+ (Xt)dBt, X0 = x 2 Rd, 0 � t � T, (2.33)

where the coe�cients b : Rd �! Rd and : Rd �! R

d� Rdare Borel measurable. Suppose

that there exists a bijection � : Rd �! R

d, which is twice continuously di�erentiable. Let�x : R

d �! L�R

d,Rd�and �xx : R

d �! L�R

d � Rd,Rd

�be the corresponding derivatives of

� and assume that

�x(y)(y) = idRd for y a.e.

as well as

��1 is Lipschitz continuous.


Require that the function b� : Rd �! R

d given by

b�(x) := �x

��1 (x)

� b(��1 (x))

+1

2�xx

��1 (x)

�" dXi=1

(��1 (x)) [ei] ,

dXi=1

(��1 (x)) [ei]

#

satis�es the conditions of Theorem 2.1, where ei, i = 1, . . . , d, is a basis of Rd. Then there existsa Malliavin di�erentiable solution Xt to (2.33).

Proof. The proof can be directly obtained from Ito’s Lemma. See [41]. �

3. Applications

3.1. The Bismut-Elworthy-Li formula. As an application we want to use Theorem 2.1 toderive a Bismut-Elworthy-Li formula for solutions v to the Kolmogorov equation

@

@tv(t, x) =

dXj=1

bj(t, x)@

@xjv(t, x) +

1

2

dXi=1

@2

@x2iv(t, x) (3.1)

with initial condition v(0, x) = �(x), where b : [0, T ]� Rd ! R

d belongs to Lqp.

It is known that, see [31] or [15], that when � is continuous and bounded there exists a solutionto (3.1) given by

v(t, x) = E[�(Xxt )], (3.2)

where v is a solution to the Kolmogorov Equation (3.1) which is unique among all boundedsolutions in the space Hq

2,p, as introduced in Theorem 2.2, with p, q > 2 satisfying (2.3). Moreover,@@xv 2 L1([0, T ]� R

d).

In the sequel, we aim at �nding a representation for @@xv without using derivatives of �. See

[38] in the case of b 2 L1([0, T ]� Rd).

Theorem 3.1 (Bismut-Elworthy-Li formula). Assume � 2 Cb(Rd) and let U be an open, bounded

subset of Rd. Then the derivative of the solution to (3.1) can be represented as

@

@xv(t, x) = E[�(Xx

t )

Z t

0

a(s)

�@

@xXx

s

��

dBs]� (3.3)

for almost all x 2 U and all t 2 (0, T ], where a = at is any bounded measurable function such thatR t

0at(s)ds = 1 and where � denotes the transposition of matrices.

Proof. The proof is similar to Theorem 2 in [41] in the case of b 2 L1([0, T ] � Rd). For the

convenience of the reader we give the full proof.Assume that � 2 C2

b (Rd) (the general case of � 2 Cb(R

d) can be proved by approximation of� in relation (3.5)) and let bn and X

n,xt be as in the previous section. If we replace b by bn in

(3.1) we have the unique solution given by

vn(t, x) = E[�(Xn,xt )].

By using Remark 2.13 we see that vn(t, x)! v(t, x) for each t and x.By [45, Page 109] we have that

DsXn,xt

@

@xXn,x

s =@

@xXn,x

t ,

where the above product is the usual matrix product. So it follows that

@

@xXn,x

t =

Z t

0

a(s)DsXn,xt

@

@xXn,x

s ds. (3.4)


Interchanging integration and di�erentiation in connection with the chain rule we �nd that

@

@xvn(t, x) = E[�0(Xn,x

t )@

@xXn,x

t ]

= E[

Z t

0

a(s)Ds�(Xn,xt )

@

@xXn,x

s ds]

= E[�(Xn,xt )

Z t

0

a(s)

�@

@xXn,x

s

��

dBs]�,

where we applied the chain rule and the duality formula for the Malliavin derivative to the lastequality.Choose ' 2 C1

0 (U). In what follows, we will prove thatZRd

@

@x'(x)v(t, x)dx = �

ZRd

'(x)E[�(Xxt )

Z t

0

a(s)

�@

@xXx

s

��

dBs]�dx. (3.5)

In fact, dominated convergence combined with Remark 2.13 givesZRd

@

@x'(x)v(t, x)dx = � lim

n!1

ZRd

'(x)E[�(Xn,xt )

Z t

0

a(s)

�@

@xXn,x

s

��

dBs]�dx

= � limn!1

ZRd

'(x)E[(�(Xn,xt )� �(Xx

t ))

Z t

0

a(s)

�@

@xXn,x

s

��

dBs]�dx

� limn!1

ZRd

'(x)E[�(Xxt )

Z t

0

a(s)

�@

@xXn,x

s

��

dBs]�dx

= � limn!1

i)n � limn!1

ii)n.

As for the �rst term we get

i)n �

ZRd

|'(x)|k@

@x�k1kXn,x

t �Xxt kL2(�;Rd)kak1

sup

k�1,s2[0,T ]

E[k@

@xXk,x

s k2Rd�d ]

!1/2

dx,

which goes to zero as n tends to in�nity by Lebesque dominated convergence theorem, Remark2.13 and Remark 2.8.For the second term, ii)n since X

xt is Malliavin di�erentiable and � 2 C2

b (Rd) it follows from

the Clark-Ocone formula that (see e.g. [45])

�(Xxt ) = E[�(Xx

t )] +

Z t

0

E[Ds�(Xxt )|Fs]dBs.

So

ii)n =

ZRd

'(x)E[�(Xxt )

Z t

0

a(s)

�@

@xXn,x

s

��

dBs]�dx (3.6)

=

ZRd

'(x)E[

�E[�(Xx

t )] +

Z t

0

E[Ds�(Xxt )|Fs]dBs

� Z t

0

a(s)

�@

@xXn,x

s

��

dBs]�dx (3.7)

=

Z t

0

a(s)

ZRd

'(x)E[Ds�(Xxt )

@

@xXn,x

s ]dxds. (3.8)

One checks by means of Lemma 2.6 that '(·)Ds�(X·t) = '(·)�0(X ·

t)DsX·t belongs to L

2(Rd��;Rd)so that for each s, the function

gn(s) =

ZRd

'(x)E[Ds�(Xxt )

@

@xXn,x

s ]dx


converges toRRd '(x)E[Ds�(X

xt )

@@xX

xs ]dx by the weak convergence of

@@xX

n,xs in L2([0, T ]�U��)

for a subsequence in virtue of Remark 2.8. Further,

|gn(s)| �

ZRd

|'(x)|kDs�(Xxt )kL2(�;Rd)k

@

@xXn,x

s kL2(�;Rd)dx

� supy2Rd, u�t, k2N

kDu�(Xyt )kL2(�;Rd)k

@

@xXk,y

u kL2(�;Rd)

ZRd

|'(x)|dx

so that Lebesgue’s dominated convergence theorem gives

limn!1

ii)n =

Z t

0

a(s)

ZRd

'(x)E[Ds�(Xxt )

@

@xXx

s ]dxds.

By reversing equations (3.6), (3.7) and (3.8) with @@xX

xs in place of

@@xX

n,xs we obtain the result. �

Appendix A. Framework

In this appendix we collect some facts from Gaussian white noise analysis and Malliavin calculus,which we shall use in Section 2 to construct strong solutions of SDE’s. See [24, 46, 33] for moreinformation on white noise theory. As for Malliavin calculus the reader may consult [45, 35, 36, 10].

A.1. Basic Facts of Gaussian White Noise Theory. A crucial step in our proof for theconstuction of strong solutions (see Section 3) relies on a generalised stochastic process in theHida distribution space which is shown to be a SDE solution. Let us �rst recall the de�nition ofthis space which is due to T. Hida (see [24]).From now on we �x a time horizon 0 < T < 1. Let A be a (positive) self-adjoint operator on

L2([0, T ]) with Spec(A) > 1. Require that A�r is of Hilbert-Schmidt type for some r > 0 and let{ej}j�0 be a complete orthonormal basis of L

2([0, T ]) in Dom(A) and let �j > 0, j � 0 be theeigenvalues of A such that

1 < �0 � �1 � ... �! 1.

Suppose that each basis element ej is a continuous function on [0, T ]. Further let O�,� 2 , bean open covering of [0, T ] such that

supj�0

��(�)j sup

t2O�

|ej(t)| <1

for �(�) � 0.In the sequel let S([0, T ]) be the standard countably Hilbertian space constructed from

(L2([0, T ]), A). See [46]. Then S([0, T ]) is a nuclear subspace of L2([0, T ]). The topologicaldual of S([0, T ]) is denoted by S 0([0, T ]). Then the Bochner-Minlos theorem entails the existenceof a unique probability measure � on B(S 0([0, T ])) (Borel �algebra of S 0([0, T ])) such thatZ

S0([0,T ])

eih!,�i�(d!) = e� 1

2k�k2L2([0,T ])

for all � 2 S([0, T ]), where h!,�i stands for the action of ! 2 S p([0, T ]) on � 2 S([0, T ]). De�ne

�i = S 0([0, T ]) , Fi = B(S 0([0, T ])) , μi = � ,

for i = 1, . . . , d. Then the product measure

μ =d�i=1μi (A.1)

on the measurable space

(�,F) :=

dY

i=1

�i,di=1

Fi

!(A.2)

is called d-dimensional white noise probability measure.


Consider the Doleans-Dade exponential

ee(�, !) = exp�h!,�i� 1

2k�k

2L2([0,T ];Rd)

�,

for ! = (!1, . . . , !d) 2 (S 0([0, T ]))d and � = (�(1), . . . ,�(d)) 2 (S([0, T ]))d, where

h!,�i :=Pd

i=1 h!i,�ii.

Now let�(S([0, T ]))d

�b�nbe the n�th completed symmetric tensor product of (S([0, T ]))d with

itself. One checks that ee(�, !) is holomorphic in � around zero. Hence, there exist generalised

Hermite polynomials Hn(!) 2��(S([0, T ]))d

�b�n�0

such that

ee(�, !) =Xn�0

1

n!

Hn(!),�

�n�

(A.3)

for � in a certain neighbourhood of zero in (S([0, T ]))d. One proves that�DHn(!),�

(n)E: �(n) 2

�(S([0, T ]))d

�b�n, n 2 N0

�(A.4)

is a total set of L2(�). Further, it can be shown that the generalised Hermite polynomials satisfythe orthogonality relationZ

S0

DHn(!),�

(n)ED

Hm(!), (m)Eμ(d!) = �n,mn!

��(n), (n)

�L2([0,T ]n;(Rd)�n)

(A.5)

for all n,m 2 N0, �(n) 2

�(S([0, T ]))d

�b�n, (m) 2

�(S([0, T ]))d

�b�mwhere

�n,m =

�1 if n = m0 else

.

Denote by bL2([0, T ]n; (Rd)�n) the space of square integrable symmetric functions f(x1, . . . , xn)with values in (Rd)�n. Then it follows from relation (A.5) that the mappings

�(n) 7�!DHn(!),�

(n)E

from�S([0, T ])

d�b�n

to L2(�) have unique continuous extensions

In : bL2([0, T ]n; (Rd)�n) �! L2(�)

for all n 2 N. These extensions In(�(n)) can be identi�ed as n-fold iterated Ito integrals of

�(n) 2 bL2([0, T ]n; (Rd)�n) with respect to a d�dimensional Wiener process

Bt =�B

(1)t , . . . , B

(d)t

�(A.6)

on the white noise space(�,F , μ) . (A.7)

We mention that square integrable functionals of Bt admit a Wiener-Ito chaos representationwhich can be regarded as an in�nite-dimensional Taylor expansion, that is

L2(�) =Mn�0

In(bL2([0, T ]n; (Rd)�n)). (A.8)

The de�nition of the Hida stochastic test function and distribution space is based on the Wiener-Ito chaos decomposition (A.8): Set

Ad := (A, . . . , A) . (A.9)

Using a second quantisation argument, the Hida stochastic test function space (S) is de�ned asthe space of all f =

Pn�0

Hn(·),�

(n)�2 L2(�) such that

kfk20,p :=

Xn�0

n!��(Ad)�n

�p�(n)

��2L2([0,T ]n;(Rd)�n)

<1 (A.10)


for all p � 0. In fact, the space (S) is a nuclear Frechet algebra with respect to multiplication offunctions and its topology is induced by the seminorms k·k0,p , p � 0. Further one shows that

ee(�, !) 2 (S) (A.11)

for all � 2 (S([0, T ]))d.On the other hand, the topological dual of (S), denoted by (S)�, is called Hida stochastic

distribution space. Using these de�nitions we ontain the Gel’fand triple

(S) ,! L2(�) ,! (S)�.

It turns out that the white noise of the coordinates of the d�dimensional Wiener process Bt, thatis the time derivatives

W it :=

d

dtBi

t, i = 1, . . . , d , (A.12)

belong to (S)�.We also recall the de�nition of the S-transform. See [48]. The S�transform of a � 2 (S)�,

denoted by S(�), is de�ned by the dual pairing

S(�)(�) = h�, ee(�, !)i (A.13)

for � 2 (SC([0, T ]))d. Here SC([0, T ]) the complexi�cation of S([0, T ]). The S�transform is a

monomorphism from (S)� to C. In particular, if

S(�) = S( ) for �, 2 (S)�

then

� = .

As an example one �nds that

S(W it )(�) = �i(t), i = 1, ..., d (A.14)

for � = (�(1), . . . ,�(d)) 2 (SC([0, T ]))d.

Finally, we recall the concept of the Wick or Wick-Grassmann product. The Wick productde�nes a tensor algebra multiplication on the Fock space and is introduced as follows: The Wickproduct of two distributions �, 2 (S)�, denoted by � � , is the unique element in (S)� suchthat

S(� � )(�) = S(�)(�)S( )(�) (A.15)

for all � 2 (SC([0, T ]))d. As an example, we getD

Hn(!),�(n)E�

DHm(!),

(m)E=DHn+m(!),�

(n)b (m)E

(A.16)

for �(n) 2�(S([0, T ]))d

�b�nand (m) 2

�(S([0, T ]))d

�b�m. The latter in connection with (A.3)

implies that ee(�, !) = exp�(h!,�i) (A.17)

for � 2 (S([0, T ]))d. Here the Wick exponential exp�(X) of a X 2 (S)� is de�ned as

exp�(X) =Xn�0

1

n!X�n, (A.18)

where X�n = X � . . . �X, provided that the sum on the right hand side converges in (S)�.


A.2. Basic elements of Malliavin Calculus. In this section we pass in review some basicde�nitions from Malliavin calculus.For convenience we consider the case d = 1. Let F 2 L2(�). Then we know from (A.8) that

F =Xn�0

DHn(·),�

(n)E

(A.19)

for unique �(n) 2 bL2([0, T ]n). Suppose thatXn�1

nn!��(n)

��2L2([0,T ]n)

<1 . (A.20)

Then the Malliavin derivative Dt of F in the direction of Bt can be de�ned as

DtF =Xn�1

nDHn�1(·),�

(n)(·, t)E. (A.21)

We denote by D1,2 the space of all F 2 L2(�) such that (A.20) holds. The Malliavin derivative

D· is a linear operator from D1,2 to L2([0, T ]��). We mention that D1,2 is a Hilbert space with

the norm k·k1,2 given by

kFk21,2 := kFk

2L2(�,μ) + kD·Fk

2L2([0,T ]��,��μ) . (A.22)

We get the following chain of continuous inclusions:

(S) ,! D1,2 ,! L2(�) ,! D

�1,2 ,! (S)�, (A.23)

where D�1,2 is the dual of D1,2.

Appendix B. Technical results

We give a list if technical results needed for the proofs of Section 2 and 3.

Lemma B.1. Let {fn}n�0 be a bounded sequence of functions in Lqp. Then, for every k 2 R

supx2Rd

supn�0

E

"exp

(k

Z T

0

|fn(s, x+Bs)|2ds

)#<1.

In particular, there exists a weak solution to SDE (2.1).

Proof. See [30, Lemma 3.2] �

Lemma B.2. Let {fn}n�0 a sequence of elements in Lp,q that converges to some f 2 Lp,q. Thenthere exists " > 1 such that

supn�0

E

"Z T

0

kfn(s,�ns )k

2"ds

#<1. (B.1)

Here �ns : x 7! Xx,n

t denotes the stochastic �ow associated to the solution of the SDE (2.1) withdrift coe�cient bn 2 C1

b (Rd).

Proof. See [19, Lemma 15]. �

We also need the following crucial lemma, which can be found in [18], Lemma 3.4.

Lemma B.3. Let Un be the solution of the PDE (2.6) with � = b = bn 2 C1b (R

n). Let Xx,nt be

the solution of the SDE (2.1) with drift coe�cient bn 2 C1b (R

d). Then the following holds true

(i) For each r > 0 there exists a function f with limn f(n) = 0 such that

supx2Br

supt2[0,T ]

kUn(t, x)� U(t, x)k � f(n)

and

supx2Br

supt2[0,T ]

krUn(t, x)�rU(t, x)k � f(n)


(ii) There exists a � 2 R for which supt2[0,T ]

x2Rd

krUn(t, x)k �1

2.

(iii) supn�0

k�Un(t, x)kLp,q <1.

(iv) As a consequence of the boundedness of Un and rUn we have

supt2[0,T ]

E [k�nt (x)k

a] � C (1 + |x|a) .

The following lemma gives a bound for the derivative of the inverse of the family of di�eomor-phisms �t. See [18], Lemma 3.5 for its proof.

Lemma B.4. Let �t,n : Rd ! R

d be the C1-di�eomorphisms de�ned as �t,n(x) := x+Un(t, x) forx 2 R

d associated to Xx,nt the solution of SDE (2.1) with drift coe�cient bn 2 C1

b (Rd). Then

supn�0

supt2[0,T ]

kr��1t,nkC(Rd) � 2.

The next result was shown in [17], Corollary 13.

Lemma B.5. Let V nt be the process de�ned in (2.18). Then for every � 2 R

supn�0

Ehe�V

nT

i� C.

Observe that the same estimate holds for any t 2 [0, T ] since V nt is an increasing process.

Appendix C.

The following result which is due to [7, Theorem 1] gives a compactness criterion for subsets ofL2(�;Rd) using Malliavin calculus.

Theorem C.1. Let {(�,A, P ) ;H} be a Gaussian probability space, that is (�,A, P ) is a prob-ability space and H a separable closed subspace of Gaussian random variables of L2(�), whichgenerate the -�eld A. Denote by D the derivative operator acting on elementary smooth randomvariables in the sense that

D(f(h1, . . . , hn)) =nX

i=1

@if(h1, . . . , hn)hi, hi 2 H, f 2 C1b (R

n).

Further let D1,2 be the closure of the family of elementary smooth random variables with respectto the norm

kFk1,2 := kFkL2(�) + kDFkL2(�;H) .

Assume that C is a self-adjoint compact operator on H with dense image. Then for any c > 0 theset

G =nG 2 D1,2 : kGkL2(�) +

��C�1DG��L2(�;H)

� co

is relatively compact in L2(�).

A useful bound in connection with Theorem C.1, based on fractional Sobolev spaces is thefollowing (see [7]):

Lemma C.2. Let vs, s � 0 be the Haar basis of L2([0, T ]). For any 0 < � < 1/2 de�ne theoperator A� on L2([0, T ]) by

A�vs = 2k�vs, if s = 2

k + j

for k � 0, 0 � j � 2k andA�T = T.

Then for all � with � < � < (1/2), there exists a constant c1 such that

kA�fk � c1

8<:kfkL2([0,T ]) +

Z T

0

Z T

0

|f(t)� f(t0)|2

|t� t0|1+2�

dt dt0

!1/29=; .


A direct consequence of Theorem C.1 and Lemma C.2 is now the following compactness criterionwhich is essential for the proof of Corollary 2.9.

Corollary C.3. Let a sequence of FT -measurable random variables Xn 2 D1,2, n = 1, 2..., besuch that there exist constants � > 0 and C > 0 with

supnE[|Xn|

2] � C,

supnEkDtXn �Dt0Xnk

2� C|t� t0|�

for 0 � t0 � t � T andsupn

sup0�t�T

EkDtXnk

2� C .

Then the sequence Xn, n = 1, 2..., is relatively compact in L2(�).

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(Sbornik) 22 (1974), 129–149.

CMA, Department of Mathematics, University of Oslo, Moltke Moes vei 35, P.O. Box 1053 Blindern,

0316 Oslo, Norway.

E-mail address: [email protected]


0316 Oslo, Norway.


Department of Mathematics,LMU, Theresienstr. 39, D-80333 Munich, Germany



0316 Oslo, Norway.


II

COMPUTING DELTAS WITHOUT DERIVATIVES

D. R. BANOS, S. DUEDAHL, T. MEYER-BRANDIS, AND F. PROSKE

This Version : May 23, 2015

Abstract. A well-known application of Malliavin calculus in Mathematical Finance is theprobabilistic representation of option price sensitivities, the so-called Greeks, as expectationfunctionals that do not involve the derivative of the pay-o� function. This allows for numericallytractable computation of the Greeks even for discontinuous pay-o� functions. However, whilethe pay-o� function is allowed to be irregular, the coe�cients of the underlying di�usion arerequired to be smooth in the existing literature, which for example excludes already simpleregime switching di�usion models. The aim of this article is to generalise this application ofMalliavin calculus to Ito di�usions with irregular drift coe�cients, whereat we here focus onthe computation of the Delta, which is the option price sensitivity with respect to the initialvalue of the underlying. To this purpose we �rst show existence, Malliavin di�erentiability, and(Sobolev) di�erentiability in the initial condition of strong solutions of Ito di�usions with driftcoe�cients that can be decomposed into the sum of a bounded but merely measurable and aLipschitz part. Furthermore, we give explicit expressions for the corresponding Malliavin andSobolev derivative in terms of the local time of the di�usion, respectively. We then turn tothe main objective of this article and analyse the existence and probabilistic representation ofthe corresponding Deltas for lookback and Asian type options. We conclude with a simulationstudy of several regime-switching examples.

Key words and phrases: Greeks, Delta, option sensitivities, Malliavin calculus, Bismut-Elworthy-Li formula, irregular di�usion coe�cients, strong solutions of stochastic di�erential equa-tions, relative L2-compactness

MSC2010: 60H10, 60H07, 60H30, 91G60.

1. Introduction

Throughout this paper, let T > 0 be a given time horizon and (�,F , P ) a complete probabilityspace equipped with a one-dimensional Brownian motion {Bt}t2[0,T ] and the �ltration {Ft}t2[0,T ]

generated by {Bt}t2[0,T ] augmented by the P -null sets. Further, we will only deal with randomvariables that are Brownian functionals, i.e. we assume F := FT .

One of the most prominent applications of Malliavin calculus in �nancial mathematics concernsthe derivation of numerically tractable expressions for the so-called Greeks, which are importantsensitivities of option prices with respect to involved parameters. The �rst paper to address thisapplication was [15], which has consecutively triggered an active research interest in this topic, seee.g. [14], [4], [1]. See also [7], [11] and references therein for a related approach based on functionalIto calculus. Suppose the risk-neutral dynamics of the underlying asset of a European option isdriven by a stochastic di�erential equation (for short SDE) of the form

dXxt = b(Xx

t )dt+ �(Xxt )dBt, Xx

0 = x 2 R ,

where b : R ! R and � : R ! R are some given drift and volatility coe�cients, respectively. Let� : R ! R denote the pay-o� function and the expectation E[�(Xx

T )] the risk-neutral price attime zero of the option with maturity T > 0. For notational simplicity we assume the discountingrate to be zero. In this paper we will focus on the Delta

@

@xE[�(Xx

T )] , (1.1)

which is a measure for the sensitivity of the option price with respect to changes of the initial valueof the underlying asset. As is well known, the Delta has a particular role among the Greeks as it

1


determines the hedge portfolio in many complete market models. If the drift b(·), the volatility �(·),and the pay-o� �(·) are ”su�ciently regular” to allow for di�erentiation under the expectation,the Delta can be computed in a straight-forward manner as

E

�@

@x�(Xx

T )

�= E[�0(Xx

T )ZT ] , (1.2)

where the �rst variation process Zt :=@@xX

xt is given by

Zt = exp

�Z t

0

�b0(Xx

s )�1

2(�0(Xx

s ))2�ds+

Z t

0

�0(Xxs ) dBs

�, (1.3)

and where �0, b0,�0 denote the derivatives of �, b,�, respectively. For example, requiring that

�, b,� are continuously di�erentiable with bounded derivatives would allow (1.2) to hold (we referto [18] for conditions on b and � that guarantee the existence of the �rst variation process), andthe expectation in (1.2) could be approximated e.g. by Monte Carlo methods. In most realisticsituations, though, straight-forward computations as in (1.2) are not possible. In that case, onecould combine numerical methods to approximate the derivative and the expectation in (1.1),respectively, to compute the Delta. However, in particular for discontinuous pay-o�s � as is thecase for a digital option this procedure might be numerically ine�cient, see for example [15]. Atthat point, the following result for lookback options obtained with the help of Malliavin calculusappears to be useful, where the option pay-o� is allowed to depend on the path of the underlyingat �nitely many time points.

Theorem 1.1 (Proposition 3.2 in [15]). Let b(·) and �(·) be continuously di�erentiable withbounded Lipschitz derivatives, �(·) > � > 0, and � : R

m ! R be such that the pay-o��(Xx

T1, . . . , Xx

Tm), T1, . . . , Tm 2 (0, T ], of the corresponding lookback option is square integrable.

Then the Delta exists and is given by

@

@xE[�(Xx

T1, . . . , Xx

Tm)] = E

"�(Xx

T1, . . . , Xx

Tm)

Z T

0

a(t)��1(Xxt )Zt dBt

#, (1.4)

where Zt is the �rst variation process given in (1.3) and a(t) is any square integrable deterministicfunction such that, for every i = 1, . . . ,m,Z Ti

0

a(s)ds = 1.

While for notational simplicity we present the above result for one-dimensional Xx we remarkthat in [15] the extension to multi-dimensional underlying asset and Brownian motion is considered.If the option is of European type, i.e. the pay-o� �(Xx

T ) depends only on the underlying at T ,then (1.4) is the probabilistic representation of the space derivative of a solution to a Kolmogorovequation which is also referred to as Bismuth-Elworthy-Li type formula in the literature due to[13], [6]. The strength of (1.4) is that the Delta is expressed again as an expectation of the pay-o�

multiplied by the so-called Malliavin weightR T

0a(t)��1(Xx

t )Zt dBt. Computing the Delta byMonte-Carlo via this reformulation then guarantees a convergence rate that is independent of theregularity of the pay-o� function � and the dimensionality. Note that the Malliavin weight isindependent of the option pay-o�, and thus the same weight can be employed in the computationsof the Deltas of di�erent options. Also, in [14] and [3] the question of how to optimally choose thefunction a(t) with respect to computational e�ciency is considered.

While the representation (1.4) succeeds to handle irregular pay-o�s by getting rid of the de-rivative of �, the regularity assumptions on the coe�cients b and � driving the dynamics of theunderlying di�usion are rather strong. Consider for example an extended Black and Scholes modelwhere the stock pays a dividend yield that switches to a higher level when the stock value passesa certain threshold. Then, again with the risk-free rate equal to zero for simplicity, the logarithmof the stock price is modelled by the following dynamics under the risk-neutral measure:

dXxt = b(Xx

t )dt+ �dBt, Xx0 = x 2 R ,

COMPUTING DELTAS WITHOUT DERIVATIVES 3

where � > 0 is constant and the drift coe�cient b : R ! R is given by

b(x) := ��11(�1,R)(x)� �21[R,1)(x),

for dividend yields �1,�2 2 R+ and a given threshold R 2 R. In [9], a (more complex) irregulardrift b is interpreted as state-dependent fees deducted by the insurer in the evolution of variableannuities instead of dividend yield. Already, this simple regime-switching model is not covered bythe result in Theorem 1.1 since the drift coe�cient is not continuously di�erentiable.

Or allow for state-dependent regime-switching of the mean reversion rate in an extendedOrnstein-Uhlenbeck process:

dXxt = b(Xx

t )dt+ �dBt, Xx0 = x 2 R ,


b(x) := ��1x1(�1,R)(x)� �2x1[R,1)(x)

for mean reversion rates �1,�2 2 R+ and a given threshold R 2 R (here the mean reversion level isset equal to zero). This type of model captures well, for instance, the evolution of electricity spotprices, which switches between so-called spike regimes on high price levels with very fast meanreversion and base regimes on normal price levels with moderate speed of mean reversion, seee.g. [5], [17], [26] and references therein. Alternatively, an extended Ornstein-Uhlenbeck processwith state-dependent regime-switching of the mean reversion level (low and high interest rateenvironments) is an interesting modi�cation of the Va�s��cek short rate model. Note that in thatcase the Delta is rather a generalised Rho, i.e. a sensitivity measure with respect to the short endof the yield curve. We observe that also these two extended Ornstein-Uhlenbeck processes are notcovered by the result in Theorem 1.1.

Motivated by these examples, this paper aims at deriving an analogous result to Theorem 1.1when the underlying is driven by an SDE with irregular drift coe�cient. More precisely, we willconsider SDE’s

dXxt = b(t,Xx

t )dt+ dBt, 0 � t � T, Xx0 = x 2 R , (1.5)

where we allow for time-inhomogeneous drift coe�cients b : [0, T ]� R ! R in the form

b(t, x) = b(t, x) + b(t, x) , (t, x) 2 [0, T ]� R , (1.6)

for b merely bounded and measurable, and b Lipschitz continuous and at most of linear growth inx uniformly in t, i.e. there exists a constant C > 0 such that

|b(t, x)� b(t, y)| � C|x� y| (1.7)

|b(t, x)| � C(1 + |x|) (1.8)

for x, y 2 R and t 2 [0, T ]. Adding the Lipschitz component b(t, x) in (1.6) is motivated by the factthat many drift coe�cients interesting for �nancial applications are of linear growths. At presentwe are not able to show our results for general measurable drift coe�cients of linear growths, butonly for those where the irregular behavior remains in a bounded spectrum. However, from anapplication point of view this class is very rhich already, and in particular it contains the regimeswitching examples from above. In (1.5) we consider a constant volatility coe�cient �(t, x) := 1,but we will see at the end of Section 3 (Theorem 3.8) that our results apply to many SDE’s withmore general volatility coe�cients which can be reduced to SDE’s of type (1.5) (which for exampleis possible for volatility coe�cients as in Theorem 1.1).

In order to be able to apply Malliavin calculus to the underlying di�usion, the �rst thing weneed to ensure is that the solution of SDE (1.5) is a Brownian functionals, i.e. we are interestedin the existence of strong solutions of (1.5).

De�nition 1.2. A strong solution of SDE (1.5) is a continuous, {Ft}t2[0,T ]-adapted process{Xx

t }t2[0,T ] that solves equation (1.5).


Remark 1.3. Note that the usual de�nition of a strong solution requires the existence of aBrownian-adapted solution of (1.5) on any given stochastic basis. However, an {Ft}t2[0,T ]-adapted solution {Xx

t }t2[0,T ] on the given stochastic basis (�,F , P,B) can be written in the formXx

t = Ft(B·) for some family of functionals Ft, t 2 [0, T ], (see e.g. [24] for an explicit form of

Ft). Then for any other stochastic basis (�, F , P , B) one gets that Xxt := Ft(B·), t 2 [0, T ], is a

B-adapted solution to SDE (1.5). So once there is a Brownian-adapted solution of (1.5) on onegiven stochastic basis, it follows that there indeed exists a strong solution in the usual sense. Thisjusti�es our de�nition of a strong solution above.

To pursue our objectives we proceed as follows in the remaining parts of the paper. In Sec-tion 2 we recall some fundamental concepts from Malliavin calculus and local time calculus whichcompose central mathematical tools in the following analysis.

We then analyse in Section 3 the existence and Malliavin di�erentiability of a unique strongsolution of SDE’s with irregular drift coe�cients as in (1.5) (Theorem 3.1). It is well known thatthe SDE is Malliavin di�erentiable as soon as the coe�cients are Lipschitz continuous (see e.g.[28]); for merely bounded and measurable drift coe�cients Malliavin di�erentiability was shownonly recently in [25], (see also [23]). Here, we extend ideas introduced for bounded coe�cients in[25] to drift coe�cients of type (1.6). Unlike in most of the existing literature on strong solutionsof SDE’s with irregular coe�cients our approach does not rely on a pathwise uniqueness argument(Yamada-Watanabe Theorem). Instead, we employ a compactness criterium based on Malliavincalculus together with local time calculus to directly construct a strong solution which in additionis Malliavin di�erentiable. Also, we are able to give an explicit expression for the Malliavinderivative of the strong solution of (1.5) in terms of the integral of b (and not the derivative of b)with respect to local time of the strong solution (Proposition 3.2). We mention that while existenceand Malliavin di�erentiability of strong solutions could be extended to analogue multi-dimensionalSDE’s as in [23], the explicit expression of the Malliavin derivative is in general only possible forone-dimensional SDE’s as considered in this paper. Moreover, in this paper we replace argumentsthat are based on White Noise analysis in [25] and [23] by alternative proofs which might makethe text more accessible for readers who are unfamiliar with concepts from White Noise analysis.

Next, we need to analyse the regularity of the dependence of the strong solution in its initialcondition and to introduce the analogue of the �rst variation process (1.3) in case of irregular driftcoe�cients. Using the close connection between the Malliavin derivative and the �rst variationprocess, we �nd that the strong solution is Sobolev di�erentiable in its initial condition (Theorem3.4). Again, we give an explicit expression for the corresponding (Sobolev) �rst variation processwhich does not include the derivative of b (Proposition 3.5).

In Section 4 we develop our main result (Theorem 4.2) which extends Theorem 1.1 to SDE’s withirregular drift coe�cients. To this end, one has to show in the �rst place that the Delta exists, i.e.that E[�(Xx

T1, . . . , Xx

Tm)] is continuously di�erentiable in x. At this point the explicit expressions

for the Malliavin derivative and the �rst variation process are essential. In the �nal representationof the Delta we then have gotten rid of both the derivative of the pay-o� � and the derivative ofthe drift coe�cient b in the �rst variation process, whence the title ”Computing Deltas withoutDerivatives” of the paper. In addition to Deltas of lookback options as in Theorem 1.1, we further

consider Deltas of Asian options with pay-o�s of the type �

�R T2

T1Xx

u dufor T1, T2 2 [0, T ] and

some function � : R ! R. In case the starting point of the averaging period of the Asian pay-o�lies in the future, i.e. T1 > 0, we are able to give analogue results to the ones of lookback options.If the averaging period starts today, i.e. T1 = 0, the Malliavin weight in the expression for theDelta would include a general Skorohod integral which is neither numerically nor mathematicallytractable in our analysis (except for linear coe�cients as in the Black and Scholes model wherethe Skorohod integral turns out to be an Ito integral). However, we are still able to state twoapproximation results for the Delta in this case.

In Section 5 we consider some examples and compute the Deltas in the concrete regime-switchingmodels mentioned above. We do a small simulation study and compare the performance to a �nitedi�erence approximation of the Delta in the same spirit as in [15].


We conclude the paper by an appendix with some technical proofs from Section 3 which havebeen deferred to the end of the paper for better readability.

Notations: We summarise some of the most frequently used notations:

• C1(R) denotes the space of continuously di�erentiable functions f : R ! R.• C1

0 ([0, T ] � R), respectively C10 (R), denotes the space of in�nitely many times di�eren-

tiable functions on [0, T ]� R, respectively R, with compact support.• For a measurable space (S,G) equipped with a measure μ, we denote by Lp(S,G) or Lp(S)

the Banach space of (equivalence classes of) functions on S integrable to some power p,p � 1.

• Lploc(R) denotes the space of locally Lebesgue integrable functions to some power p, p � 1,

i.e.RU|f(x)|pdx < 1 for every open bounded subset U � R.

• W 1,ploc (R) denotes the subspace of L

ploc(R) of weakly (Sobolev) di�erentiable functions such

that the weak derivative f 0 belongs to Lploc(R), p � 1.

• For a progressive process Y· we denote the Doleans-Dade exponential of the correspondingBrownian integral (if well de�ned) by

EZ t

0

b(u, Yu)dBu

�:= exp

Z t

0

b(u, Yu)dBu �1

2

Z t

0

b2(u, Yu)du

�, t 2 [0, T ]. (1.9)

• For Z 2 L2(�,FT ) we denote the Wiener-transform of Z in f 2 L2([0, T ]) by

W(Z)(f) := E

"ZE Z T

0

f(s)dBs

!#.

• We will use the symbol . to denote less or equal than up to a positive real constant C > 0not depending on the parameters of interest, i.e. if we have two mathematical expressionsE1(�), E2(�) depending on some parameter of interest � then E1(�) . E2(�) if, and onlyif, there is a positive real number C > 0 independent of � such that E1(�) � CE2(�).

2. Framework

Our main results centrally rely on tools from Malliavin calculus as well as integration withrespect to local time both in time and space. We here provide a concise introduction to the mainconcepts in these two areas that will be employed in the following sections. For deeper informationon Malliavin calculus the reader is referred to i.e. [28, 21, 22, 10]. As for theory on local timeintegration for Brownian motion we refer to i.e. [12, 29].

2.1. Malliavin calculus. Denote by S the set of simple random variables F 2 L2(�) in the form

F = f

Z T

0

h1(s)dBs, . . . ,

Z T

0

hn(s)dBs

!, h1, . . . , hn 2 L2([0, T ]), f 2 C1

0 (Rn).

The Malliavin derivative operator D acting on such simple random variables is the process DF ={DtF, t 2 [0, T ]} in L2(�� [0, T ]) de�ned by

DtF =nX

i=1

@if

Z T

0

h1(s)dBs, . . . ,

Z T

0

hn(s)dBs

!hi(t).

De�ne the following norm on S:

kFk1,2 := kFkL2(�) + kDFkL2(�;L2([0,T ])) = E[|F |2]1/2 + E

"Z T

0

|DtF |2dt#1/2

. (2.1)

We denote by D1,2 the closure of the family of simple random variables S with respect to the

norm given in (2.1), and we will refer to this space as the space of Malliavin di�erentiable randomvariables in L2(�) with Malliavin derivative belonging to L2(�).


In the derivation of the probabilistic representation for the Delta, the following chain rule forthe Malliavin derivative will be essential:

Lemma 2.1. Let ' : Rm ! R be continuously di�erentiable with bounded partial derivatives.Further, suppose that F = (F1, . . . , Fm) is a random vector whose components are in D

1,2. Then'(F ) 2 D

1,2 and

Dt'(F ) =mXi=1

@i'(F )DtFi, P � a.s., t 2 [0, T ].

The Malliavin derivative operator D : D1,2 ! L2(� � [0, T ]) admits an adjoint operator � =D� : Dom(�) ! L2(�) where the domain Dom(�) is characterised by all u 2 L2(� � [0, T ]) suchthat for all F 2 D

1,2 we have

E

"Z T

0

DtF utdt

#� CkFk1,2,

where C is some constant depending on u.For a stochastic process u 2 Dom(�) (not necessarily adapted to {Ft}t2[0,T ]) we denote by

�(u) :=

Z T

0

ut�Bt (2.2)

the action of � on u. The above expression (2.2) is known as the Skorokhod integral of u andit is an anticipative stochastic integral. It turns out that all {Ft}t2[0,T ]-adapted processes in

L2(�� [0, T ]) are in the domain of � and for such processes ut we have

�(u) =

Z T

0

utdBt,

i.e.the Skorokhod and Ito integrals coincide. In this sense, the Skorokhod integral can be consideredto be an extension of the Ito integral to non-adapted integrands.

The dual relation between the Malliavin derivative and the Skorokhod integral implies thefollowing important formula:

Theorem 2.2 (Duality formula). Let F 2 D1,2 and u 2 Dom(�). Then

E

"F

Z T

0

ut�Bt

#= E

"Z T

0

utDtFdt

#. (2.3)

The next result, which is due to [8] and central in proving existence of strong solutions in thefollowing, provides a compactness criterion for subsets of L2(�) based on Malliavin calculus.

Proposition 2.3. Let Fn 2 D1,2, n = 1, 2..., be a given sequence of Malliavin di�erentiable

random variables. Assume that there exist constants � > 0 and C > 0 such that

supn

E[|Fn|2] � C,

supn

E�|DtFn �Dt0Fn|2

�� C|t� t0|�

for 0 � t0 � t � T , and

supn

sup0�t�T

E�|DtFn|2

�� C .

Then the sequence Fn, n = 1, 2..., is relatively compact in L2(�).

We conclude this review on Malliavin calculus by stating a relation between the Malliavin de-rivative and the �rst variation process of the solution of an SDE with smooth coe�cients thatis essential in the derivation of Theorem 1.1. We give the result for the case when the volatil-ity coe�cient is equal to 1, but the analogue result is valid for more general smooth volatilitycoe�cients. Assume the drift coe�cient b(t, x) in the SDE (1.5) ful�ls the Lipschitz and lineargrowth conditions (1.7)-(1.8). Then it is well known that there exists a unique strong solution


Xxt , t 2 [0, T ], to equation (1.5) that is Malliavin di�erentiable, and that for all 0 � s � t � T the

Malliavin derivative DsXxt ful�ls, see e.g. [28, Theorem 2.2.1]

DsXxt = 1 +

Z t

s

b0(u,Xxu)DsX

xudu, (2.4)

where b0 denotes the (weak) derivative of b with respect to x.Further, under these assumptions the strong solution is also di�erentiable in its initial condition,

and the �rst variation process @@xX

xt , t 2 [0, T ], ful�ls (see e.g. [18] for di�erentiable coe�cients

and [2] for an extension to Lipschitz coe�cients)

@

@xXx

t = 1 +

Z t

0

b0(u,Xxu)

@

@xXx

udu. (2.5)

Solving equations (2.4) and (2.5) thus yields the following proposition.

Proposition 2.4. Let Xxt , t 2 [0, T ], be the unique strong solution to equation (1.5) when b(t, x)

ful�ls the Lipschitz and linear growth condition (1.7)-(1.8). Then Xxt is Malliavin di�erentiable

and di�erentiable in its initial condition for all t 2 [0, T ], and for all s � t � T we have

DsXxt = exp

�Z t

s

b0(u,Xxu)du

�(2.6)

and

@

@xXx

t = exp

�Z t

0

b0(u,Xxu)du

�. (2.7)

As a consequence,

@

@xXx

t = DsXxt

@

@xXx

s , (2.8)

where all equalities hold P -a.s.

2.2. Integration with respect to local-time. Let now Xx be a given (strong) solution to SDE(1.5). In the sequel we need the concept of stochastic integration over the plane with respect tothe local time LXx

(t, y) of Xx. For Brownian motion, the local time integration theory in timeand space has been introduced in [12]. We extend this local time integration theory to moregeneral di�usions of type (1.5) by resorting to the Brownian setting under an equivalent measurewhere Xx is a Brownian motion. To this end, we notice the following fact that is extensively usedthroughout the paper.

Remark 2.5. The Radon-Nikodym density

dQ

dP= E

�

Z T

0

b(s,Xxs )dBs

!de�nes a probability measure Q equivalent to P under which Xx is Brownian motion starting in x.Indeed, because b is of at most linear growth we obtain by Gronwall’s inequality as in the proof ofLemma A.1 a constant Ct,x > 0 such that |Xx

t | � Ct,x(1 + |Bt|). One can thus �nd a equidistantpartition 0 = t0 < t1... < tm = T such that

E

�exp

�Z ti+1

ti

b2(s,Xxs )ds

�� E

�exp

�Z ti+1

ti

�C1 + C2|Bs|+ C3|Bs|2

ds

�< 1

�

for all i = 0, ...,m�1, where C1, C2 and C3 are some positive constants. Then it is well-known, seee.g. [16, Corollary 5.16], that Q is an equivalent probability measure under which Xx is Brownianmotion by Girsanov’s theorem.


We now de�ne the feasible integrands for the local time-space integral with respect to LXx

(t, y)by the Banach space (Hx, k·k) of functions f : [0, T ]� R �! R with norm

kfkx =2

Z T

0

ZR

f2(s, y)1p2�s

exp

�|y � x|2

2s

�dyds

!1/2

+

Z T

0

ZR

|y � x| |f(s, y)| 1

sp2�s

exp

�|y � x|2

2s

�dyds.

We remark that this space of integrands is the same as the one introduced in [12] for Brownianmotion (i.e. the special case when the Xx is a Brownian motion), except that we have in a straightforward manner generalised the space in [12] to the situation when the Brownian motion hasarbitrary initial value x.

We denote by f� : [0, T ]� R �! R a simple function in the form

f�(s, y) =X

1�i�n�1,1�j�m�1

fij1(yi,yi+1](y)1(sj ,sj+1](s),

where (sj)1�j�m is a partition of [0, T ] and (yi)1�i�n and (fij)1�i�n,1�j�m are �nite sequencesof real numbers. It is readily checked that the space of simple functions is dense in (Hx, k·k). Thelocal time-space integral of an simple function f� with respect to LXx

(dt, dy) is then de�ned byZ T

0

ZR

f�(s, y)LXx

(ds, dy) :=

:=X

1�i�n�11�j�m�1

fij(LXx

(sj+1, yi+1)� LXx

(sj , yi+1)� LXx

(sj+1, yi) + LXx

(sj , yi)).

Lemma 2.6. For f 2 Hx let fn, n � 1, be a sequence of simple functions converging to f in

Hx. ThenR T

0

RRfn(s, y)L

Xx

(ds, dy), n � 1, converges in probability. Further, for any otherapproximating sequence of simple functions the limit remains the same.

Proof. De�ne FXx

n :=R T

0

RRfn(s, x)L

Xx

(ds, dx). Now consider the equivalent measure Q from

Remark 2.5 under which Xx is Brownian motion. De�ne FXx

:=R T

0

RRf(s, x)LXx

(ds, dx) to bethe time-space integral of f with respect to the local time of Brownian motion Xx under Q, whichexists as an L1(Q)-limit of FXx

n , n � 1 by the Brownian local time integration theory introducedin [12] (since fn, n � 1 converge to f in Hx). We show that FXx

n , n � 1 converge in probability toFXx

under P . Indeed,

E[1 ^ |FXx

� FXx

n |] =E

"�1 ^ |FBx

� FBx

n |E Z T

0

b(s, Bxs )dBs

!#

�E

24E Z T

0

b(s, Bxs )dBs

!1+"351/(1+")

E

��1 ^ |FBx

� FBx

n | 1+"

"

� "1+"

�C"E[�1 ^ |FBx

� FBx

n |]

"1+"

n!1�! 0 , (2.9)

where, in analogy to the notation FXx

and FXx

n above, the notation FBx

and FBx

n refers to thecorresponding integrals with respect to local time of Brownian motion Bx under P , and where inthe �rst equality we have used that (FBx

, FBx

n ) has the same law under P as (FXx

, FXx

n ) underQ. The inequalities follow by Lemma A.1 for some " > 0 suitably small. Further, by [12] weknow that FBx

n , n � 1 converge to FBx

in L1(P ), which implies the convergence in (2.9). HenceFXx

n , n � 1 converge to FXx

in the Ky-Fan metric d(X,Y ) = E[1 ^ |X � Y |], X,Y 2 L0(�),which characterises convergence in probability. Finally, again by [12], FXx

is independent of theapproximating sequence fn, n � 1. �


De�nition 2.7. For f 2 Hx the limit in Lemma 2.6 is called the time-space integral of f with

respect to LXx

(dt, dx) and is denoted byR T

0

RRf(s, y)LXx

(ds, dy). Further, for any t 2 [0, T ] we

de�neR t

0

RRf(s, y)LXx

(ds, dy) :=R T

0

RRf(s, y)I[0,t](s)L

Xx

(ds, dy).

Remark 2.8. We notice that the drift coe�cient b(t, x) in (1.6), which is of linear growth in xuniformly in t, is in Hx, and thus the local time integral of b(t, x) with respect to LXx

(dt, dy) existsfor any x 2 R.

If Xx is a Brownian motion B· we have the following decomposition due to [12] that we employin the construction of strong solutions, and that also constitutes the foundation in the constructionof the local time integral in [12].

Theorem 2.9. Let f 2 H0. ThenZ t

0

ZR

f(s, y)LBx

(ds, dy) =

= �

Z t

0

f(s, Bxs )dBs +

Z T

T�t

f(T � s, bBxs )dWs �

Z T

T�t

f(T � s, bBxs )

bBs

T � sds,

(2.10)

where bBt = BT�t, 0 � t � T is time-reversed Brownian motion, and W·, de�ned by

bBt = BT +Wt �

Z t

0

bBs

T � sds,

is a Brownian motion with respect to the �ltration of bB·.

We conclude this subsection by stating three further identities for the local time integral of ageneral di�usions Xx which will be useful later on.

Lemma 2.10. Let f 2 Hx be Lipschitz continuous in x. Then for all t 2 [0, T ]

�

Z t

0

ZR

f(s, y)LXx

(ds, dy) =

Z t

0

f 0(s,Xxs )ds. (2.11)

where f 0 denotes the (weak) derivative of f(t, y) with respect to y.If f 2 Hx is time homogeneous (i.e. f(t, y) = f(y) only depends on the space variable) and

locally square integrable, then for any t 2 [0, T ]Z t

0

ZR

f(s, x)LXx

(ds, dx) = �[f(·, Xx), Xx]t. (2.12)

and

�

Z t

0

ZR

f(s, y)LXx

(ds, dy) = 2F (Xxt )� 2F (x)� 2

Z t

0

f(Xxs )dX

xs (2.13)

where F is a primitive function of f and [b(·, Xx· ), X

x· ]t is the generalised covariation process

[f(·, Xx· ), X

x· ]t := P � lim

m!1

mXk=1

�f(tmk , Xx

tmk)� f(tmk�1, X

xtmk�1

)�

Xxtk

�Xxtk�1

where for every m {tmk }mk=1 is a partition of [0, t] such that limm

supk=1,...,m

|tmk � tmk�1| = 0. Note that

(2.13) can be considered as a generalised Ito formula.

Proof. If Xx = x+B, then identities (2.11)-(2.13) are given in [12]. For general Xx, we considerthe identities under the equivalent measure Q from Remark 2.5. Then, by the construction of thelocal time integral outlined in Lemma 2.6, the integrals in the identities are the ones with respectto Brownian motion Xx, for which we know the identities hold by [12] (where such identities aregiven in the case x = 0 but one can easily extend them to the case of the Brownian motion startingat an arbitrary x 2 R). �


3. Existence, Malliavin, and Sobolev differentiability of strong solutions

In this section we prepare the necessary theoretical grounds to develop the probabilistic rep-resentation of Deltas. Being notationally and technically rather heavy, the proofs of this sectionare deferred to Appendix A for an improved ow and readability of the paper. We �rst study theexistence and Malliavin di�erentiability of a unique strong solution of SDE (1.5) before we turnto the di�erentiability of the strong solution in its initial condition and the corresponding �rstvariation process. We state the �rst main result of this section:

Theorem 3.1. Suppose that the drift coe�cient b : [0, T ] � R ! R is in the form (1.6). Thenthere exists a unique strong solution {Xx

t }t2[0,T ] to SDE (1.5). In addition, Xxt is Malliavin

di�erentiable for every t 2 [0, T ].

The proof of Theorem 3.1 employs several auxiliary results presented in Appendix A. The mainsteps are:

(1) First, we construct a weak solution Xx to (1.5) by means of Girsanov’s theorem, that iswe introduce a probability space (�,F , P ) that carries some Brownian motion B and acontinuous process Xx such that (1.5) is ful�lled. However, a priori Xx is not adapted tothe �ltration {Ft}t2[0,T ] generated by Brownian motion B.

(2) Next, we approximate the drift coe�cient b = b + b by a sequence of functions (whichalways exists by standard approximation results)

bn := bn + b, n � 1, (3.1)

such that {bn}n�1 � C10 ([0, T ]� R) with supn�1 kbnk1 � C < 1 and bn ! b in (t, x) 2

[0, T ] � R a.e. with respect to the Lebesgue measure. By standard results on SDE’s, weknow that for each smooth coe�cient bn, n � 1, there exists a unique strong solution Xn,x

·to the SDE

dXn,xt = bn(t,X

n,xt )dt+ dBt, 0 � t � T, Xn,x

0 = x 2 R . (3.2)

We then show that for each t 2 [0, T ] the sequenceXn,xt converges weakly to the conditional

expectation E[Xxt |Ft] in the space L2(�;Ft) of square integrable, Ft-measurable random

variables.(3) By Proposition 2.4 we know that for each t 2 [0, T ] the strong solutions Xn,x

t , n � 1, areMalliavin di�erentiable with

DsXn,xt = exp

�Z t

s

b0n(u,Xn,xu )du

�, 0 � s � t � T, n � 1, (3.3)

where b0n denotes the derivative of bn with respect to x. We will use representation (3.3)to employ a compactness criterion based on Malliavin calculus to show that for everyt 2 [0, T ] the set of random variables {Xn,x

t }n�1 is relatively compact in L2(�;Ft), whichthen allows to conclude that Xn,x

t converges strongly in L2(�;Ft) to E[Xxt |Ft]. Further

we obtain that E[Xxt |Ft] is Malliavin di�erentiable as a consequence of the compactness

criterion.(4) In the last step we show that E[Xx

t |Ft] = Xxt , which implies that Xx

t is Ft-measurableand thus a strong solution. Moreover, we show that this solution is unique.

Notation: In the following we sometimes include the drift coe�cient b into the sequence {bn}n�0

by putting b0 := b0 + b := b+ b = b.

The next important result is an explicit representation of the Malliavin derivative of the strongsolution Xx

t , t 2 [0, T ]. For smooth coe�cients b we can explicitly express the Malliavin deriva-tive in terms of the derivative of b as stated in (3.3). For general, not necessarily di�erentiablecoe�cients b, we are still able to give an explicit formula which now only involves the coe�cientb in a local time integral:


Proposition 3.2. For 0 � s � t � T , the Malliavin derivative DsXxt of the unique strong solution

Xxt to equation (1.5) has the following explicit representation:

DsXxt = exp

��

Z t

s

ZR

b(u, y)LXx

(du, dy)

�P-a.s., (3.4)

where LXx

(du, dy) denotes integration in space and time with respect to the local time of Xx, seeSection 2.2 for de�nitions.

Next, we turn our attention to the study of the strong solution Xxt as a function in its initial

condition x for SDE’s with possible irregular drift coe�cients. The �rst result establishes Holdercontinuity jointly in time and space.

Proposition 3.3. Let Xxt , t 2 [0, T ] be the unique strong solution to the SDE (1.5). Then for

all s, t 2 [0, T ] and x, y 2 K for any arbitrary compact subset K � R there exists a constant

C = C(K, kbk1, kb0k1) > 0 such that

E�|Xx

t �Xys |2

�� C(|t� s|+ |x� y|2).

In particular, there exists a continuous version of the random �eld (t, x) 7! Xxt with Holder

continuous trajectories of order � < 1/2 in t 2 [0, T ] and � < 1 in x 2 R.

If the drift coe�cient b is regular, then we know by Proposition 2.4 that Xxt is even di�erentiable

as a function in x. The �rst variation process @@xX

x· is then given by (2.7) in terms of the derivative

of the drift coe�cient and is closely related to the Malliavin derivative by (2.8). In the followingwe will derive analogous results for irregular drift coe�cients, where in general the �rst variationprocess will now exist in the Sobolev derivative sense. Let U � R be an open and bounded subset.The Sobolev space W 1,2(U) is de�ned as the set of functions u : R ! R, u 2 L2(U) such that itsweak derivative belongs to L2(U). We endow this space with the norm

kuk1,2 = kuk2 + ku0k2where u0 stands for the weak derivative of u 2 W 1,2(U). We say that the solution Xx

t , t 2 [0, T ],is Sobolev di�erentiable in U if for all t 2 [0, T ], X ·

t belongs to W 1,2(U), P -a.s. Observe that ingeneral X ·

t is not in W 1,2(R), e.g. take b � 0.

Theorem 3.4. Let b : [0, T ] � R ! R be as in (1.6). Let Xxt , t 2 [0, T ] be the unique strong

solution to the SDE (1.5) and U � R an open, bounded set. Then for every t 2 [0, T ] we have

(x 7! Xxt ) 2 L2(�,W 1,2(U)).

We remark that using analogue techniques as in [27] one could even establish that the strongsolution gives rise to a ow of Sobolev di�eomorphisms. This, however, is beyond the scope ofthis paper.

Similarly as for the Malliavin derivative, we are able to give an explicit representation forthe �rst variation process in the Sobolev sense that does not involve the derivative of the driftcoe�cient by employing local time integration.

Proposition 3.5. Let b : [0, T ] � R ! R be as in (1.6). Then the �rst variation process (inthe Sobolev sense) of the strong solution Xx

t , t 2 [0, T ] to SDE (1.5) has the following explicitrepresentation

@

@xXx

t = exp

��

Z t

0

ZR

b(u, y)LXx

(du, dy)

�dt� P � a.s. (3.5)

As a direct consequence of Proposition 3.5 together with Proposition 3.2 we obtain the followingrelation between the Malliavin derivative and the �rst variation process, which is an extension ofProposition 2.4 to irregular drift coe�cients and which is a key result in deriving the desiredexpression for the Delta.


Corollary 3.6. Let Xxt , t 2 [0, T ], be the unique strong solution to (1.5). Then the following

relationship between the spatial derivative and the Malliavin derivative of Xxt holds:

@

@xXx

t = DsXxt

@

@xXx

s P � a.s. (3.6)

for any s, t 2 [0, T ], s � t.

Remark 3.7. Note that by Lemma 2.10 the Malliavin derivative in (3.4) and the �rst variationprocess in (3.5) can be expressed in various alternative ways. Firstly, we observe that by formula

(2.11) the local time integral of the regular part b in b can be separated and rewritten in the form

�

Z t

s

ZR

b(u, y)LXx

(du, dy) = �

Z t

s

ZR

b(u, y)LXx

(du, dy) +

Z t

s

b0(u,Xxu)du a.s. (3.7)

If in addition b(t, ·) is locally square integrable and continuous in t as a map from [0, T ] toL2loc(R) or even time-homogeneous, then by Lemma 2.10 also the local time integral associated

to the irregular part b can be reformulated in terms of the generalised covariation process as in(2.12) or in terms of the generalised Ito formula as in (2.13), respectively. In particular, thesereformulations appear to be useful for simulation purposes.

We conclude this section by giving an extension of all the results seen so far to a class of SDE’swith more general di�usion coe�cients.

Theorem 3.8. Consider the time-homogeneous SDE

dXxt = b(Xx

t )dt+ �(Xxt )dBt, Xx

0 = x 2 R, 0 � t � T, (3.8)

where the coe�cients b : R �! R and � : R �! R are Borel measurable. Require that there existsa twice continuously di�erentiable bijection : R �! R with derivatives

0 and 00 such that

0(y)�(y) = 1 for a.e. y 2 R,

as well as

�1 is Lipschitz continuous.

Suppose that the function b� : R �! R given by

b�(x) := 0 ��1(x)

b(�1 (x)) +

1

200 �

�1(x)

�(�1 (x))2

satis�es the conditions of Theorem 3.1. Then there exists a Malliavin di�erentiable strong solutionXx

· to (3.8) which is (locally) Sobolev di�erentiable in its initial condition.

Proof. The proof is obtained directly from Ito’s formula. See [25]. �

4. Existence and derivative-free representations of the Delta

We now turn the attention to the study of option price sensitivities with respect to the initialvalue of an underlying asset with irregular drift coe�cient. Notably, we will consider lookbackoptions with path-dependent (discounted) pay-o� in the form

�(XxT1, . . . , Xx

Tm) (4.1)

for time points T1, . . . , Tm 2 (0, T ], some function � : Rm ! R, and where the evolution of theunderlying price process under the risk-neutral pricing measure is modelled by the strong solutionXx of SDE (1.5) with possibly irregular drift b as in (1.6). In particular, for m = 1 the pay-o�(4.1) is associated to a European option with maturity T1. Then the current option price is givenby E

��(Xx

T1, . . . , Xx

Tm)�and the main objective of this section is to establish existence and a

derivative-free, probabilistic representation of the Delta

@

@xE��(Xx

T1, . . . , Xx

Tm)�.


After having analysed lookback options, we will also address the problem of computing Deltas ofAsian options with (discounted) path-dependent pay-o� in the form

�

Z T2

T1

Xxu du

!(4.2)

for T1, T2 2 [0, T ] and some function � : R ! R.We start with a preliminary result which shows that in case of a smooth pay-o� function with

compact support the Delta exists for a large class of path dependent options that includes bothlookback as well as Asian options.

Lemma 4.1. Let Xxt , t 2 [0, T ], be the strong solution to SDE (1.5) and {Xn,x

t }n�1 the cor-responding approximating strong solutions of SDE (3.2). Let � 2 C1

0 (Rm) and consider thefunctions

un(x) := E

"�

Z T

0

Xn,xu μ1(du),

Z T

0

Xn,xu μ2(du), . . . ,

Z T

0

Xn,xu μm(du)

!#(4.3)

and

u(x) := E

"�

Z T

0

Xxuμ1(du),

Z T

0

Xxuμ2(du), . . . ,

Z T

0

Xxuμm(du)

!#(4.4)

where μ1, . . . , μm are �nite measures on [0, T ] independent of x 2 R. Consider also the function

u(x) := E

"mXi=1

@i�

Z T

0

Xxuμ1(du),

Z T

0

Xxuμ2(du), . . . ,

Z T

0

Xxuμm(du)

!Z T

0

@

@xXx

uμi(du)

#(4.5)

where @@xX

x is the �rst variation process of Xx introduced in (3.5). Then

un(x)n!1��! u(x) for all x 2 R,

andu0n(x)

n!1��! u(x)

uniformly on compact subsets K � R, where u0n denotes the derivative. As a result, we obtain that

u 2 C1(R) with u0 = u. In particular, we obtain the result for lookback options by taking μi = �tithe Dirac measure concentrated on ti, i = 1, . . . ,m, and for Asian options by taking m = 1 andμ1 = du.

Proof. First of all, observe that the expression in (4.5) is well-de�ned. This can be seen by usingCauchy-Schwarz inequality, the fact that � 2 C1

0 (Rm), and Corollary A.9.It is readily checked that un(x) ! u(x) for all x 2 R since � is smooth by using the mean-value

theorem and the fact that Xn,xt ! Xx

t in L2(�) as n ! 1 for every t 2 [0, T ] (see Theorem A.6).We introduce the following short-hand notation for them-dimensional random vector associated

to a process Y :

h(Y·,T ) :=

Z T

0

Yuμ1(du) ,

Z T

0

Yuμ2(du) , . . . ,

Z T

0

Yuμm(du)

!.

For the smooth coe�cients bn we have un 2 C1(R), n � 1, and since @i� are bounded for alli = 1, . . . ,m and by dominated convergence we have

u0n(x) = E

"mXi=1

@i��h(Xn,x

·,T ) Z T

0

@

@xXn,x

u μi(du)

#.

Moreover, we can take integration with respect to μi(du), i = 1, ...m, outside the expectation.Thus

u0n(x) =

mXi=1

Z T

0

E

�@i�

�h(Xn,x

·,T ) @

@xXn,x

u

�μi(du).


Hence

|u0n(x)� u(x)| =

mXi=1

Z T

0

E

�@i�

�h(Xn,x

·,T ) @

@xXn,x

u � @i��h(Xx

·,T ) @

@xXx

u

�μi(du)

=:mXi=1

Z T

0

Fn,i(u, x)μi(du)

where Fn,i(u, x) denotes the expectation in the integral. We will show that for any i = 1, . . . ,mand compact subset K � R,

sup(u,x)2[0,T ]�K

|Fn,i(u, x)| n!1��! 0.

Indeed, by plugging in expression (3.5) for the �rst variation process and Girsanov’s theorem weget

|Fn,i(u, x)| ��Eh@i�

�h(Bx

·,T ) exp

��

Z u

0

ZR

bn(v, y)LBx

(dv, dy)

�E Z T

0

bn(u,Bxu)dBu

!

� @i��h(Bx

·,T ) exp

��

Z u

0

ZR

b(v, y)LBx

(dv, dy)

�E Z T

0

b(u,Bxu)dBu

!�i��

��Eh@i�

�h(Bx

·,T ) E Z T

0

b(u,Bxu)dBu

!

�

exp

��

Z u

0

ZR

bn(v, y)LBx

(dv, dy)

�� exp

��

Z u

0

ZR

b(v, y)LBx

(dv, dy)

�� i��+

��Eh@i�

�h(Bx

·,T ) exp

��

Z u

0

ZR

bn(v, y)LBx

(dv, dy)

�

�

E Z T

0

bn(u,Bxu)dBu

!� E

Z T

0

b(u,Bxu)dBu

!!i��:= In + IIn

Here, we will show estimates for IIn, for In the argument is analogous. Similarly to how we obtainthe estimate II1n + II2n in the proof of Lemma A.5, using inequality |ex � 1| � |x|(ex + 1) we get

IIn .Eh|@i�

�h(Bx

·,T ) ||Un| exp

��

Z u

0

ZR

bn(v, y)LBx

(dv, dy)

�E Z T

0

bn(u,Bxu)dBu

!i+ E

h|@i�

�h(Bx

·,T ) ||Un| exp

��

Z u

0

ZR

bn(v, y)LBx

(dv, dy)

�E Z T

0

b(u,Bxu)dBu

!i=: II1n + II2n ,

where

Un :=

Z T

0

(bn(u,Bxu)� b(u,Bx

u))dBu �1

2

Z T

0

(b2n(u,Bxu)� b2(u,Bx

u))du.

We will now show that II1n ! 0 as n ! 1 uniformly in x on a compact subset K � R. Theconvergence of II2n then follows immediately, too. Denote p = 1+"

" with " > 0 from Lemma A.1and use Holder’s inequality with exponent 1 + " on the Doleans-Dade exponential, then employ


formula (2.11) on b in bn = bn + b and use Cauchy-Schwarz inequality successively. As a result,

II1n .E

24E Z T

0

bn(u,Bxu)dBu

!1+"351/(1+")

E�|@i� �

h(Bx·,T�)

|�1/(2p) E[|Un|8p]1/(8p)

� E

"exp

��4p

Z u

0

ZR

bn(v, y)LBx

(dv, dy)

�1/(4p))E

�exp

�8p

Z u

0

b0(v,Bxv )dv

��1/(8p).

The �rst and fourth factor are bounded uniformly in n � 0 and x 2 K by Remark A.2 and Lemma

A.3, respectively. The second and and �fth factor can be controlled since @i�, i = 1, . . . ,m and b0

are uniformly bounded. It remains to show that

supx2K

E[|Un|8p] n!1��! 0

for any compact subset K � R.Using Minkowski’s inequality, Burkholder-Davis-Gundy’s inequality and Holder’s inequality we

can write

E[|Un|8p] .Z T

0

E[|bn(u,Bxu)� b(u,Bx

u)|8p]du+

Z T

0

E[|b2n(u,Bxu)� b2(u,Bx

u)|8p]du. (4.6)

Then write the integrand of the �rst term in (4.6) as


u)|8p] =1p2�u

ZR

|bn(u, y)� b(u, y)|8pe� (y�x)2

2u dy.

Using Cauchy-Schwarz inequality on |bn(u, y)� b(u, y)|8pe�y2

4u we obtain

E[|bn(u,Bxu)�b(u,Bx

u)|8p] �

�1p2�u

e�x2

2u

ZR

|bn(u, y)� b(u, y)|16pe� y2

2u dy

�1/2 ZR

e�y2

2u+2 xyu dy

�1/2

.

Then for each u 2 [0, T ], by taking the supremum over x 2 K and by dominated convergence, weget

supx2K


u)|8p] n!1��! 0 ,

and hence the result follows. Similarly, one can argue for the second term in (4.6).In sum,

sup(u,x)2[0,t]�K

|Fn,i(u, x)| n!1��! 0

for all i = 1, . . . ,m and hence u0n(x)

n!1��! u(x) uniformly on compact sets K � R, and thus

u 2 C1(R) with u0 = u. �

We come to the main result of this paper, which extends Theorem 1.1 to lookback optionswritten on underlyings with irregular drift coe�cients. In particular, when plugging in expression(3.5) for the �rst variation process, we see that the formula for the Delta in (4.8) below involvesneither the derivative of the pay-o� function � nor the derivative of the drift coe�cient b. Weobtain this result for pay-o� functions � 2 Lq

w(Rm), where

Lqw(R

m) :=

�f : Rm ! R measurable :

ZRm

|f(x)|q w(x)dx < 1�

(4.7)

for the weight function w de�ned by w(x) := exp(� 12T |x|2), x 2 R

m, and where the exponent qdepends on the drift b. Note that all pay-o� functions of practical relevance are contained in thesespaces.

Theorem 4.2. Let Xx be the strong solution to SDE (1.5) and � : Rm ! R a function in

L4pw (Rm), where p > 1 is the conjugate of 1 + " for " > 0 in Lemma A.1. Then, for any 0 < T1 �

· · · � Tm � T , the priceu(x) := E

��(Xx

T1, . . . , Xx

Tm)�


of the associated lookback option is continuously di�erentiable in x 2 R, and its derivative, i.e.the Delta, takes the form

u0(x) = E

"�(Xx

T1, . . . , Xx

Tm)

Z T

0

a(s)@

@xXx

s dBs

#(4.8)

for any bounded measurable function a : R ! R such that, for every i = 1, . . . ,m,Z Ti

0

a(s)ds = 1. (4.9)

Proof. Assume �rst � 2 C10 (Rm). Then by Lemma 4.1 with μi = �ti , i = 1, . . . ,m, we know that

u(x) = E��(Xx

T1, . . . , Xx

Tm)�is continuously di�erentiable with derivative

u0(x) :=mXi=1

E

�@i�(X

xT1, . . . , Xx

Tm)@

@xXx

Ti

�.

Now, by Corollary 3.6, we have for any i = 1, . . . ,m

@

@xXx

Ti= DsX

xTi

@

@xXx

s for all s � Ti . (4.10)

Also recall that DsXxTi

= 0 for s � Ti. So, for any function a : R ! R satisfying (4.9) we have

@

@xXx

Ti=

Z T

0

a(s)DsXxTi

@

@xXx

s ds.

As a result,

u0(x) =mXi=1

E

"@i�(X

xT1, . . . , Xx

Tm)

Z T

0

a(s)DsXxTi

@

@xXx

s ds

#

= E

"Z T

0

a(s)Ds�(XxT1, . . . , Xx

Tm)@

@xXx

s ds

#,

where in the last step we could use the chain rule for the Malliavin derivative backwards, see Lemma2.1, since �(Xx

T1, . . . , Xx

Tm) is Malliavin di�erentiable due to Theorem 3.1. Then a(s) @

@xXxs is an

Fs-adapted Skorokhod integrable process by Corollary A.9 with p = 2, so the duality formula forthe Malliavin derivative (see Theorem 2.2) yields

u0(x) = E

"�(Xx

T1, . . . , Xx

Tm)

Z T

0

a(s)@

@xXx

s dBs

#.

Finally, we extend the result to a pay-o� function � 2 L4pw (Rm). By standard argu-

ments we can approximate � by a sequence of functions �n 2 C10 (Rm), n � 0, such that

�n ! � in L4pw (Rm) as n ! 1. Now de�ne un(x) := E[�n(X

xT1, . . . , Xx

Tm)] and u(x) :=

E[�(XxT1, . . . , Xx

Tm)R T

0a(s) @

@xXxs dBs]. Then

|u0n(x)� u(x)| =

��E"�

�n(XxT1, . . . , Xx

Tm)� �(Xx

T1, . . . , Xx

Tm) Z T

0

a(s)@

@xXx

s dBs

#�� E

h��n(XxT1, . . . , Xx

Tm)� �(Xx

T1, . . . , Xx

Tm)��2i1/2 E "Z T

0

|a(s) @

@xXx

s |2ds#1/2

� CE

"��n(BxT1, . . . , Bx

Tm)� �(Bx

T1, . . . , Bx

Tm)��2 E

Z T

0

b(u,Bxu)dBu

!#1/2,

where we have used Cauchy-Schwarz inequality, Ito’s isometry, Corollary A.9 and Girsanov’stheorem in this order. Then we apply Holder’s inequality with 1+ " for a small enough " > 0 and


use Lemma A.1 to get

|u0n(x)� u(x)| �

� CEh��n(B

xT1, . . . , Bx

Tm)� �(Bx

T1, . . . , Bx

Tm)��2 1+"

"

i "2(1+")

E

24E Z T

0

b(u,Bxu)dBu

!1+"35

12(1+")

� CEh��n(B

xT1, . . . , Bx

Tm)� �(Bx

T1, . . . , Bx

Tm)��2 1+"

"

i "2(1+")

.

For the last quantity, denote by Pt(y) :=1p2�t

e�y2/(2t), y 2 R the density of Bt, and set T0 := 0

and y0 := x. Recall that 0 < T1 � · · · � Tm. Using the independent increments of the Brownianmotion we rewrite

E

" ��n(BxT1, . . . , Bx

Tm)� �(Bx

T1, . . . , Bx

Tm)��2 1+"

"

#

=

ZRm

|�n(y1, . . . , ym)� �(y1, . . . , ym)|2 1+""

mYi=1

PTi�Ti�1(yi � yi�1)dy1 · · · dym.

Furthermore, with t� := mini=1,...,m�1(ti+1 � ti)

E

" ��n(BxT1, . . . , Bx

Tm)� �(Bx

T1, . . . , Bx

Tm)��2 1+"

"

#

� (2�t�)�m/2

ZRm

|�n(y1, . . . , ym)� �(y1, . . . , ym)|2 1+""

mYi=1

e�

y2i

4(Ti�Ti�1)

� e�

y2i

4(Ti�Ti�1)+

yiyi�1Ti�Ti�1

�y2i�1

2(Ti�Ti�1) dy1 · · · dym.

By applying Cauchy-Schwarz inequality we obtain

E

" ��n(BxT1, . . . , Bx

Tm)� �(Bx

T1, . . . , Bx

Tm)��2 1+"

"

#

� (2�t�)�m/2

ZRm

|�n(y1, . . . , ym)� �(y1, . . . , ym)|4 1+"" e�

|y|2

2T dy1 · · · dym�1/2

�

ZRm

mYi=1

e�

y2i

2(Ti�Ti�1)+

2yiyi�1Ti�Ti�1

�y2i�1

(Ti�Ti�1) dy1 · · · dym!1/2

=: In · II.For the second factor we have

II � e�x2

T

ZRm

e�y12T +

xy1T

mYi=2

e�(yi�yi�1)2

2T dy1 · · · dym!1/2

and hence

supx2K

II < 1.

Thus, since factor In converges to 0 by assumption, we can approximate u uniformly in x 2 R oncompact sets by smooth pay-o� functions. So u 2 C1(R) and u0 = u. �

Next, we consider Asian options with pay-o� given by (4.2). If T1 > 0 we are able to give theanalogous result to Theorem 4.2 by approximating the Asian pay-o� with lookback pay-o�s:

Corollary 4.3. Let Xx be the strong solution to SDE (1.5) and � : R ! R a function in L4pew (R)

where ew is de�ned in (4.15) further below and where p > 1 is the conjugate of 1 + " for " > 0 in


Lemma A.1. Then for any T1, T2 2 (0, T ] with T1 < T2, the price

u(x) = E

"�

Z T2

T1

Xxudu

!#of the associated Asian option is continuously di�erentiable in x 2 R, and its derivative, i.e. theDelta, takes the form

u0(x) = E

"�

Z T2

T1

Xxs ds

!Z T1

0

a(s)@

@xXx

s dBs

#(4.11)

for any bounded measurable function a : R ! R such thatZ T1

0

a(s)ds = 1. (4.12)

Proof. Assume �rst that � 2 C1(R), and consider a series of partitions of [T1, T2] with vanishingmesh, i.e. let {T1 = tm0 < tm1 < . . . < tmm = T2}1m=1 with limm!1 supi=1,...,m(tmi �tmi�1) = 0. Thenwe may write the integral using Riemann sums as followsZ T2

T1

Xxs ds = lim

m!1

Xi=1,...,m

Xxtmi

(tmi � tmi�1).

Then

�

Z T2

T1

Xxs ds

!= lim

m!1�

0@ Xi=1,...,m

Xxtmi

(tmi � tmi�1)

1A =: limm!1

�m(Xxtm1

, . . . , Xxtmm

).

By Theorem 4.2 we have

u0(x) = limm!1

E

"�m(Xx

tm1, . . . , Xx

tmm)

Z T

0

am(s)@

@xXx

s dBs

#where am is a bounded measurable function such that

R tmi0

am(s)ds = 1 for each i = 1, . . . ,m.Then

u0(x) = limm!1

E

"�m(Xx

tm1, . . . , Xx

tmm)

Z T

0

am(s)@

@xXx

s dBs

#

= E

"�

Z T2

T1

Xxs ds

!Z T1

0

a(s)@

@xXx

s dBs

#,

where a is a function such thatR T1

0a(s)ds = 1.

For a general pay-o� �, we approximate � in L4pw (R) by a sequence of functions {�n}n�0 �

C10 (R) and de�ne u(x) := E[�(

R T2

T1Xx

s ds)] and u(x) := Eh�

�R T2

T1Xx

s ds R T1

0a(s) @

@xXxs dBs

i.

Consider un(x) = E[�n(R T2

T1Xx

s ds)]. Finally, similarly as in Theorem 4.2 one has un(x) ! u(x)as n ! 1 for all x 2 R and

|u0n(x)� u(x)| . E

"��n

Z T2

T1

Bxs ds

!� �

Z T2

T1

Bxs ds

! ��2p#1/p

,

which goes to zero uniformly in x 2 K on compact sets K � R as n ! 1 by using the fact thatR T2

T1Bx

s ds has a Gaussian distribution with mean x(T2 � T1) and varianceT 32 �T 3

1

3 � (T2 � T1)T21

which explains the weight ew. �

Remark 4.4. From the proof of Corollary 4.3 it follows that the Delta (4.11) of an Asian optioncan be approximated by the Delta

E

"�

mXi=1

Xxti (ti � ti�1)

!Z T2

0

a(s)@

@xXx

s dBs

#(4.13)


of a lookback option for a �ne enough partition T1 = t0 < t1 < · · · < tm = T2, whereR ti0

a(s)ds = 1for each i = 1, . . . ,m.. From a numerical point of view, this might make a di�erence since thefunction a in (4.13) can be chosen to have support on the full segment [0, T2], while in (4.11) thefunction a can only have support on [0, T1].

If the averaging period of the Asian option starts today, i.e. T1 = 0, then formula (4.11)does not hold anymore. Instead, one can derive alternative closed-form expressions for the Asiandelta for smooth di�usion coe�cients, see e.g. [15] and [3], which potentially can be generalisedto irregular drift coe�cients. However, except for linear coe�cients (Black & Scholes model),these expressions involve stochastic integrals in the Skorokhod sense which are, in general, hardto simulate. Instead, we here propose to enlarge the state space by one dimension and to considera perturbed Asian pay-o�. In that case we are able to derive a probabilistic representation forthe corresponding Delta that only includes Ito integrals. More precisely, we consider the (strong)solution to the perturbed two-dimensional SDE

dXxt = b(t,Xx

t )dt+ dBt, Xx0 = x 2 R,

dY �,x,yt = Xx

t dt+ �dWt , Y�,x,y0 = y 2 R, 0 � t � T, (4.14)

for � > 0, where W is a one-dimensional Brownian motion independent of B. The idea is nowto consider the perturbed Asian pay-o� with averaging period [0, T2], T2 2 (0, T ] as a Europeanpay-o� on Y �,x,y

T2:

�

Z T2

0

Xxs ds

! �(Y �,x,0

T2) = �

Z T2

0

Xxs ds+ �WT2

!.

We then get the following result, where we now consider the slightly di�erently weighted pay-o�function space

Lqw(R) :=

�f : R ! R measurable:

ZR

|f(x)|qw(x)dx < 1�

for the weight function w de�ned by

w(x) = exp

�

|x|22T2 (T 2

2 /3 + 1)

�, x 2 R. (4.15)

Theorem 4.5. Let Y �,x,y· be the second component of the strong solution to (4.14) and � 2 L4p

w (R),where p > 1 is the conjugate of 1+" for " > 0 in Lemma A.1. For a given maturity time T2 2 (0, T ]and 0 < � � 1, the price

u�(x) := E[�(Y �,x,0T2

)]

of the associated perturbed Asian option is continuously di�erentiable in x 2 R, and its derivative,i.e. the Delta, takes the form

u0�(x) = E

"�(Y �,x,0

T2)

Z T

0

a(s)@

@xXx

s dBs + ��1

Z T

0

a(s)

Z s

0

@

@xXx

udu dWs

!#, (4.16)

where a : [0, T ] �! R is a bounded measurable function such thatR T

0a(s)ds = 1.

Proof. The proof is a straight forward generalization of the proof of Theorem 4.2 to the (partic-ularly simple) two-dimensional extension (4.14) of the underlying SDE. Therefore, we here onlygive the main steps.

First observe that the strong solution (Xxt , Y

�,x,yt ) is clearly di�erentiable in y, and by Theo-

rem 3.4 also (weakly) di�erentiable in x, and we get

Dx,y

Xx

t

Y �,x,yt

�=

@@xX

xt 0R t

0@@xX

xudu 1

�,

for all t 2 [0, T ], where Dx,y denotes the derivative.


Assume �rst � 2 C10 (R). Then it follows from Lemma 4.1 that E[�(Y �,x,y

T2)] is continuously

di�erentiable in (x, y) with

Dx,yE[�(Y �,x,yT2

)] = E

"0

�0

(Y �,x,yT2

)

��

Dx,y

Xx

T2

Y �,x,yT2

�#,

where indicates the transposition of a matrix.On the other hand, if we denote by D the Malliavin derivative in the direction of (B,W ), it

follows by means of the estimate in (A.12) and Corollary 3.6 that Y �,x,yT2

is Malliavin di�erentiableand that for 0 � s � T

Ds

Xx

T2

Y �,x,yT2

�1 00 �

��1

Dx,y

Xx

s

Y �,x,ys

�= Dx,y

Xx

T2

Y �,x,yT2

�(4.17)

dx� ds� P�a.e. Then, using (4.17), the chain rule from Lemma 2.1 and the duality relation forthe Malliavin derivative, we see that

Dx,yE[�(Y �,x,yT2

)] = E

"0

�0

(Y �,x,yT2

)

�� Z T

0

a(s)Ds

Xx

T2

Y �,x,yT2

�1 00 �

��1

Dx,y

Xx

s

Y �,x,ys

�ds

#

= E

"�(Y �,x,y

T2)

Z T

0

a(s)

1 00 ��1

�Dx,y

Xx

s

Y �,x,ys

��

d

Bs

Ws

�#.

Thus

@

@xE[�(Y �,x,y

T2)] = E

"�(Y �,x,y

T2)

Z T

0

a(s)@

@xXx

s dBs + ��1

Z T

0

a(s)

Z s

0

@

@xXx

udu dWs

!#for all x, y, � > 0.

For general � 2 L4pw (R) one pursues an approximation argument analogously to the one in the

proof of Theorem 4.2, where we now use the Gaussian distribution ofR T2

0Bx

s ds+ �WT2with mean

xT2 and variance T 32 /3 + �2T2, which explains the weight (4.15) for 0 < � � 1. �

Finally, we address the question whether both (4.11) for T1 ! 0 as well as (4.16) for � ! 0 areindeed approximations for the Delta of the Asian option with averaging period starting in 0. Wehere give an a�rmative answer for a class of pay-o� functions � in spaces of the type

W 1,qew (R) :=

�f 2 W 1,q

loc (R);

ZR

|f(x)|q ew(x)dx+

ZR

|f 0(x)|q ew(x)dx < 1�

for some q > 1, where f 0 denotes the weak derivative of f and the weight function ew is de�ned in(4.15). See [?] for more information on weighted Sobolev spaces.

Theorem 4.6. Let Xx be the strong solution to SDE (1.5) and � 2 W 1,4pw (R), where p > 1 is the

conjugate of 1+ " for " > 0 in Lemma A.1. Further, require that the points of discontinuity of thedistributional derivative �

0

are contained in a Lebesgue null set and that the following conditionsare satis�ed Z

R

ZR

sup�>0

|�(y)� �(y � �z)|2p ew(y)PT (z)dydz < 1 (4.18)

and ZR

ZR

sup�>0

|�0(y)� �0(y � �z)|4p ew(y)PT (z)dydz < 1, (4.19)

where Pt(z) =1p2�t

exp(� 12tz

2), t > 0, z 2 R is the Gaussian kernel. Then

u(x) := E

"�

Z T2

0

Xxs ds

!#


is continuously di�erentiable in x 2 R, and

u0(x) = lim�!0

E

"�

�Y �,x,0T2

Z T

0

a(s)@

@xXx

s dBs + ��1

Z T

0

a(s)

Z s

0

@

@xXx

udu dWs

!#, (4.20)

as well as

u0(x) = limT1!0

E

"�

Z T2

T1

Xxudu

!Z T1

0

a(u)@

@xXx

udBu

#. (4.21)

Proof. By Theorem 4.5 we have that u� 2 C1(R) for all � > 0. Hence,

@

@xE[�(Y �,x,0

T2)] = E

��

0

(Y �,x,0T2

)@

@xY �,x,0T2

�

for all � > 0, dx-a.e. Let J � R be a compact set. Then, using the same line of reasoning just asin the proof of Theorem 4.2, using Cauchy-Schwarz inequality, Girsanov’s theorem, and LemmaA.1 we �nd the estimates

supx2J

�� @@xE[�(Y �,x,0T2

)]�@

@xE[�(Y 0,x,0

T2)]

�� C

E

�ZR

��0

(y)� �0

(y � �WT2)��4p w(y)dy

��1/(4p)

and

supx2J

��E[�(Y �,x,0T2

)]� E[�(Y 0,x,0T2

)]�� K

E

�ZR

|�(y)� �(y � �WT2)|2p w(y)dy

��1/(2p)

for constants C, K depending only on T2, J , p (and not on �).Finally, using dominated convergence in connection with (4.18) and (4.19), the proof follows.

To prove (4.21) de�ne uT1(x) := E[�(R T2

T1Xx

udu)]. Since � 2 L4pew (R), we have by Corollary 4.3

that uT1 2 C1(R) for every T1 > 0. Moreover, since � 2 W 1,4pew (R) we have

u0T1(x) = E

"�

0 Z T2

T1

Xxudu

!Z T2

T1

@

@xXx

udu

#.

Consequently, for every compact J � R we have

supx2J

��@

@xE

"�

Z T2

T1

Xxudu

!#�

@

@xE

"�

Z T2

0

Xxudu

!#�� sup

x2J

��E"

�0 Z T2

T1

Xxudu

!� �

0 Z T2

0

Xxudu

!!Z T2

T1

@

@xXx

udu

#��+ sup

x2J

��E"�

0 Z T2

0

Xxudu

!Z T1

0

@

@xXx

udu

#��=:A1 +A2

where A1 and A2 denote the respective summands. It is clear that A2 goes to 0 uniformly in xon J as T1 ! 0. To show the corresponding convergence for A1, similar computations as in thebeginning of the proof, using Cauchy-Schwarz inequality, Girsanov’s theorem, Lemma A.1, andthat �0 2 L4p

ew (R), give for some constant C" > 0

A1 =C" supx2J

E

24��0 Z T1

0

Bxudu

!��4p351/(4p)

� C"k�0kL4pew (R)

ZR

e� z2

2T1(T21 /3+1) dz

T1!0��! 0 .

Hence (4.21) follows. �

Example 4.7. We conclude this section by verifying the conditions in Theorem 4.6 for a pay-o�function that is used in the next section. Consider the function � : R �! [0,1) given by

�(y) = exp(�y)(C exp(y)�K)+,


where C, K > 0 are constants and (x)+ := max(x, 0) for x 2 R. We immediately see that

� 2 W 1,4ploc (R) \ L4p

w (R) and that

�0

(y) = � exp(�y)(C exp(y)�K)+ + C1[log(K/C),1)(y) dx� a.e.

On the other hand we have that

sup�>0

��0

(y)� �0

(y � �z)��4p � M(

��0

(y)��4p + sup

�>0

��0

(y � �z)��4p

� M((2C +K exp(|y|))4p + (2C +K exp(|y|+ |z|))4p).So condition (4.19) is ful�lled. In the same way one veri�es condition (4.18). Hence � satis�esthe assumptions of the previous theorem.

5. Examples and Simulations

We complete this paper by applying the results from Section 4 to the computation of theDeltas in the regime-switching examples mentioned in the Introduction. More complex examplesof state-dependent drift coe�cients (see e.g. [9]) can be treated following the same principles. Toimplement the methodology, we �rst employ Remark 3.7 and observe that all drift coe�cients from

the regime switching examples in the Introduction can be written in the form b(t, x) = b(x)+ b(x)

as in (1.6) such that identity (2.11) holds for b(x) and identity (2.13) holds for b(x). We thus getthe following rewriting of the �rst variation process (3.5):

@

@xXx

t = exp

�2(Xx

t )� 2(x)� 2

Z t

0

b(Xxs ) dX

xs +

Z t

0

b0(Xxu)du

�, (5.1)

where (·) := b(0)+R ·0b(y) dy is a primitive of b(·). This form is convenient for simulation purposes.

5.1. Black & Scholes model with regime-switching dividend yield. Consider an extendedBlack & Scholes model where the stock pays a dividend yield that switches to a higher level whenthe stock value passes a certain threshold R 2 R+. That is, under the risk-neutral measure thestock price S is given by the SDE

Ss0t = s0 +

Z t

0

b(Ss0u )Ss0

u du+

Z t

0

�Ss0u dBu ,


b(x) := ��11(�1,R)(x)� �21[R,1)(x),

for dividend yields �1,�2 2 R+. We are interested in computing the Delta of a European optionwritten on the stock with given pay-o� function � : R ! R and maturity T :

@

@s0E[�(Ss0

T )] .

In order to �t the computation of the Delta in our framework, we rewrite the stock price with thehelp of Ito’s formula as

Ss0T = e�X

ln(s0)/�

T ,

where Xxt is the solution of the SDE

Xxt = x+

Z t

0

b(Xxu)du+Bt , (5.2)

withb(x) := ��11(�1,R)(x)� �21[R,1)(x)�

�

2,

and �1 := �1

�, �2 := �2

�, R := ln(R)

�. We see that SDE (5.2) is in the required form (1.5) with

b(t, x) = �(�2 � �1)1[R,1)(x) and b(t, x) = ��1 ��

2 . With � := � � exp ��· we thus get by thechain rule

@

@s0E[�(Ss0

T )] =@

@s0E[�(X

ln(s0)/�T )] =

1

s0�· @

@xE[�(Xx

T )] |x= ln(s0)�

.


If � 2 L4pw (R) we know by Theorem 4.2 that the Delta exists, and we can compute @

@xE[�(XxT )]

by (4.8) to obtain

@

@s0E[�(Ss0

T )] = E

"�(Ss0

T )

Z T

0

a(s)

s0�

@

@xX ln(s0)/�

s dBs

#(5.3)

for any bounded measurable function a : R ! R such thatR T

0a(s)ds = 1 , and where @

@xXxs is

given by (5.1) with b0 = 0 and

(x) :=

Z x

0

b(y) dy = �(�2 � �1)(x�R)1[R,1)(x).

We now consider the Delta for a call option, i.e. �(x) := (x �K)+, and for a digital option, i.e.�(x) := 1{x�K}, for some strike price K > 0. It is easily seen that in both cases � 2 L4p

w (R).To compute (5.3) by Monte Carlo, Xx is approximated by an Euler scheme (see [30], Theorem3.1 on the Euler scheme approximation for coe�cients b which are non-Lipschitz due to a set ofdiscontinuity points with Lebesgue measure zero). As in [15] we compare the performance of (5.3)to the approximation of the Delta by a �nite di�erence scheme combined with Monte Carlo:

@

@s0E[�(Ss0

T )] E[�(Ss0+�

T )]� E[�(Ss0��

T )]

2�, (5.4)

for � su�ciently small. We set the parameters T = 1, s0 = 100, �1 = 0.05, �2 = 0.15, R = 108,� = 0.1 and K = 94. Our �ndings are analogue to the ones in [15]: for the continuous call optionpay-o� function the approximation (5.4) seems to be more e�cient (see Figure 1), whereas forthe discontinuous pay-o� function of a digital option, the approximation (5.3) via the Malliavinweight exhibits considerably better convergence (see Figure 2).

Figure 1. Delta of a European Call Option Black & Scholes model with regime-switching dividend yield.


Figure 2. Delta of a European Digital Option under the Black & Scholes modelwith regime-switching dividend yield.

5.2. Electricity spot price model with regime-switching mean-reversion rate. Typically,electricity spot prices exhibit a mean-reverting behaviour with at least two di�erent regimes ofmean-reversion: a spike regime with very strong mean-reversion on exceptionally high price levelsand a base regime with moderate mean-reversion on regular price levels. These features can becaptured by modelling the electricity spot price S (under a risk-neutral pricing measure) by anextended Ornstein-Uhlenbeck process with regime-switching mean-reversion rate:

Ss0t = s0 +

Z t

0

b(Ss0u )du+ �Bt , (5.5)

where the drift coe�cient is given by

b(x) := ��1x1(�1,R)(x)� �2x1[R,1)(x) (5.6)

for mean reversion rates �1,�2 2 R+, a given spike price threshold R 2 R, and � > 0. In orderto guarantee positive prices, one could alternatively model the log-price by (5.5), or one couldintroduce another regime with high mean-reversion as soon as the price falls below zero (we recallthat short periods of negative electricity prices have been observed).

Since electricity is a ow commodity, derivatives on spot electricity are written on the averageprice of the delivery of 1 kWh over a future period [T1, T2], i.e. the underlying is of the typeR T2

T1Ss0t dt for T1 > 0. The most liquidly traded electricity derivatives are futures and forwards. In

that case the pay-o� is linear and the computation of the Delta can be reduced to the computationof the Deltas of European type options:

@

@s0E

"1

T2 � T1

Z T2

T1

Ss0t dt

#=

1

T2 � T1

Z T2

T1

@

@s0E[Ss0

t ] dt .

For derivatives with non-linear pay-o� �, the Delta

@

@s0E

"�

Z T2

T1

Ss0t dt

!#is of Asian type.

Again, in order to �t the computation of the Delta in our framework we rewrite the stock pricewith the help of Ito’s formula as

Ss0t = �X

s0/�t ,

where Xx is the solution of the SDE

Xxt = x+

Z t

0

b(Xxu)du+Bt, (5.7)

with

b(x) := ��11(�1,R)(x) + �21[R,1)(x)

x, (5.8)


Figure 3. Three versions of the functions for a(s) from Remark 4.4.

where R = R/�. We see that the SDE (5.7) is in the required form (1.5) with b(x) =

� (�2 � �1)R1[R,1)(x) and b(x) = b(x) � b(x). As in the previous example, by the chain rulewe get that

@

@s0E

"�

Z T2

T1

Ss0t dt

!#=

@

@s0E

"�

Z T2

T1

Xs0/�t dt

!#=

1

�

@

@xE

"�

Z T2

T1

Xxt dt

!#��x=

s0�

(5.9)

with � := �� ·�. Note that in this example the �rst variation process @@xX

xs is given by (5.1) with

(x) :=

Z x

0

b(y) dy = � (�2 � �1)R(x�R)1[R,1)(x)

and Z t

0

b0(u,Xxu)du = ��1

Z t

0

1(�1,R)(Xxu)du� �2

Z t

0

1[R,1)(Xxu)du.

We compare the performance of the formula for the Asian Delta in Corollary 4.3 with theapproximation presented in Remark 4.4 and with a �nite di�erence approximation analogous to(5.4) when � is a call option pay-o� and a digital option pay-o�, respectively. Obviously, in bothcases the pay-o� in terms of Xx

· ful�ls the assumptions in Theorem 4.2. In the approximationpresented in Remark 4.4 an optimal (in the sense that it minimises the variance of the Malliavinweight) choice for a(s) could improve the convergence rate of the method. In the simulations wecompared the following possible choices for a(s):

a1(s) :=

(1t1

if 0 � s � t1

0 if t1 < s � T2

a2(s) :=

8>><>>:1t1

if 0 � s � t1

k ifj

s�T1

T2�T1· 2m

k� 0 mod 2 and t1 < s � T2

�k ifj

s�T1

T2�T1· 2m

k� 1 mod 2 and t1 < s � T2

a3(s) :=

(1t1

if 0 � s � t1�� s�T1

T2�T1· m

2 � 1�j

s�T1

T2�T1· m

2 �12

k�� k if t1 < s � T2

,

see Figure 3. However, the di�erent choices of function a above did not produce relevant di�erencesin the results. Note, that implementing the approximation from Remark 4.4 with function a1(s)is essentially the same as the implementing the Delta from Corollary 4.3. We thus only comparethe Delta from Corollary 4.3 with a �nite di�erence scheme for parameters: T1 = 0.4, T2 = 1,s0 = 100, �1 = 0.2, �2 = 0.4, R = 101, � = 5 and K = 87. We remark that if T1 approaches zero,


Figure 4. Delta of an Asian Call Option under the Electricity spot price modelwith regime-switching mean-reversion rate.

Figure 5. Delta of an Asian Digital Option under the Electricity spot pricemodel with regime-switching mean-reversion rate.

the variance of the Malliavin weight increases, and thereby the Monte Carlo method becomes lesse�ective. As for the European option in Subsection 5.1, also for these Asian type options the �nitedi�erence method seems to be more e�cient for the continuous call option pay-o�, see Figure 4,whereas for the digital option pay-o�, the approximation through the Malliavin weight providesbetter convergence, see Figure 5.

5.3. Generalised Black & Scholes model with regime-switching short rate. Consider ageneralised Black & Scholes model where under the risk-neutral measure the stock price Ss0· isgiven by

Ss0t = s0 +

Z t

0

rr0u Ss0u du+

Z t

0

�Ss0u dBu , (5.10)

and the stochastic short rate rr0· is given by an extended Va�s��cek model where the mean-reversionlevel switches between a high interest rate regime and a low interest rate regime when the shortrate passes a certain threshold R 2 R:

rr0t = r0 +

Z t

0

b(rr0u )du+B�t , (5.11)

where B�t = eBt +

p1� 2Bt and the drift coe�cient is given by

b(x) := ��(x�m11(�1,R)(x)�m21[R,1)(x)) (5.12)

for a mean-reversion rate � 2 R+ and mean-reversion levels m1,m2 2 R, and where eB is a

Brownian motion independent of B, i.e. we allow for a correlation coe�cient 0 �p1� 2 <

1 with the stock price. Note that we set the volatility coe�cient in (5.11) equal to one fornotational simplicity. We see that the drift of the SDE (5.11) is in the required form (1.5) with


b(x) = �� (m1 �m2)1[R,1)(x) and b(x) = �� (x�m1). Further, we mention that the SDE (5.11)has a Malliavin di�erentiable unique strong solution with respect to the �ltration Ft, 0 � t � T ,

generated by the Brownian motions eB· and B·. Moreover, there exists an �� with probability

mass 1 such that for all ! 2 �� and 0 � t � T : (x 7�! rx(t, !)) 2 \p>0W

1,ploc (R). The proofs of

these properties are essentially the same as in Section 3. For example, Girsanov’s theorem in the

previous proofs is applied to the Brownian motion B�t := eBt +

p1� 2Bt, 0 � t � T .

Now consider the price of a European option with pay-o� function � written on the stock atmaturity T :

Ehe�

R T0

rr0s ds�

�s0e

R T0

rr0s ds+�BT� 12�

2Ti

.

In this example we are interested in computing the generalised Rho

@

@r0Ehe�

R T0

rr0s ds�

�s0e

R T0

rr0s ds+�BT� 12�

2Ti

, (5.13)

that is, the sensitivity of the option with respect to the initial value r0 of the short rate (i.e. asensitivity with respect to movements of the short end of the yield curve). We see that (5.13)has the form of a Delta with respect to an Asian pay-o� in the short rate rr0· which, however,additionally depends on the factor BT .

Although the extension of the results in Section 4 is straight forward to this simple two-dimensional setting, we can still remain in the one-dimensional setting from Section 4 by con-

sidering the Malliavin derivative eDs only with respect to Brownian motion eB· and by applyingrelation (3.6) from Corollary 3.6 in the form

@

@r0rr0t =

1

eDsr

r0t

@

@r0rr0s for all s � t . (5.14)

We here intend to analyse the performance of the approximation (4.20) from Theorem 4.6 foran Asian Delta. Under the corresponding assumptions from Theorem 4.6 for the pay-o� function

�

Z T

0

rr0t dt, BT

!:= exp

(�

Z T

0

rr0t dt

)�

s0 exp

(Z T

0

rr0t dt+ �BT �1

2�2T

)!,

and by following the argument in the proof of Theorem 4.6 we then obtain that the function

u(r0) := E

"�

Z T

0

rr0t dt, BT

!#is continuously di�erentiable in r0 2 R, and that

@

@r0u(r0) = lim

n!1E

"�

Z T

0

rr0s ds+ n�1WT , BT

! Z T

0

a(s)

@

@r0rr0s d eBs + n

Z T

0

a(s)

Z s

0

@

@r0rr0u du

�dWs

!#(5.15)

where a : R ! R is as in Theorem 4.6. Note that in this example the �rst variation process @@r0

rr0sis given by (5.1) with

(x) :=

Z x

0

b(y) dy = �� (m1 �m2) (x�R)1[R,1)(x)

and Z t

0

b0(u,Xxu)du = ��t.

We compare the performance of the approximation of the generalised Rho @@r0

u presented in

(5.15) with a �nite di�erence approximation analogous to (5.4) when � is a call option pay-o�,see Figure 6. The parameters are T = 1, s = 2, � = 0.1, � = 0.3, m1 = 0.5, m2 = 1.2, R = 1.4and K = exp(0.4) and we choose a(s) = 1/T . Note that for a call option pay-o� � we knowfrom Example 4.7 that the assumptions in Theorem 4.6 are ful�lled. Further, we also compute


the Delta of a digital pay-o�, see Figure 7, even though the conditions of Theorem 4.6 are notsatis�ed. Our conjecture is that the result of Theorem 4.6 also holds for discontinuous pay-o�s, and

the simulation reinforces that. As n from Theorem 4.6 increases �

�R T

0rr0(s)ds+ n�1WT , BT

becomes a better approximation of ��R T

0rr0(s)ds,BT

but at the same time the variance of the

Malliavin weight increases, thus, the convergence of the Monte Carlo simulation becomes slower.The experience of several simulations is that n 20 gives the best balance between these twoopposite impacts. However, we can see that in both cases the �nite di�erence method seemsconsiderably more e�cient.

Figure 6. Approximation: Generalised Rho of a European Call Option underthe Generalised Black & Scholes model with regime-switching short rate.

Figure 7. Approximation: Generalised Rho of a European Digital Option underthe Generalised Black & Scholes model with regime-switching short rate.

Appendix A. Proofs of results in Section 3

In this appendix we recollect the proofs of the results in Section 3.

A.1. Some auxiliary results. We start by giving some auxiliary technical lemmata which pro-vide relevant estimates that will be progressively used throughout some proofs in the sequel.

Lemma A.1. Let b : [0, T ]�R ! R be a function of at most linear growth, i.e. |b(t, x)| � C(1+|x|)for some C > 0, all x 2 R and t 2 [0, T ]. Then for any compact subset K � R there exists an" > 0 such that

supx2K

E

24E Z T

0

b(u,Bxu)dBu

!1+"35 < 1 (A.1)

where Bxt := x+Bt.


Proof. Indeed, write

E

24E Z T

0

b(u,Bxu)dBu

!1+"35 =E

"exp

(Z T

0

(1 + ")b(u,Bxu)dBu �

1

2

Z T

0

(1 + ")b2(u,Bxu)du

)#

=E

"exp

(Z T

0

(1 + ")b(u,Bxu)dBu �

1

2

Z T

0

(1 + ")2b2(u,Bxu)du

+1

2

Z T

0

"(1 + ")b2(u,Bxu)du

)#

=E

"exp

(1

2

Z T

0

"(1 + ")b2(u,X",xu )du

)#

where in the last step X",x denotes a weak solution of the SDE(dX",x

t = (1 + ")b(t,X",xt )dt+ dBt, t 2 [0, T ]

X",x0 = x,

which is obtained from Girsanov’s theorem in the same way as in the �rst step of Subsection A.2in equation (A.8). Observe that, since b has at most linear growth, we have

|X",xt | � |x|+ C(1 + ")

Z t

0

(1 + |X",xu |)du+ |Bt|

for every t 2 [0, T ]. Then Gronwall’s inequality gives

|X",xt | � (|x|+ C(1 + ")T + |Bt|) eC(1+")T , (A.2)

and due to the sublinearity of b and the estimate (A.2) we can �nd a constant C",T dependingonly on ", T such that lim"&0 C",T < 1 and

|b(u,X",xu )| � C",T (1 + |x|+ |Bt|) .

As a result,

Ehexp

�"(1+")

Z T

0

b2(u,X",xu )du

�i� E

"exp

("(1 + ")C2

",T

Z T

0

(1 + |x|+ |Bu|)2 du)#

� eC",TT (1+|x|)2E

"exp

(2C",T (1 + |x|)

Z T

0

|Bu|du+ C",T

Z T

0

|Bu|2du)#

where C",T := "(1 + ")C2",T > 0 is a constant such that lim"&0 C",T = 0. Clearly, from the above

expression we can see that for every compact set K � R we can choose " > 0 small enough suchthat

supx2K

Ehexp

�"(1 + ")

Z T

0

b2(u,X",xu )du

�i< 1.

�

Remark A.2. From Lemma A.1 it follows immediately that if the approximating functions bn,n � 1 are as in (3.1) then for any compact subset K � R, one can �nd an " > 0 such that

supx2K

supn�0

E

24E Z T

0

bn(u,Bxu)dBu

!1+"35 < 1, (A.3)

where we recall that b0 := b.


Lemma A.3. Let f : [0, T ]�R ! R be a bounded measurable function. Then for every t 2 [0, T ],� 2 R and compact subset K � R we have

supx2K

E

�exp

��

Z t

0

ZR

f(s, y)LBx

(ds, dy)

��< 1 (A.4)

where LBx

(ds, dy) denotes integration with respect to local-time of the Brownian motion Bxt :=

Bt+x in both time and space, see Section 2 or [12] for more information on local-time integration.

Proof. By virtue of decomposition (2.10) from the Section 2 and Cauchy-Schwarz inequality twicewe have

E

�exp

��

Z t

0

ZR

f(s, y)LBx

(ds, dy)

��E

�exp

��2�

Z t

0

f(s, Bxs )dBs

��1/2

� E

"exp

(4�

Z T

T�t

f(T � s,BxT�s)dWs

)#1/4

� E

"exp

(�4�

Z T

T�t

f(T � s,BxT�s)

BT�s

T � sds

)#1/4=: I · II · III.

where Wt :=R t

0BT�s

T�s ds+BT�t�BT is a Brownian motion with respect to the �ltration generated

by B·. For factor I, Holder’s inequality gives

E

�exp

�� 2�

Z t

0

f(s, Bxs )dBs

��

� E

�EZ t

0

(�4�f(s, Bxs ))dBs

��1/2E

�exp

�Z t

0

(8�2f2(s, Bxs ))ds

��1/2

= E

�exp

�Z t

0

(8�2f2(s, Bxs ))ds

��1/2

� C,

where C > 0 is independent of x since f is bounded. Analogously, we obtain a bound for II.Finally, III follows from

E

"exp

(k

Z T

0

|Bs|s

ds

)#< 1 (A.5)

for any k 2 R, see Lemma A.4 below. �

Lemma A.4. Let B be a one-dimensional Brownian motion on [0, T ]. Then for any integer p � 1and 0 � " < 1/(4p)

E

"��Z T

0

|Bu|1+"

u1+"du

��p#

< 1. (A.6)


Proof. Indeed,

E

"��Z T

0

|Bu|1+"

u1+"du

��p#

�E

" sup

u2[0,T ]

|Bu|"!p ��

Z T

0

|Bu|u1+"

du

��p#

�E

"sup

u2[0,T ]

|Bu|2p"#1/2

E

24��Z T

0

|Bu|u1+"

du

��2p351/2

�CE

24��Z T

0

|Bu|u1+"

du

��2p351/2

for a positive constant C > 0. Now, set d := 2p then we may write

E

24��Z T

0

|Bu|u1+"

du

��2p35 =

Z T

0

d)· · ·Z T

0

E [|Bu1| · · · |Bud

|]u1+"1 · · ·u1+"

d

du1 · · · dud

= d!

Z0<u1<···<ud<T

E [|Bu1| · · · |Bud

|]u1+"1 · · ·u1+"

d

du1 · · · dud (A.7)

where the last equality follows from the fact that the integrand is a symmetric function.Then for a centered random Gaussian vector (Z1, . . . , Zd) with covariances Cov(Zi, Zj) = �i,j ,

i, j = 1, . . . , d we have the following estimate that can be found in [20, Theorem 1]

E[|Z1 · · ·Zd|] �0@X

�2Sd

dYj=1

�j,�(j)

1A1/2

where Sd denotes the set of permutations of (1, . . . , d). Applying the above inequality to theintegral in (A.7)

Z0<u1<···<ud<T

E [|Bu1| · · · |Bud

|]u1+"1 · · ·u1+"

d

du1 · · · dud �X�2Sd

Z0<u1<···<ud<T

dYj=1

uj ^ u�(j)

u1+"j u1+"

�(j)

!1/2

du1 · · · dud.

Given a permutation � 2 Sd we have that, if 0 < u1 < u2 < · · ·ud < T then

dYj=1

uj ^ u�(j)

u1+"j u1+"

�(j)

!1/2

=u�1/21 · · ·u�d/2

d

u1+"1 · · ·u1+"

d

where the �i’s, depend on � and have the property thatPd

i=1 �i = d and �i 2 {0, 1, 2} for alli = 1, . . . , d. Moreover, observe that �1 � 1 independently of � since u1^u�(1) = u1 for all � 2 Sd.So, if we now integrate iteratively we obtainZ

0<u1<···<ud<T

E [|Bu1| · · · |Bud

|]u1+"1 · · ·u1+"

d

du1 · · · dud �X�2Sd

1Qdj=1

�12

Pji=1 �i � j"

T d( 12�")

if, and only if 12

Pji=1 �i � j" > 0 for all j = 1, . . . , d which holds by just observing that

1

2

jXi=1

�i >�1

2� d

1

2d� j

1

2d

for every j = 1, . . . , d where we used �1 � 1. So it su�ces to take " � 0 such that " < 12d . �


A.2. Proof of Theorem 3.1. We now develop the proof of Theorem 3.1 according to the four-step scheme outlined in Section 3. In order to construct a weak solution of (1.5) in the �rst

step, let (�,F , eP ) be some given probability space which carries a Brownian motion eB, and put

Xxt := eBt+x, t 2 [0, T ]. As we already noted in Remark 2.5, it is well-known, see e.g. [16, Corollary

5.16], that for sublinear coe�cients b the Radon-Nikodym derivative dPdP

:= E�R T

0b(u,Xx

u)deBu

de�nes an equivalent probability measure P under which the process

Bt := Xxt � x�

Z t

0

b(s,Xxs )ds, t 2 [0, T ], (A.8)

is a Brownian motion on (�,F , P ). Hence, because of (A.8), the pair (Xx, B) is a weak solutionof (1.5) on (�,F , P ). The stochastic basis that we operate on in the following is now given by the�ltered probability space (�,F , P, {Ft}t2[0,T ]), which carries the weak solution (Xx, B) of (1.5),where {Ft}t2[0,T ] denotes the �ltration generated by Bt, t 2 [0, T ], augmented by the P -null sets.

Next, we prove that for given t 2 [0, T ] the sequence of strong solutions {Xn,xt }n�1 of the SDE’s

(3.2) with regular coe�cients bn from (3.1) converges weakly in L2(�;Ft) to E[Xxt |Ft].

Lemma A.5. Let bn : [0, T ]�R ! R be a sequence of functions approximating b a.e. as in (3.1)and Xn,x

t the corresponding strong solutions to (3.2), n � 1. Then for every t 2 [0, T ] and function' 2 L2p

w (R) where the space L2pw (R) is de�ned as in (4.7) with p being the conjugate exponent of

1 + ", " > 0 from Lemma A.1, we have

'(Xn,xt )

n!1��! E['(Xx

t )|Ft]

weakly in L2(�;Ft).

Proof. First of all, we shall see that '(Xn,xt ), E['(Xx

t )|Ft] 2 L2(�;Ft), n � 0. Indeed, Girsanov’stheorem, Remark A.2 and the fact that ' 2 L2p

w (R) imply that for some constant C" > 0 with" > 0 small enough we have

supn�0

E[|'(Xn,xt )|2] � C"E[|'(x+Bt)|2

1+"" ]

"1+" = C"

1p2�t

ZR

|'(x+ z)|2 1+"" e�

|z|2

2T dz < 1. (A.9)

To show that

E ['(Xn,xt )Z]

n!1��! E[E['(Xx

t )|Ft]Z]

for any Z 2 L2(�;Ft) it su�ces to show

W(Xn,xt )(f)

n!1��! W(E[Xx

t |Ft)](f)

for every f 2 L2([0, T ])Indeed, by Girsanov’s theorem we can write

E

"'(Xn,x

t )� E['(Xxt )|Ft]

�E Z T

0

f(u)dBu

!#=

=E

"'(Bx

t )

E Z T

0

(bn(u,Bxu) + f(u))dBu

!� E

Z T

0

(b(u,Bxu) + f(u))dBu

!!#

=E

"'(Bx

t )E Z T

0


!

�

E Z T

0


!,E Z T

0


!� 1

!#


Then, using inequality |ex � 1| � |x|(ex + 1) we have

E

"'(Xn,x

t )� E['(Xxt )|Ft]

�E Z T

0

f(u)dBu

!#

�E

"|'(Bx

t )| |Un| E Z T

0


!#

+ E

"|'(Bx

t )| |Un| E Z T

0


!#:= In + IIn

where

Un :=

Z T

0


u))dBu �1

2

Z T

0

[(bn(u,Bxu) + f(u))2 � (b(u,Bx

u) + f(u))2]du.

For the term In, Holder’s inequality with exponents p = 1+"" and q = 1 + " and then again for

p = q = 2 yields

In � Eh|'(Bx

t )Un|1+""

i "1+"

E

24E Z T

0


!1+"35

11+"

� Eh|'(Bx

t )|21+""

i "2(1+")

Eh|Un|2

1+""

i "2(1+")

E

24E Z T

0

(bn(u,Bxu) + f(u)) dBu

!1+"35

11+"

=: I1 · I2n · I3n,where I1, I2n and I3n are the respective factors above and " > 0 is such that I3n is bounded uniformlyin n � 0 (see Remark A.2). We can then control the �rst factor I1 due to the fact that ' 2 L2p

w (R)as it is shown in (A.9).

Finally, for the second factor I2n de�ne p := 2 1+"" . Then using Minkowski’s inequality,

Burkholder-Davis-Gundy’s inequality and Holder’s inequality we can write

(I2n)p =E

"��Z T

0


u))dBu �1

2

Z T

0


u) + f(u))2]du

��p#

� 2p�1E

"��Z T

0


u))dBu

��p#

+ 2p�2E

"��Z T

0


u) + f(u))2]du

��p#

. 2p�1E

24 Z T

0

|bn(u,Bxu)� b(u,Bx

u)|2 du!p/2

35+ 2p�2T p�1

Z T

0

Eh��(bn(u,Bx

u) + f(u))2 � (b(u,Bxu) + f(u))2

��2pi du. 2p�1T p/2�1

Z T

0

E [|bn(u,Bxu)� b(u,Bx

u)|p] du

+ 2p�2T p�1

Z T

0

Eh��(bn(u,Bx

u) + f(u))2 � (b(u,Bxu) + f(u))2

��2pi duand by dominated convergence we obtain I2n ! 0 as n ! 1. Similarly, we obtain the result forIIn. �


We now turn to the third step of our scheme to prove Theorem 3.1. The next theorem gives theL2(�;Ft)-convergence of the sequence of strong solutions Xn,x

t to the limit E[Xxt |Ft] which, in

addition, is Malliavin di�erentiable. The technique used in this result is the compactness criteriongiven in Proposition 2.3 due to [8].

Theorem A.6. Let bn : [0, T ]� R ! R, n � 1, be as in (3.1) and Xn,x· the corresponding strong

solutions to (3.2). Then, for each t 2 [0, T ]

Xn,xt

L2(�;Ft)��! E[Xx

t |Ft] (A.10)

as n ! 1. Moreover, the right-hand side of (A.10) is Malliavin di�erentiable.

Proof. The main step is to show relative compactness of {Xn,xt }n�1 by applying Proposition 2.3.

Let t 2 [0, T ], 0 � s � s0 � t and a compact set K � R be given. Using the explicit represen-tation introduced in (3.3), Girsanov’s theorem, the mean-value theorem, Holder’s inequality withexponent 1+" for a su�ciently small " > 0 and Cauchy-Schwarz inequality successively we obtain

E�(DsX

n,xt �Ds0X

n,xt )2

�=

=E

24exp�2 Z t

s0b0n(u,B

xu)du

� exp

(Z s0

s

b0n(u,Bxu)du

)� 1

!2

E Z T

0

bn(u,Bxu)dBu

!35�E

"exp

�2

Z t

s0b0n(u,B

xu)du

� sup

0��1exp

(�

Z s0

s

b0n(u,Bxu)du

)!2

�

Z s0

s

b0n(u,Bxu)du

!2

E Z T

0

bn(u,Bxu)dBu

!#

�E

"exp

�21 + "

"

Z t

s0b0n(u,B

xu)du

�sup

0��1exp

(21 + "

"�

Z s0

s

b0n(u,Bxu)du

)

�

��Z s0

s

b0n(u,Bxu)du

��2 1+"

"# "

1+"

E

24E Z T

0

bn(u,Bxu)dBu

!1+"35

11+"

�E

�exp

�41 + "

"

Z t

s0(b0n(u,B

xu) + b0(u,Bx

u))du

�� "2(1+")

� E

"sup

0��1exp

(81 + "

"�

Z s0

s

(b0n(u,Bxu) + b0(u,Bx

u))du

)# "4(1+")

� E

24��Z s0

s

(b0n(u,Bxu) + b0(u,Bx

u))du

��8 1+"

"

35"

4(1+")

E

24E Z T

0

bn(u,Bxu)dBu

!1+"35

11+"

=: I1n · I2n · I3n · I4n,where I1n, I

2n, I

3n and I4n denote the respective factors shown above.

Here, by Remark A.2, " > 0 is chosen such that

supx2K

supn�0

I4n < 1.

For I1n and I2n we use Cauchy-Schwarz inequality and the fact that b0 is bounded and get

I1n . E

�exp

�41 + "

"

Z t

s0b0n(u,B

xu)du

�� "2(1+")

=: II1n


and

I2n . E

"sup

0��1exp

(81 + "

"�

Z s0

s

b0n(u,Bxu)du

)# "4(1+")

=: II2n.

For I3n, Minkowski’s inequality and the boundedness of b0 give

I3n � E

24��Z s0

s

b0n(u,Bxu)du

��8 1+"

"

+

��Z s0

s

b0(u,Bxu)du

��8 1+"

"

35"

4(1+")

. E

24��Z s0

s

b0n(u,Bxu)du

��8 1+"

"

35"

4(1+")

+ kb0k21T |s0 � s|

� II3n + kb0k21T |s0 � s|.Now we want to get rid of the derivatives b0n in II1n, II

2n and II3n. In order to do so, we use

integration with respect to the local time of the Brownian motion, see Theorem 2.9 in the Section2 or e.g. [12] for more information about local-time integration. We obtain

E�(DsX

n,xt �Ds0X

n,xt )2

�.E

�exp

��4

1 + "

"

Z t

s0

ZR

bn(u, y)LBx

(du, dy)

�� "2(1+")

� E

"sup

0��1exp

(�8

1 + "

"�

Z s0

s

ZR

bn(u, x)LBx

(du, dy)

)# "4(1+")

�

E

24��Z s0

s

ZR

bn(u, x)LBx

(du, dy)

��8 1+"

"

35"

4(1+")

+ kb0k|s0 � s|!.

Observe that factors II1n and II2n can be controlled uniformly in n � 1 and x 2 K by virtueof Lemma A.3. Now, denote p" := 41+"

" . Then for factor II3n we use representation (2.11) fromTheorem 2.9 in connection with (2.10) in Section 2 and apply Minkowski’s inequality, Burkholder-Davis-Gundy’s inequality and Holder’s inequality with exponent ("0 + 2)/"0 for a suitable "0 > 0in order to obtain

II3n �E

24��Z s0

s

bn(u,Bxu)dBu �

Z T�s

T�s0bn(T � u, Bx

u)dWu +

Z T�s


u)Bu

T � udu

��2p"351/p"

.E

" Z s0

s

|bn(u,Bxu)|2du

!p"#1/p"

+ E

" Z T�s

T�s0|bn(T � u, Bx

u)|2du!p"

#1/p"

+ E

24��Z T�s


u)Bu

T � udu

��2p"351/p"

. |s0 � s|"0/("0+2)E

24 Z s0

s

|bn(u,Bxu)|"

0+2du

! 2p""0+2

351/p"

+ |s0 � s|"0/("0+2)E

24 Z T�s

T�s0|bn(T � u, Bx

u)|"0+2du

! 2p""0+2

351/p"

+ |s0 � s|2"0/("0+2)E

24��Z T�s

T�s0

��bn(T � u, Bxu)

Bu

T � u

��("0+2)/2

du

��4p""0+2

351/p"

.


The last expectation is bounded by taking "0 < 28p"�1 and applying Lemma A.4.

Altogether, we can �nd a constant C > 0 such that

supx2K

supn�1

E�(Ds0X

n,xt �DsX

n,xt )2

�� C|s0 � s|"0/("0+2) (A.11)

for 0 � s0 � s � t where 0 < "0/("0 + 2) < 1.Similarly, one also obtains

supx2K

sup0�s�t

supn�1

E�(DsX

n,xt )2

�� C (A.12)

for a constant C > 0.Then (A.9) with ' = id, (A.11), (A.12) together with Proposition 2.3 imply that the set

{Xn,xt }n�1 is relatively compact in L2(�;Ft). Since the sequence of solutions X

n,xt also converges

weakly to E[Xxt |Ft] due to Lemma A.5 with ' = id, by uniqueness of the limit we have that

Xnk,xt

L2(�;Ft)��! E[Xx

t |Ft]

for a subsequence nk, k � 0.In fact, one observes that the L2(�;Ft)-convergence holds for the whole sequence. Indeed,

assume by contradiction, that there exists a subsequence nj , j � 0, such that there is an " > 0with E[|Xnj ,x

t �Xxt |2] > " for all j � 0. Then {bnj

}j�0 is a sequence of approximating coe�cientsas required in (3.1). Thus, by the previous results there exists a subsequence njm , m � 0, suchthat Xnjm ,x ! Xx in L2(�;Ft), which gives rise to a contradiction.

Moreover, since the sequence of Malliavin derivatives {DsXn,xt }n�1 is bounded uniformly in n

in the L2([0, T ] � �)-norm because of (A.12), we also have that the limit E[Xxt |Ft] is Malliavin

di�erentiable, see for instance [28, Lemma 1.2.3]. �

Remark A.7. Note that we have proved the estimates (A.11) and (A.12) uniformly in x 2 K fora compact set K even though this is not needed to apply Proposition 2.3. We will, however, usethis uniform bounds later on in the proofs of Lemma A.8 and Theorem 3.4.

We are now ready to complete the proof of Theorem 3.1 by use of the previous steps.

Proof of Theorem 3.1. It remains to prove that Xxt is Ft-measurable for every t 2 [0, T ] and by

Remark 1.3 it then follows that there exists a strong solution in the usual sense that is Malliavindi�erentiable. Indeed, let ' be a continuous bounded function, then by Theorem A.6 we have, fora subsequence nk, k � 0, that

'(Xnk,xt ) ! '(E[Xx

t |Ft]), P � a.s.

as k ! 1.On the other side, by Lemma A.5 we also have

'(Xn,xt ) ! E ['(Xx

t )|Ft]

weakly in L2(�;Ft). By the uniqueness of the limit we immediately have

' (E[Xxt |Ft]) = E ['(Xx

t )|Ft] , P � a.s.

for all continuous, bounded functions ', which implies thatXxt is Ft-measurable for every t 2 [0, T ].

To show uniqueness, assume that we have two strong solutions Xx and Y x to the SDE (1.5).Then using the Cameron-Martin formula shows that

W(Xxt )(h) = E[Xx

t (h)],

for h 2 L2([0, T ]) where we recall that W(Xxt )(h) denotes the Wiener transform, and the process

Xxt (h), 0 � t � T satis�es the SDE

dXxt (h) = (b(t,Xx

t (h)) + h(t))dt+ d bBt, Xx0 (h) = x (A.13)


for a Brownian motion bBt, 0 � t � T. In the same way, the process Y xt (h), 0 � t � T solves

(A.13). On the other hand, it follows from the linear growth of the drift coe�cient b that Xxt (h)

and Y xt (h), 0 � t � T , are unique in law (see e.g. Proposition 3.10 in [16]). Hence

W(Xxt )(h) = W(Y x

t )(h)

for all t, h. Thus Xx and Y x are indistinguishable. �

A.3. Proof of Proposition 3.2: By equation (3.3) and formula (2.11), we can write for regularcoe�cients bn

DsXn,xt = exp

��

Z t

s

ZR

bn(u, y)LXn,x

(du, dy)

�.

Then, since Xn,xt , n � 0 is relatively compact in L2(�;Ft) and kDsX

n,xt kL2([0,T ]��) is bounded

uniformly in n � 0 due to the proof of Theorem A.6 we know that the sequence DsXn,xt , n � 0

converges weakly to DsXxt in L2([0, T ] � �), see [28, Lemma 1.2.3]. Therefore, it is enough to

check that our candidate is the weak limit. So we must prove that

*W

exp

n�R t

·RRbn(u, y)L

Xn,x

(du, dy)o� exp

n�R t

·RRb(u, y)LXx

(du, dy)o�

(f), g

+L2([0,T ])

! 0

as n ! 1 for every f 2 L2([0, T ]) and g 2 C10 ([0, T ]). It su�ces to show that the Wiener

transform goes to zero.Then, as we did for Lemma A.5, using Girsanov’s theorem we have

��EhE�R T

0f(u)dBu

�exp

n�R t

s

RRbn(u, y)L

Xn,x

(du, dy)o� exp

n�R t

s

RRb(u, y)LXx

(du, dy)oi ��

=

��E�exp

��

Z t

s

ZR

bn(u, y)LBx

(du, dy)

�E Z T

0


!

� exp

��

Z t

s

ZR

b(u, y)LBx

(du, dy)

�E Z T

0


! ��

��E�

exp

��

Z t

s

ZR

bn(u, y)LBx

(du, dy)

�� exp

��

Z t

s

ZR

b(u, y)LBx

(du, dy)

��

� exp

�Z t

s

b0(u,Bxu)du

�E Z T

0


! ��+

��E�

E Z T

0


!� E

Z T

0


!!

� exp

��

Z t

s

ZR

bn(u, y)LBx

(du, dy)

�exp

�Z t

s

b0(u,Bxu)du

��=: In + IIn.

For term In we de�ne p := 1+"" for a suitable " > 0 and then apply Holder’s inequality with

exponent 1 + " on the stochastic exponential. Then we apply Cauchy-Schwarz inequality and

bound the factor with kb0k1, and �nally we use inequality |ex � 1| � |x|(ex + 1). As a result we


obtain

In =

��E�exp

��

Z t

s

ZR

b(u, y)LBx

(du, dy)

�exp

��

Z t

s

ZR

(bn(u, y)� b(u, y))LBx

(du, dy)

�� 1

�

� exp

�Z t

s

b0(u,Bxu)du

�E Z T

0


! ��.E

"exp

��2p

Z t

s

ZR

b(u, y)LBx

(du, dy)

�

�

��exp

��

Z t

s

ZR


(du, dy)

�� 1

�2p ��#1/(2p)

� E

�E Z T

0


!1+" �1/(1+")

.E

" ��Z t

s

ZR


(du, dy)

��2p

exp

��

Z t

s

ZR

bn(u, y)LBx

(du, dy)

�

+ exp

��

Z t

s

ZR

b(u, y)LBx

(du, dy)

��2p#1/(2p)

where in the last inequality we choose " > 0 small enough so that the stochastic exponential isbounded due to Lemma A.1. Then Minkowski’s inequality gives

(In)2p .E

�|Vn|2p exp

��2p

Z t

s

ZR

bn(u, y)LBx

(du, dy)

��

+ E

�|Vn|2p exp

��2p

Z t

s

ZR

b(u, y)LBx

(du, dy)

�� (A.14)

where

Vn :=

Z t

s

ZR


(du, dy).

Then Cauchy-Schwarz inequality and Lemma A.3 give

E

�|Vn|2p exp

�� 2p

Z t

s

ZR

bn(u, y)LBx

(du, dy)

�� (A.15)

� Eh|Vn|4p

i1/2E

�exp

��4p

Z t

s

ZR

bn(u, y)LBx

(du, dy)

��1/2

. Eh|Vn|4p

i1/2.

Finally, using representation (2.10) in the Section 2, Minkowski’s inequality, Burkholder-Davis-Gundy’s inequality in the �rst two terms and Holder’s inequality in the last term we obtain


E�|Vn|p

�= E

"�� Z t

s


u))dBu +

Z T�s

T�t

(bn(T � u, Bxu)� b(T � u, Bx

u))dWu

�

Z T�s

T�t


u))Bu

T � udu

��p#

� E

" ��Z t

s


u))dBu

��p

+ E

" ��Z T�s

T�t


u))dWu

��p #

+ E

" ��Z T�s

T�t


u))Bu

T � udu

��p #

� E

" �Z t

s

|bn(u,Bxu)� b(u,Bx

u)|2du�p/2

+ E

"Z T�s

T�t

|bn(T � u, Bxu)� b(T � u, Bx

u)|2du#p/2 #

+ E

" ��Z T�s

T�t


u))Bu

T � udu

��p #

.

By dominated convergence, all terms converge to zero as n ! 1. In order to justify that the thirdterm also converges to 0 one needs to use the estimate in Lemma A.4. The second term in (A.14)is estimated in the same way. Similarly, one can also bound IIn.

Lemma A.8. Let bn : [0, T ] � R ! R, n � 0 be as in (3.1) and Xn,xt the corresponding strong

solutions with drift coe�cients bn. Then, for any compact subset K � R and p � 1

supn�1

supx2K

supt2[0,T ]

E

�@

@xXn,x

t

�p�� CK,p

for a constant CK,p > 0 depending on K and p. Here, @@xX

n,xt is the �rst variation process of

Xn,xt , n � 1 (see Proposition 2.4).

Proof. The proof of this result relies on the proof of (A.12) in Theorem A.6 by observing that@@xX

n,xt = D0X

n,xt by Proposition 2.4. Then following exactly the same steps as in Theorem A.6

we see that all computations can be done for an arbitrary power p � 1. Finally, from the term II1nin the proof of Theorem A.6 one can see that supn�1 supx2K supt2[0,T ] E

h�@@xX

n,xt

pi< 1. �

A.4. Proof of Proposition 3.3: First, start observing that, for any given p � 1, we have

E [|Xn,xt |p] . |x|p +

Z t

0

Eh|bn(u,Xn,x

u )|pidu+

Z t

0

Eh|b(u,Xn,x

u )|pidu+ E [|Bt|p]

. |x|p + |t|p + C

Z t

0

E [|Xn,xu |p] du

due to the uniform boundedness of bn, the continuity of b and Holder continuity of the Brownianmotion. Then, Gronwall’s inequality gives

supn�1

E [|Xn,xt |p] � C. (A.16)

Now, assume that 0 � s < t � T . Then

Xn,xt �Xn,y

s = x� y +

Z t

0

bn(u,Xn,xu )du�

Z s

0

bn(u,Xn,yu )du+Bt �Bs

= x� y +

Z t

s

bn(u,Xn,xu )du+

Z s

0

(bn(u,Xn,xu )� bn(u,X

n,yu ))du+Bt �Bs.


Now since bn has linear growth together with (A.16), the uniform boundedness of bn and Holdercontinuity of the Brownian motion yield

E�|Xn,x

t �Xn,ys |2� . |x� y|2 + |t� s|+ E

"��Z s

0


n,yu ))du

��2#.

Then we use the fact that Xn,s,·t is a stochastic ow of di�eomorphisms (see e.g. [18]), the mean

value theorem and Lemma A.8 in order to obtain

E

"��Z s

0


n,yu ))du

��2#

= |x� y|2E"��Z s

0

Z 1

0

b0n(u,Xn,x+�(y�x)u )

@

@xXn,x+�(y�x)

u )d�du

��2#

� C|x� y|2Z 1

0

E

"��Z s

0

b0n(u,Xn,x+�(y�x)u )

@

@xXn,x+�(y�x)

u )du

��2#d�

= C|x� y|2Z 1

0

E

"�� @@xXn,x+�(y�x)s � (1� �)

��2#d�

� C|x� y|2 sups2[0,T ]x2K

E

"�� @@xXn,xs

��2#

� C|x� y|2.Altogether

Eh|Xn,x

t �Xn,ys |2

i� C

�|t� s|+ |x� y|2 for a �nite constant C > 0 independent of n.

To conclude, we use Fatou’s lemma applied to a subsequence and the fact that Xn,xt ! Xx

t inL2(�) as n ! 1 due to Theorem A.6.

A.5. Proof of Theorem 3.4. First of all, observe that for any smooth function with compactsupport ' 2 C1

0 (U,R) and t 2 [0, T ], the sequence of random variables

hXnt , 'i :=

ZU

Xn,xt '(x)dx

converges weakly in L2(�) to hXt, 'i by using the Wiener transform following the same steps asin Lemma A.5.

Then for all measurable sets A 2 F , ' 2 C10 (R) and using Cauchy-Schwarz inequality we have

E[1AhXnk,xt �Xx

t , '0i] � k'0kL2(U)|U |1/2

sup

x2supp(U)

E�1A(X

nk,xt �Xx

t )2�!1/2

< 1

where the last quantity is �nite by Proposition 3.3. Then by Theorem A.6 we see that

limk!1

E[1AhXnk,xt �Xx

t , '0i] = 0.

In addition, by virtue of Lemma A.8 we have that

supn�0

EkXn,xt k2W 1,2(U) < 1,

that is x 7! Xn,xt is bounded in L2(�,W 1,2(U)). As a result, the sequence Xn,x

t is weakly relativelycompact in L2(�,W 1,2(U)), see e.g. [19, Theorem 10.44], and therefore there exists a subsequencenk, k � 0 such that Xnk,x

t converges weakly to some element Yt 2 L2(�,W 1,2(U)) as k ! 1. Letus denote by Y 0

t the weak derivative of Yt.Then

E[1AhXxt , '

0i] = limk!1

E[1AhXnk,xt , '0i] = � lim

k!1E[1Ah @

@xXnk,x

t , 'i] = �E[1AhY 0t , 'i].


So

hXt, '0i = �hY 0

t , 'i, P � a.s. (A.17)

Finally, we need to show that there exists a measurable set �0 � � with full measure such thatX ·

t has a weak derivative on this subset. To this end choose a sequence {'n} in C1(R) dense inW 1,2(U). Choose a measurable subset �n of � with full measure such that (A.17) holds on �n

with ' replaced by 'n. Then �0 := \n�1�n satis�es the desired property.

Corollary A.9. Let b : [0, T ] � R ! R as in (1.6) and Xxt the corresponding strong solution of

(1.5). Then, for any compact subset K � R and p � 1

supx2K

supt2[0,T ]

E

�@

@xXx

t

�p�� CK,p

for a constant CK,p > 0 depending on K and p. Here, @@xX

xt is the �rst variation process of Xx

t ,(see Proposition 3.5).

Proof. This is a direct consequence of Lemma A.8 in connection with Fatou’s lemma. �

A.6. Proof of Proposition 3.5: By Theorem 3.4 we know that the sequence {Xn,xt }n�0 con-

verges weakly to Xxt in L2(�,W 1,2(U)). Therefore, it is enough to check that our candidate is the

limit of @@xX

n,xt in the weak topology of L2(U � �) for any open bounded U � R, i.e.Z

U

Wexp

��

Z t

0

ZR

bn(u, y)LXn,x

(du, dy)

�� exp

��

Z t

0

ZR

b(u, y)LXx

(du, dy)

��(f)g(x)dx

converges to 0 as n ! 1 for every f 2 L2([0, T ]) and g 2 C10 (U). This can be shown following

exactly the same steps as in Proposition 3.2 by integrating In and IIn against g(x) over x 2 U .The only di�erence here is that we need all bounds to be uniformly in x 2 U . At the end, oneneeds to show that

supn�0

supx2supp(U)

Eh|Vn|2pe�2p

R t0

RRbn(u,y)L

Bx(du,dy)

i< 1

where

Vn :=

Z t

0

ZR


(du, dy)

which holds by Lemma A.3 and the fact that bn, n � 0, is uniformly bounded. For IIn one canfollow similar steps and use Remark A.2.

Acknowledgement: We would like to thank Aaron Folly for support with the simulations inSection 5.

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D. R. Banos: Department of Mathematics, University of Oslo, Moltke Moes vei 35, P.O. Box 1053

Blindern, 0316 Oslo, Norway.


S. Duedahl: Department of Mathematics, University of Oslo, Moltke Moes vei 35, P.O. Box 1053

Blindern, 0316 Oslo, Norway.


T. Meyer-Brandis: Department of Mathematics, LMU, Theresienstr. 39, D-80333 Munich, Germany


F. Proske: CMA, Department of Mathematics, University of Oslo, Moltke Moes vei 35, P.O. Box

1053 Blindern, 0316 Oslo, Norway.


III

IMPLEMENTATION OF STOCHASTIC YIELD CURVE DURATION AND

PORTFOLIO IMMUNIZATION STRATEGIES

SINDRE DUEDAHL

June 19, 2015

Abstract. In this paper we propose an implementation method for a new concept of sto-chastic duration which can be used to measure the sensitivity of complex bond portfolioswith respect to the �uctuations of the yield surface. Our approach relies on a �rst orderapproximation of a chaos expansion in the direction of the yield surface, whose dynamicsis described by the Musiela equation. Using the latter technique, we obtain an in�nite-dimensional generalization of the classical Macaulay duration, which can be interpreted asthe derivative of a �rst order approximation of a Taylor series on locally convex spaces.

1. Introduction

Asset and liability management (ALM) is the �nancial risk management of insurance com-panies, banks and any �nancial institution. The latter comprises risk assessment in all di-rections, e.g. policy setting, structuring of the bank’s or insurance’s repricing and maturityschedules, selecting �nancial hedge positions, capital budgeting, and internal measurementsof pro�tability. Further it pertains to contingency planning in the sense that the �nancialinstitution has to analyze the impact of unexpected changes (e.g. interest rates, competitiveconditions, economic growth or liquidity) and how it will react to those changes.

Portfolios managed e.g. by pension funds are usually of high complexity and stochasticallydepend on the entire term structure of interest rates R(t, x) or yield surface, dynamicallyin time. Therefore an accurate risk management of interest rates necessitates the study ofstochastic models for interest rates R(t, x) in time t and space x (”time-to-maturity”), thatis the avarage rate at (future) time t with respect to the time period [t, t+ x], to analyze theinterest rate risk and sensitivity of bond portfolios.

One way to model the stochastic �uctuations of the yield surface (t, x) 7! R(t, x) is basedon the so-called Musiela equation, which is a special type of a stochastic partial di�erentialequation (SPDE). In this model (see e.g. [CT]) it is assumed that

R(t, x) =1

x

Z x

0ft(s)ds,

where the forward (interest rate) curves ft, 0 � t � T satisfy the Musiela equation, that isft, 0 � t � T is the mild solution to the SPDE

dft =d

dxft + �(t, ft)dt+ �(t, ft)dWt, f0 = f, (1.1)

where � : [0, T ]�H �! H, � : [0, T ]�H �! L2(H,H) are Borel measurable functions andWt is a cylindrical Wiener process in H on a �ltered probability space�

�,F , {Ft}0�t�T , μ�. (1.2)

1

2 S. DUEDAHL

Here the �ltration {Ft}0�t�T is μ�completed and generated byW . Further, L2(H,H) denotesthe space of Hilbert-Schmidt operators from H into itself.

A crucial aspect of asset liability management is the measurement of the sensitivity andrisk analysis of bond portfolios with respect to the stochastic �uctuation of the yield surface.A widely spread method in banks and insurances to measure changes of bond portfolio valueswith respect to the stochastic �uctuation of the yield surface is the concept of modi�edduration which was introduced by Macaulay in 1938 ([Mac]). The de�nition of this concepthowever is based on the �rst order Taylor expansion approximation of bond values and requiresthe unrealistic assumption of parallel shifts of (piecewise) �at interest rates dynamically intime. The latter approach, but also other techniques based on fair prices of interest ratederivatives (see e.g. [Chen]), are therefore not suitable for complex hedging portfolios ofbonds, since the portfolio weights with respect to the hedged positions usually depend onthe whole term structure of interest rates and hence are time-dependent functionals of the(stochastic) yield surface. In order to overcome this problem, one could use the concept ofstochastic duration in [KPR] to measure the yield surface sensitivity of bond portfolios. Herethe stochastic duration, which can be considered a generalization of the classical duration ofMacaulay, is de�ned as a Malliavin derivative in the direction of the (centered) forward curveft, 0 � t � T in the Musiela equation (1.1) under a certain change of measure and conditionson the �ltration {Ft}0�t�T .

Since the concept of stochastic duration, which enables a more accurate interest rate man-agement and which could be e.g. used to devise new premium calculation principles forlife insurance policies with ”stochastic” technical interest rates, it is necessary to developnumerical methods or approximation schemes for its estimation.

In this paper we aim at proposing a numerical approach to estimate the stochastic durationin [KPR] in the more general setting of mild solutions to (1.1) by using a �rst order chaosexpansion approximation of bond portfolio values as functionals of the forward curve ft, 0 �

t � T. This idea is in line with the classical Macaulay de�nition of duration and correspondsto a �rst order Taylor series approximation on locally convex spaces in in�nite dimensions(see e.g. [HKPS]). This approximation may be also compared to the approach of Jamshidian([Jam]) with respect to the stochastic modeling of large multi-currency portfolios by meansof a Gaussian distribution as an application of the central limit theorem. In this context it isworth mentioning that the second order chaos expansion approximation of the bond portfoliovalue, which gives a more realistic portfolio modeling and which we don’t consider in thispaper, actually corresponds to the application of a non-central limit theorem (see [APP].)

Furthermore, using the above techniques we want to generalize the concept of immunizationstrategies for bond portfolios as introduced in [HW] to the case of non-�at stochastic interestrates.

The paper is organized as follows:In Section 2 we pass in review some basic facts from in�nite dimensional interest rate

modeling and Malliavin calculus for Gaussian �elds. Moreover, adopting the ideas in [KPR]we introduce the concept of stochastic duration in the setting of mild solutions to (1.1).

Finally, in Section 3 we want to discuss an implemention method for the estimation ofstochastic duration and the concept of portfolio immunization strategies.

3

2. Framework

We recall in this section some mathematical preliminaries.Consider the SPDE

dft = (Aft + �(t, ft))dt+ �(t, ft)dWt, (2.1)

where A is the generator of a strongly continuous semigroup St, t � 0 on H, � : [0, T ] �H ! H,� : [0, T ] � H ! L2(H,H) are Borel measurable functions, Wt, 0 � t � T is acylindrical Wiener process in H on a �ltered probability space (�,F , {Ft}0�t�T , μ). We needthe following concept of solution to (2.1):

De�nition 2.1 (Mild solutions). An Ft-adapted process ft on��,F , {Ft}0�t�T , μ

�is said

to be a mild solution to (2.1) if (see [GM]):

(1) μ(R T

0 kftkHdt < 1) = 1,

(2) μ�R T

0 (k�(t, ft)kH + k�(t, ft)k2L2(H,H))dt < 1

�= 1, and

(3) for all t � T , μ�a.s.,

ft = Stf0 +

Z t

0St�s�(s, fs)ds+

Z t

0St�s�(s, fs)dWs. (2.2)

Remark 2.2. If the coe�cients � and � in (2.1) satisfy the Lipschitz condition

k�(t, x)� �(t, y)kH + k�(t, x) + �(t, y)kL2(H,H) � Kkx� ykH , x, y 2 H

for a constant K < 1, then there exists a unique mild solution ft, 0 � t � T to (2.1).Moreover, for all p > 2 we have that

E

"sup

t2[0,T ]kXtk

pH

#< Cp

�1 + kX0k

pH

�

for a constant Cp < 1.

In the sequel, we choose H to be the following weighted Sobolev space Hw (see [CT]):

De�nition 2.3. Let w : [0,1) ! (0,1) be an increasing function such thatZ1

0

x2

w(x)dx < 1.

Then the space Hw de�ned as

Hw =

(f : [0,1) ! |fabsolutely continuous and

Z1

0

�� ddxf(x)��2

w(x)dx < 1

)is a Hilbert space with the inner product

hf, giHw= f(0)g(0) +

Z1

0

d

dxf(x)

d

dxg(x)dx.

The space H = Hw exhibits the following important properties which we want to usethroughout the paper:

4 S. DUEDAHL

(1) The evaluation functional

�x : H ! R; f 7! f(x)

is a continuous linear functional.(2) The integration functional

Ix : H ! ; f 7!

Z x

0f(s)ds

is a continuous linear functional.(3) The di�erential operator A = d

dxis the generator of the strongly continuous semigroup

given by the left shift operator St : H ! H de�ned by

St(f)(x) = f(t+ x)

for f 2 H,x � 0.

In what follows, we assume that A = ddx.

In order to rule out arbitrage opportunities, we shall also require that the drift coe�cient �in (2.1) satis�es the following generalized Heath-Jarrow-Morton (HJM) no-arbitrage condition(see [CT]):

�(t, g) =Xl�1

�(l)(t, g) ·

Z·

0�(l)(t, g)(u)du+

Xl�1

�(l)(t, g) · �(l)(t, g), 0 � t � T, g 2 Hw

in L2(�;Hw), where �(l) : � � [0, T ] � Hw ! , l � 1 is a sequence of predictable (risk

premium) processes and where �(l)(t, g) = �(t, g)(el) for an orthonormal basis (el)l�1 of Hw.Assuming that �(t, x) is always invertible, we may rewrite (2.2) as

ft = Stf0 +

Z t

0St�s�(s, fs)dcWs, (2.3)

where

cWt := Wt +

Z t

0�(s, fs)

�1�(s, fs)ds.

By the in�nite-dimensional Girsanov theorem, which can be applied if e.g. the Novikov con-

dition E[exp(12R T

0 k�(s, fs)�1�(s, fs)k

2Hw

ds)] < 1 holds, there exists a measure �, equivalent

to μ, under which cWt is a cylindrical Wiener process.In the following, we shall also require that �(s, f) = �(s) for all f 2 Hw, 0 � s � T. Thus,

in this case the centered forward curve ft, 0 � t � T given by

ft = ft � Stf0

becomes a centered Gaussian random �eld in time t and time-to-maturity X· under r.We shall also assume the following condition. There exists a unique strong solution Xt, 0 �

t � T to the SDE

dXt = �(t,Xt)dt+ dWt, X0 = x 2 Hw, 0 � t � T.

The latter condition in connection with the properties of the left shift operator St and thedi�usion coe�cient � actually ascertains that the �ltrations generated by W· and f· coincide.

5

Using the above conditions, we can now introduce the concept of stochastic duration as aMalliavin derivative with respect to the centered forward curve ft.

2.1. Malliavin calculus for Gaussian �elds. We now de�ne the Skorohod integral andMalliavin derivative with respect to the Gaussian processXt, according to [Man]. Let {Xt, 0 �

t � T} be a centered Gaussian process on (�,F , μ), let C(t, s) be the covariance function ofX·, and let K(C) be the reproducing kernel Hilbert space (RKHS) of C. Moreover, let H(X)be the closed linear subspace of L2(�,F , P ) spanned by {Xt, 0 � t � T}. If h 2 K(C), thereis a unique element Yh 2 H(X) such that

h(t) = E[XtYh].

De�nition 2.4 (First-order stochastic integral). I1(h) = Yh.

I1 is an isometry of K(C) into L2(�,F , μ), and is called the stochastic integral of orderone. In order to de�ne higher-order integrals, let {e�|� 2 J} be an orthonormal basis inK(C). Because of isometry it is su�cient to de�ne Ip+1 for functions of the form

h(t1, . . . , tp, tp+1) = e�1(t1) . . . e�p+1(tp+1).

De�nition 2.5 (Higher-order stochastic integral). Let u1, . . . , un be the n � p + 1 distinctelements of {e�1 . . . , e�p+1}. For 1 � i � n, let pi be the number of times the element ui wasrepeated in the sequence, and de�ne

Ip+1(h) = �ni=1Hpi(I1(ui)),

where Hp is the pth Hermite polynomial.

For every integer p � 0, let K�p be the symmetric tensor product of p copies of K.

Lemma 2.6. L2(�,F , μ) =L

1

p=0 Ip(K�p).

Proof. This is Lemma 2.4 in [Man]. �

Theorem 2.7 (Chaos decomposition). It follows that every random variable V in this L2-space may be expressed as an in�nite sum

V =1Xp=0

Ip(gp),

where gp 2 K�p. This representation is known as the chaos decomposition of V with respectto f .

Now let V be a process in L2(�, K). For every p, let gp(·, ·) now be a function in K�(p+1),such that for every t, gtp(·) := gp(·, t) 2 K�p, and such that for all t (gp is symmetric in the�rst p variables),

Vt =1Xp=0

Ip(gtp).

De�nition 2.8 (Skorohod integral). IfP

1

p=0 Ip+1(gp) =P

1

p=0 Ip+1( bgp) converges in L2, thissum is de�ned as the Skorohod integral of V with respect to the Gaussian process f and isdenoted by If (V ).

6 S. DUEDAHL

Lemma 2.9. If (V ) 2 L2(�) if and only if

1Xp=1

(p+ 1)!k bfpk2K�(p+1) < 1,

and in this case

1Xp=1

(p+ 1)!k bfpk2K�(p+1) = kIf (V )k2L2(�).

Proof. This is Lemma 3.3 of [Man]. �

De�nition 2.10 (Malliavin derivative). For an element G =P

1

p=0 Ip(gp) of L2(�), if

1Xp=0

p · p!kgpk2K�p < 1,

the process D·G given by DtG =P

1

p=0 pIp�1(gp(·, t)) is in L2(�, K) and we have (see

[Man]):

E

��1Xp=0

pIp�1(gp(·, t))

��2

K

=1Xp=0

p · p!kgpk2K�p .

In this case we say that G is Malliavin di�erentiable, and we call DtG the Malliavin de-rivative of G, with respect to the Gaussian process Xt, 0 � t � T .

De�nition 2.11 (Stochastic duration). Let G be a square integrable functional of the centered

forward curve f with respect to the risk-neutral measure �. Assume that G is Malliavindi�erentiable with respect to f . Then the stochastic duration of G is the random �eld D

G

given by

DG·= D·G 2 L2(�,FT , �,K).

Remark 2.12. The Malliavin derivative D can indeed be regarded as a sensitivity measurewith respect to the stochastic �uctuations of the (centered) forward curve. The latter, however,is a consequence of the relationship between the Malliavin derivative and stochastic GateauxK-derivative (see [Man]): If G = G(f·) 2 L2(�) and if

G(f· + � · �)�G(f·)

�(2.4)

converges in L2(�) as � & 0 for � 2 K, then DG exists and the limit in (2.4) coincideswith (D·G,�)K . The probability measures μ and � are equivalent. Therefore we may interpretDG for a portfolio value G at time T as a sensitivity measure with respect to the stochasticnon-linear shifts of the (centered) yield surface.

We may also be interested to derive an estimate of the instantaneous movement of theportfolio value as a ”directional derivative” given by the scalar product

hDG, iK .

7

By substituting di�erent curves for we may get an overview of the e�ects on the portfolioof the various possible outcomes of the short-term movements of interest rates at di�erentparts of the maturity spectrum. This method exhibits a radically increased degree of �exibilityas compared to the classical method of Hull and White, where one was restricted to the studyof �at or piecewise-�at interest rates, and the dependence on time-to-maturity was not takeninto account. In the next Section, we will describe a method of estimating the stochasticduration from market data, and then extend to our setting the method of Hull and White([HW]) of constructing immunization strategies, which facilitate the reduction of interest-raterelated risk by dynamically rebalancing the portfolio with instruments which counteract theinterest-rate sensitivity measured by duration.

3. Computation of Stochastic Duration and Immunization Strategies

3.1. Implementation scheme for the stochastic duration. Consider now a square in-tegrable adapted (portfolio) process G� , 0 � � T . Then it follows from Theorem 2.7 that

G� =X��0

Ik(f�

�),� � 0. (3.1)

In the next step, we aim at approximating the chaos decomposition in (3.1) by the �rsthomogeneous chaos I1(g

�

1 ), that is we assume that

G� = I0(f�

0 ) + I1(f�

1 ),

where I0(g�

0 ) is a real number. On the other hand, it follows from the de�nition of stochastic

integrals with respect to f· and the properties of the left shift operator that

I1(f�

1 ) =

Z�

0g�s � S��s � �(s)dWs

for continuous linear functionals g�s , 0 � s � on Hw with

f �

1 (t, x) = E� [I1(f�

1 ) · f(t, x)]

for all t, x. Hence, using Girsanov’s theorem, we get that

G� = I0(f�

0 ) +

Z�

0g�s � S��s(�s)ds+

Z�

0g�s � S��s � �(s)dWs

under the original probability measure μ. Denote by ek, k � 1 an orthonormal basis of Hw.Then, we �nally approximateZ

�

0g�s � S��sdWs =

Xk�1

Z�

0g�s � S��s � �(s)dW

ks

by Z�

0g�s � S��s � �(s)dW

1s ,

where W k·denotes the k-the component of W . So our numerical estimation scheme will rely

on the stochastic process

Z� = I0(f�

0 ) +

Z�

0g�s � S��s(�s)ds+

Z�

0g�s � S��s � �(s)dBs,

8 S. DUEDAHL

where Bt, 0 � t � T is a one-dimensional Wiener process.On the other hand, by using the HJM-condition, we may similarly approximate the drift

coe�cient �(t, g) by

�(t)(e1) ·

Z·

0�(t)(e1)(u)du+ �(t)(e1) · �

(1)(t, g)

for 0 � t � T, g 2 Hw.In the following, let us assume that �(1)(·) = �(·)(e1) is the volatility function of the

one-dimensional Vasicek model for short rates, that is

�(s)(e1)(x) = � · e�a·x,

where a � 0 is the mean reversion and � > 0 the volatility.Applying the Malliavin operator to the approximating process Z() yields a �rst-order

approximation of the duration

DG� DZ� = g�

·.

The task is then to estimate the functional g� . We take as input the observed portfoliovalues �1, . . . , �n at a series of time points 1, . . . , n, which correspond to Z(1), . . . , Z(n) inour model.

To allow numerical implementation, we shall assume that 7! I0(g�

0 ) is absolutely contin-uous. Further, we shall introduce a discretized version of the functional g� :

g�s (·) =mXi=1

ai()bi(s, ·) (3.2)

where ai(·) is absolutely continuous and bi(·, ·) is given by

bi(s, ·) =

KXj=1

�ij(s)�xj(·)

for bounded and measurable functions �ij . Recall that �x : Hw ! is the evaluationfunctional for x > 0.

Furthermore, we approximate �ij and the weak derivative of ai by step functions:

�ij(s) =MXl=2

�ijl (tl�1,tl](s),

ai(t) =

Z t

0hi(s)ds,

hi(s) =

MXl=2

il (tl�1,tl](s).

Hence, using our assumptions, we see that

9

g�s � S��s � �(s)(e1) =mXi=1

� · ai()e�a�

KXj=1

�ij(s)e�axj

=

mXi=1

ai()�i(s), (3.3)

where

ai() := � · ai()e�a�

and

�i(s) :=KXj=1

�ij(s)ea(s�xj).

We now need to derive some quantity from the model process Z which takes scalar valuesand may be compared to observable market data. A natural candidate is the quadraticvariation

[Z,Z]� , 0 � � T.

By applying integration by parts in connection with (3.3) toZ�

0g�s � S��s � �(s)(e1)dBs =

mXi=1

ai(s)

Z�

0�i(s)dBs,

we get that

Z� = A� +

Z�

0b(s)dBs,

where

b(s) :=mXi=1

ai(s)�i(s)

and where A� , 0 � � T is a continuous adapted bounded variation process. So it followsthat

[Z,Z]� =

Z�

0b2(s)ds. (3.4)

The observation Y� from market data, which corresponds to [Z,Z]� , is approximately

Y� =nX

l=1

(�l � �l�1)2

for 1 < 2 < · · · < n � .However, in practice observations of Y� , 0 � � T are noisy, i.e. we have

10 S. DUEDAHL

Y� = [Z,Z]� + BY�

=

Z�

0h(s, (�ijl), ( il))ds+ BY

�, 0 � � T, (3.5)

where BY�, 0 � � T is a one-dimensional Wiener process independent of B·, 0 � � T ,

2 and

h(s, (�ijl), ( il)) := b2(s). (3.6)

In order to estimate the parameters �ijl and il from market data Y� , 0 � � T , weemploy nonlinear �ltering theory. See e.g: [Xiong] and the references contained therein formore information on nonlinear �ltering theory.

In applying nonlinear �ltering techniques, we assume that the observation process is givenby (3.5) and the observation function by (3.6). Set = 1 for convenience.

Further, suppose that the signal process X� , 0 � � T has components satis�es the SDE

dX� =

(dX

ijl� = 0 i = 1, . . . ,m, j = 1, . . . ,K, l = 2, . . . ,M

dX il�= 0 i = 1, . . . ,m, l = 2, . . . ,M.

where X0 is independent of B·, BY·.

We may here for convenience assume that X0 is a vector of i.i.d variables which are e.g.uniformly or normally distributed. In what follows we want to determine the optimal �lter

E[f(X� )|FY�], 0 � � T,

where {FYt }0�t�T is the �ltration generated by the observation process Y·, and where f :

d ! , d = m · (K + 1) · (M � 1) is a Borel measurable function.It follows from the Kallianpur-Striebel formula (see e.g. [Xiong]) that

E[f(X� )|FY�] =

EQ[M�f(X� )|FY�]

EQ[M� |FY�]

, (3.7)

where

M� = exp

�Z�

0h(s,Xs)dYs �

1

2

Z�

0h2(s,Xs)ds

, 0 � � T,

and where Y· is a Wiener process independent of X· under a Girsanov transform Q.Since Y· is independent of X· under Q we get the representation

EQ[M�f(X� )|FY�] = EPX [M�f(X� )] a.e., (3.8)

where PX denotes a probability measure with respect to X· on a separate sample space.The latter however enables us to use Monte Carlo techniques, i.e. the strong law of large

numbers to approximate (3.8) by

1

R

RXr=1

M r�f(Xr

�) (3.9)

for ”large” R, where Xr·, r = 1, . . . , R are i.i.d. copies of X· and where

11

M r�:= exp

�Z�

0h(s,Xr

s )dYs �1

2

Z�

0h2(s,Xr

s )ds

, 0 � � T.

By choosing projections for f in (3.9) in connection with (3.7) we �nally obtain �lter estimatesfor the parameters �ijl, il.

We implemented the method in MATLAB and as an illustration we reproduce in Figure 1a plot of the resulting duration surface from a simulation example with �ctional market dataand � = 0.01, a = 0.4.

Figure 1. A plot of the duration as a function of , s for � = 0.01, a = 0.4

3.2. Delta hedge. Using our implementation scheme for the stochastic duration, we �nallywant to discuss portfolio immunzation strategies against interest rate risk based on the so-called delta-hedge, which was studied in ([HW]) in the case of piecewise �at interest rates.Our aim is to generalize the concept of a delta hedge for piecewise �at interest rates to thecase of stochastic yield surfaces based on the above implementation scheme. To this end,consider a bond portfolio with value G� at time point . We now want to hedge againstthe �uctuations of the yield surface by constructing a delta hedge by means of interest ratederivatives (e.g. swaps, caps, bond options, ...) with values H�

i , i = 1, . . . ,m. The delta hedgecorresponds to the adapted stochastic process b� = (b�1 , . . . , b

�

m), 0 � � T such that

12 S. DUEDAHL

Dt,xG = Dt,x

mXk=1

b�kH�

k

!for all .For convenience, let us now assume that b� , 0 � � T is a deterministic process. Then we

see that

Dt,xG =mXk=1

b�kDt,xH�

k . (3.10)

Since in general there is no strategy b� , 0 � � T satisfying (3.10), one may resort to thefollowing minimization problem:

minb�1 ,...,b

�

m

E

"��DG� �

mXk=1

b�kDH�

k

��K

#Now, using our implementation scheme, we can regard DG� , DH�

k , k = 1, . . . ,m as deter-ministic functions and obtain the following optimization problem

minb�1 ,...,b

�

m

��DG� �

mXk=1

b�kDH�

k

��K

Here one may choose the optimization constraint given by

�� Dt,xG� �

mXk=1

b�kDt,xH�

k � �

for all t, x and some � > 0.

Appendix A. MATLAB code

clear

global M R K T m t xj sigma a

T = 2;

t = linspace(0.1, T, M);

xj = linspace(0, 10, K);

sigma = 0.01;

a = 0.4;

R = 500;

m = 3;

K = 3;

M = 4;

t = linspace(0.1, T, M);

xj = linspace(0, 10, K);

X_r{1} = rand(m, K, M-1, R);

13

X_r{2} = rand(m, M-1, R);

[s, tau] = meshgrid(0.1:0.05:T);

z = dur(X_r, s, tau);

surf(s, tau, z)

% -------------------------------------------------------------------------

function z = dur(X_r, s, tau)


N1 = size(tau, 1);

N2 = size(tau, 2);

assert(N1 == size(s, 1))

assert(N2 == size(s, 2))

z = zeros(N1, N2);

for(j = 1:N1)

for(k = 1:N2)

X = dur_param(X_r, tau(j,k));

fprintf(’computing, %3.2f%%\n’, 100 * (((j-1)*N2+k) / (N1 * N2)))

bet_tau = X{1};

gam_tau = X{2};

for(ii = 1:m)

z(j,k) = z(j,k) + aitildefunc(gam_tau, ii, tau(j,k))*betaitildefunc(bet_tau, ii,

end

end

end

end

% -------------------------------------------------------------------------

14 S. DUEDAHL

function X = dur_param(X_r, tau)

global M R K T m t xj sigma a taui

assert(size(X_r{2},1) == m)

assert(size(X_r{2},2) == M-1)

assert(size(X_r{2},3) == R)

X = cell(1,2);

X{1} = zeros(m,K,M-1);

X{2} = zeros(m,M-1);

denom = 0;

Xr = cell(1,2);

for r = 1:R

Xr{1} = X_r{1}(:,:,:,r);

Xr{2} = X_r{2}(:,:,r);

X{1} = X{1} + mtaufunc(Xr, tau).*X_r{1}(:,:,:,r);

X{2} = X{2} + mtaufunc(Xr, tau).*X_r{2}(:,:,r);

denom = denom + mtaufunc(Xr, tau);

end

X{1} = X{1}./denom;

X{2} = X{2}./denom;

end

% -------------------------------------------------------------------------

function mtau = mtaufunc(X, tau)


b = @(s)bfunc(X,s);

N = max(10, ceil(10*tau));

D = 100000;

logmtau = 0;

for i = 1:N

logmtau = logmtau + D*(tau/N)*((b(i*tau/N))^2-0.5*(b(i*tau/N))^4);

15

end

mtau = exp(logmtau);

end

% -------------------------------------------------------------------------

function b = bfunc(X,s)


b= 0;

for ii = 1:m

b = b + aitildefunc(X{2}, ii, s).*betaitildefunc(X{1}, ii, s);

end

end

% -------------------------------------------------------------------------

function betaitilde = betaitildefunc(bet, ii, s)


betaitilde = 0;

for j = 1:K

for l = 2:M

betaitilde = betaitilde + bet(ii,j,l-1) .* (t(l-1)<s) .* (t(l)>=s) .* exp(a*(s-xj(j))

end

end

end

16 S. DUEDAHL

% -------------------------------------------------------------------------

function aitilde = aitildefunc(gam, ii, s)


ai = 0;

for l = 2:M

ai = ai + gam(ii,l-1) .* (min(t(l),s)-min(t(l-1),s));

end

aitilde = sigma .* ai .* exp(-a.*s);

end

References

[APP] Azmoodeh, E. Peccati, G. Convergence towards linear combinations of chi-squared random variables:

a Malliavin-based approach. arXiv: 1409.5551[math.PR] (2014).[CT] Carmona, R., Tehranchi, M.: Interest Rate Models: an In�nite Dimensional Stochastic Analysis Per-

spective. Springer Finance (2006).[Chen] Chen, L.: Interest Rate Dynamics, Derivatives Pricing, and Risk Management. Lecture notes in eco-

nomics and mathematical systems, no. 435, Springer (1996).[GM] Gawarecki, L., Mandrekar, V.: Ito-Ramer, Skorohod and Ogawa integrals with respect to Gaussian

processes and their interrelationship. In: Chaos Expansions, Multiple Wiener-Ito Integrals, and TheirApplications. Ed.: Perez-Abreu, V, Houdre, C. CRC. Press London, p. 349-373 (1993).

[HKPS] Hida, T., Kuo, H.H., Pottho�, J., Streit, L. White Noise - An In�nite Dimensional Calculus.

Kluwer(1993).[HW] Hull, J., White, A.: The optimal hedge of interest rate sensitive securities. Research note, University of

Toronto (1994).[Jam] Jamshidian, F., Zhu, Y. : Scenario simulation: Theory and methodology. Finance and Stochastics I,

43-67 (1997).[KPR] P.C. Kettler, F. Proske, M. Rubtsov Sensitivity With Respect to the Yield Curve: Duration in a Sto-

chastic Setting. In Inspired by �nance. The Musiela Festschrift (2014) Springer-Verlag.[Mac] Macaulay, F. The movements of interest rates, bond yields and stock prices in the United States since

1856. New York: National Bureau of Economic Research (1938).[Man] V. Mandrekar, S. Zhang Skorohod Integral and Di�erentiation for Gaussian Processes In J.K Ghosh,

S.K. Mitra, K.R. Parthasarathy and B.L.S. Prakasa Rao (eds.) Statistics and Probability: A Raghu Raj

Bahadur Festschrift Wiley Eastern Limited, Publishers, (1993), 395-410.[Xiong] Xiong, I. An Introduction to Stochastic Filtering Theory. Oxford University Press (2008).

Institute of Computer Science-II, University of Bonn, Friedrich-Ebert-Alle 144, 53113 Bonn,

Germany.


IV

Modeling and Estimation of StochasticTransition Rates in Life Insurance withRegime Switching Based on Generalized

Cox ProcessesErik Bølviken1, Sindre Duedahl2 and Frank Proske1

Abstract

In this paper we aim at modeling stochastic transition rates of stateprocesses in life insurance by using generalized Cox processes. A feature ofour non-Gaussian model is that it can be used to capture ”regime switching”effects of data which may be due to regulatory changes in insurance marketsor external ”shocks” caused e.g. by an economical crisis, natural disasters orepidemics. We propose a method how to estimate the unknown parametersof our model for stochastic transition rates from insurance data by usingnon-linear filtering techniques for Levy processes. As a result we also obtainan explicit formula for the unnormalized density of a filtering problem withsingular coefficients.

Key words and phrases: Life insurance, Stochastic Transition Rates,Levy processes, non-linear filtering

AMS 2000 classification: 60G51; 60G35; 60H15; 60H40; 60H15; 91B70

1 Introduction

An important challenge in the risk analysis and risk management of life in-surance companies worldwide has been the accurate modeling of transitionrates as e.g. mortality rates or disability transition rates in the calcula-tion of insurance premiums. Compared to financial risk of technical interestrates longevity risk e.g. , which is due to increasing life expectancy of policy

1Centre of Mathematics for Applications (CMA), Department of Mathematics, Uni-versity of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway.

E-mail address: [email protected], [email protected] of Computer Science II, Friedrich-Ebert-Alle 144, University of Bonn, 53113

Bonn, Germany.E-mail address: [email protected]

1

holders and pensioners, is a source of insurance risk, which has been system-atically underestimated for many years. A reason for the negligence of thistype of risk in the insurance business has also been due to the use of deter-ministic models for mortality rates as e.g. the classical Gompertz-Makehammodel. The latter models however, which cannot capture the uncertainty ofthe future dynamics of mortality rates, have led to a miscalculation of insur-ance premiums with respect to defined-benefit pension plans and annuities,from which many insurance companies have suffered substantial losses.

In order to overcome the deficiencies of deterministic models for transi-tion rates, there have been various attempts in the literature in recent yearsto decribe the dynamics of future transition rates rates by using stochasticmodels. See e.g.the models of Lee, Carter [26] or Cairns, Blake, Dowd [7] inthe case of mortality rates.

In this paper we want to study a non-Gaussian stochastic model forstochastic transition rates, which allows for the modeling of ”regime switch-ing” effects of data or more precisely ”regime switching” effects of the jumpbehaviour or the tails of the distribution of data which may be due to differ-ent types of influence factors as e.g. regulatory changes in insurance marketsor external ”shocks” caused by a financial or political crisis, natural disastersor epidemics.

To be more specific, we consider in the following a cadlag stochasticprocess Zt, 0 ≤ t ≤ T with a finite state space S on some probability space(Ω,F , P ), which is used as a model for the state of the insured dynamicallyin time. Further, we denote by Nik(t) the process which counts the numberof transitions from state i to k of the state process Zt, 0 ≤ t ≤ T in the timeinterval (0, t]. In a regular insurance model with a Markovian state processit is well known that

Nik(t)−∫ t

0μik(s)ds, 0 ≤ t ≤ T

is a P -martingale with respect to the natural filtration{FZ

t

}0≤t≤T

, where

μik(s) is the transition rate at time s with respect to a transition from i toj. See e.g. [23].

One of the deficiencies of such a model as mentioned is that the deter-ministic transition rates may not capture the actual future transition rates.

Therefore it is reasonable to assume a stochastic model for the transitionrates μik(t), 0 ≤ t ≤ T :

In the sequel, let μik(t, x), 0 ≤ t ≤ T be the transition rate at timet of an insured aged x years with respect to a transition from state i to

2

state j, i, j ∈ S. In particular, the state space S of the insured in thecase of a permanent disability insurance consists of the states ∗ (”alive”), �(”permanently disabled”) and † (”dead”).

In order to estimate stochastic transition rates from insurance data onemay think of μik(t, x), 0 ≤ t ≤ T as a result of a ”parametrization” of thedeterministic transition rates by means of an unknown ”parametrizationprocess” Xt,0 ≤ t ≤ T .

More precisely, if S = {∗, �, †} one could assume that

μ∗�(t, x) = Y(1)t + 10Y

(2)t +Y

(3)t x,

μ∗†(t, x) = μ�†(t, x) = Y(4)t + 10Y

(5)t +Y

(6)t x,

where Yt = (Y(1)t , ..., Y

(6)t ), 0 ≤ t ≤ T is a generalized Cox process given by

dYt = h(t,Xt)dt+ dBYt +

∫R6

ςNλ(dt, dς)

and Xt, 0 ≤ t ≤ T the unknown ”parametrization process” modeled by thestochastic differential equation (SDE)

dXt = b(Xt)dt+ σ(Xt)dBXt (1)

for Borel functions h, b and σ, where BYt ∈ R

6, BXt ∈ R

d are independentBrownian motions and where Nλ is the jump measure of a ”generalized Coxprocess” with a predictable compensator μ given by

μ(dt, dς, ω) = λ(t,Xt, ς)dtν(dς) (2)

for a Levy measure ν and a Borel function λ.More generally, we may assume in this paper that stochastic transition

rates μik(t, x), 0 ≤ t ≤ T, i, k ∈ S are described by a stochastic Gompertz-Makeham model GM(r, s) given by

μik(t, x) = h1,rik (t, x) + exp(h2,sik (t, x)), (3)

where h1,rik (t, x), h2,sik (t, x) are time-dependent stochastic polynomials of de-gree r and s, respectively, that is

h1,rik (t, x) =

r∑l=0

Y(l)t xl

3

and

h2,sik (t, x) =

s∑l=0

Y(r+1+l)t xl

for all i, k ∈ S.In order to estimate the unknown ”parametrization” process Xt, 0 ≤ t ≤

T from (indirectly) observed insurance data

Yt = (Y(0)t , ..., Y

(r)t , Y

(r+1)t , ..., Y

(r+s)t )∗, 0 ≤ t ≤ T, (4)

where ∗ denotes transposition, one can apply non-linear filtering techniquesfor Levy processes as proposed in [30] to the signal process Xt ∈ R

n, 0 ≤t ≤ T and the observation process Yt ∈ R

m, 0 ≤ t ≤ T :

dXt = b(Xt)dt+ σ(Xt)dBXt , (5)


∫Rm

ςNλ(dt, dς), (6)

where m = r + s+ 2.Using the latter non-Gaussian filtering framework, we want to model

stochastic transition rates, which are subject to regime switching effects ofinsurance data. In modeling this phenomenon one could e.g. assume thatthe ”parametrization” process Xt, 0 ≤ t ≤ T is described by

dXt = b(Xt)dt+ dBXt , (7)

where the drift coefficient b : Rn −→ Rn is a discontinuous vector field. An

example of such a discontinuous vector field is

b(t, x) =

{a1 , if ‖x‖ ≥ τa2 else

.

Here the vectors a1, a2 ∈ Rn stand for the different regime switching states

the parametrization process Xt will assume, if it exceeds a certain thresholdτ at time t, that is ‖Xt‖ ≥ τ , or not.

Another example of such a drift coefficient in the case n = 1, whichexhibits the feature of mean-reversion in connection with regime switchingeffects is

b(x) =

{a(b1 − x) , if x ≥ τa(b2 − x) else

.

for a, b1, b2 ≥ 0. In this case the parametrization process Xt may be inter-preted as a mean-reverting process with a mean reversion coefficients a anddifferent long-run average levels b1, b2 depending on the threshold τ .

4

The parameters a, b1, b2 and the threshold τ in the above examples area priori unknown and will be estimated from insurance data by using non-linear filtering techniques.

The non-linear filtering problem for our model is to find the least squareestimate to the (possibly transformed) signal process Xt at time t, giventhe history of the observation process up to time t, that is to determine theconditional expectation

E[f(Xt)∣∣FY

t

],

where f is a given Borel function and where FYt is the σ−algebra, generated

by {Ys, 0 ≤ s ≤ t}.

One of the objectives of this paper is the derivation of an explicit rep-resentation of the unnormalized conditional density with respect to the op-timal filter of the filter problem (5) and (6), when the drift coefficient bin (5) is merely (bounded and) Borel measurable. In solving this problem,we explicitly construct a (weak) solution to a stochastic partial differentialequation given by the Duncan-Mortensen-Zakai or shortly Zakai equation forthe conditional unnormalized density, which can be regarded as a weak so-lution to a stochastic Fokker-Planck equation with singular coefficients. See[25] in the deterministic case. Our method relies on a representation formulaof the unnormalized conditional density found in [30] in the case of regularcoefficients and finite Levy measures, which we want to invoke in connec-tion with an approximation argument and local time techniques. As a resultwe give an explicit representation of the unnormalized conditional densityassociated with the least square estimate of the unknown parametrizationprocess Xt of the generalized Cox process (6) in our model for the dynamicsof stochastic transition rates. In contrast to [30] we do not require in thispaper that b is regular in the sense of Lipschitz continuity or that the Levymeasure ν in (2) is finite.

We remark that non-linear filtering has been intensively studied in theliterature since the 1960’s. See e.g. Lipster and Shiryaev [27], Kallianpur[20], Fleming and Rishel [13], Xiong [38] and the references therein. See alsothe innovation approach for the conditional density of the filter process byFujisaki, Kallianpur and Kunita (see e.g.[20]). As for solutions of the Zakaiequation in the Gaussian case we refer the reader to Zakai [39], Gyongy,Krylov [17], [18], Pardoux [33], [34], Kunita [24]. See also [11], [14] or theworks [28], [30], which give a generalization of results in [11], [4] to thenon-Gaussian case.

The main objective of this paper is to introduce a model for the dynamics

5

of stochastic transition rates which is able to describe ”regime switching”effects of the jump or tail distribution behaviour of e.g. observed mortalityrates or transition rates in disability insurance by using the generalized Coxprocess (6) in the framework of non-linear filtering for Levy processes, wherethe signal process, that is the parametrization process Xt of (6) is modeledby a SDE with singular coefficients.

A popular model for stochastic transition rates in the case of mortalityrates was proposed by Lee, Carter [26]. In this discrete-time model the errorterms are Gaussian distributed. A generalization of the Lee-Carter modelis the Gaussian two-factor stochastic mortality model by Cairns, Blake andDowd [7], which is used to describe the different behaviours of mortalityrates at lower and higher ages. Reasons for the success of these modelsin life insurance is the simplicity of their implementation and their predic-tion reliability in forecasting mortality rates under ”usual” circumstances.However, a disadvantage of these models is that they cannot capture e.g.the observed skewness and (semi-) heavy tailed innovation distributions ofdata coming from cohort effects or short term catastrophic events as e.g. theTsunami in 2004. In recent years there have been therefore several attemptsto tackle this problem in the literature. In order to model heavy-tailed distri-butions of mortality data Giacometti et al. [15] generalized the Lee-Cartermodel by modeling the distributional behaviour of the error terms by in-finitely divisible distributions in the case of Normal Inverse Gaussian laws.Another model in this direction, which is based on non-Gaussian distribu-tions for error terms in the framework of [35], is the paper of Wang et al.[37]. See also the approach in [32] based on Markov regime switching modelsor [8], where the authors employ jump diffusions to describe age-adjustedmortality rates.

Contrary to our model (5) and (6), however, the above mentioned modelscannot be used to model the rather complex phenomenon of the occurrenceof changing types of jumps or types of heavy-tailedness of distributions ofreal data as a result of different types of ”external” shocks. The reason forthis is that these models are finite-dimensional models (in discrete time).Our model can be regarded as an infinite dimensional model for stochas-tic transition rates, since one of the unknown parameters is given by theparametrization process Xt, 0 ≤ t ≤ T . In this paper we use the powerfultool of non-linear filtering for Levy processes to effeciently estimate this pro-cess from constantly updated observations. Therefore we may expect thatour approach is more flexible than those mentioned and also suitable forthe modeling of other types of stochastic transition rates beyond mortalityrates.

6

Our paper is organized as follows:

In Section 2 we introduce the framework of our paper and derive anexplicit representation of the unnormalized conditional density associatedwith the least square estimate of the parametrization process Xt, 0 ≤ t ≤ Tby constructing an explicit (weak) solution of a Zakai equation with singularcoefficients. Further, we study the regularity of the obtained solution. Usingthe results of Section 2, we finally want to discuss in Section 3 variousspecifications of our model and its implementation in life insurance basedon Monte-Carlo simulation.

2 Framework and Main Results

In this Section we want to introduce the mathematical framework of ourgeneral model for stochastic transition rates and to discuss the estimationof the unknown parameters or parameter processes of the the model fromconstantly updated observations in connection with a non-linear filteringproblem for Levy processes. In solving this problem we derive an explicitrepresentation of the optimal filter of the filtering problem by constructing a(weak) Lp−solution of the Zakai equation for the unnormalized conditionaldensity of the filter process with initial Levy noise and singular coefficients.

In what follows we consider a Levy process Lt ∈ Rm, 0 ≤ t ≤ T, that is

a stochastically continuous process with stationary independent incrementsstarting in zero defined on a filtered complete probability space

(Ω∗,F∗, π∗) , {F∗t }0≤t≤T ,

where {F∗t }0≤t≤T is a π∗−augmented filtration generated by L·.

We may here assume from now on that Lt, 0 ≤ t ≤ T is a cadlag process,that is a process, whose paths are right continuous paths and have existingleft limits.

By the Levy-Ito theorem the Levy process Lt = (L(1)t , ..., L

(m)t ), 0 ≤ t ≤

T can be uniquely decomposed as

L(i)t =

l∑k=1

aikB(k)t + bit+

∫ t+

0

∫Rm0

zi1{‖z‖≥1}N(ds, dz)

+

∫ t+

0

∫Rm0

zi1{‖z‖<1}N(ds, dz),

0 ≤ t ≤ T, i = 1, ...,m,

7

whereBt = (B(k)t )1≤k≤l ∈ R

l, 0 ≤ t ≤ T is a Brownian motion, (aik)1≤i≤m,1≤k≤l ∈Rm×l, (bi)1≤i≤m ∈ R

m and N(ds, dz) = N(ds, dz)−dsν(dz) the compensatedPoisson random measure associated with the Levy process L·. Here ν is aσ−finite measure on the Borel sets B(Rm

0 ), Rm0 := R

m \ {0}, referred to asLevy measure, which satisfies the integrability condition∫

Rd0

1 ∧ ‖z‖2 ν(dz) < ∞

for the Euclidean norm ‖·‖. See e.g. [36] or [5] for more information on Levyprocesses.

In what follows we want to estimate the unknown ”parametrization”process Xt, 0 ≤ t ≤ T from the observed insurance data (4) by analyzingthe non-linear filtering problem



∫Rm

ςNλ(dt, dς), (9)

for the signal process Xt ∈ Rn and the observation process Yt ∈ R

m, 0 ≤ t ≤T, n,m ∈ N on a complete probability space (Ω,F , μ) , where the Brownianmotion BY

t ∈ Rn is independent of the Brownian motion BX

t ∈ Rm and the

integer valued random measure Nλ, whose predictable compensator μ withrespect to a augmented filtration F = {Ft}0≤t≤T (generated by BX· , BY· , Nλ)is given by

μ(dt, dς, ω) = λ(t,Xt, ς)dtν(dς) (10)

for the Levy measure ν of Lt ∈ Rm and a Borel function λ. Further the

initial condition X0 in (8) is a random variable, which is independent ofBX

t , BYt and Nλ.

In order to guarantee a unique strong solution to the system (8) and (9),we require for the time being that the continuous coefficients b : Rn −→Rn, σ : Rn −→ R

n×n, h : [0, T ]× Rn −→ R

n and λ : [0, T ]× Rn × R

m0 −→ R

fulfill a linear growth and Lipschitz condition, that is

‖b(x)‖+ ‖σ(x)‖+ ‖h(t, x)‖+∫Rm0

|λ(t, x, ς)| ν(dς) ≤ C(1 + ‖x‖) (11)

8

and

‖b(x)− b(y)‖+ ‖σ(x)− σ(y)‖+ ‖h(t, x)− h(t, y)‖ (12)

+

∫Rm0

|λ(t, x, ς)− λ(t, y, ς)| ν(dς)

≤ C ‖x− y‖

for all x, y, t and a constant C < ∞, where ‖·‖ stands for a vector or matrixnorm.

For the convenience of the reader we now want to give a derivation of theZakai equation for the unnormalized filter of the non-linear filtering problem(8), (9). See e.g. [2] or [38] in the case of Wiener noise driven obervationprocesses.

For this purpose denote by πt : Ω × B(Rn) −→ [0,∞) the regular con-ditional probability measure of the signal process Xt given the σ−algebraFYt , generated by {Ys, 0 ≤ s ≤ t} and the null sets N . Then

E[f(Xt)∣∣FY

t

]= 〈πt, f〉

for all f ∈ Cb(Rn) (space of bounded continuous functions), where 〈πt, f〉 :=∫

Rn f(x)πt(ω, dx).Suppose that the function λ : [0, T ]×R

n ×Rm0 −→ R is strictly positive

and consider the density process

Λt : = exp{m∑i=1

∫ t

0−hi(s,Xs)dB

Y,is − 1

2

∫ t

0‖h(s,Xs)‖2 ds (13)

+

∫ t

0

∫Rm0

− log λ(s,Xs, ς)Nλ(ds, dς)

+

∫ t

0

∫Rm0

(λ(s,Xs, ς)− 1)dsν(dς)},

0 ≤ t ≤ T,

where BYt = (BY,1

s , ..., BY,ms )∗ and h(t, x) = (h1(t, x), ..., hm(t, x))∗ (∗ trans-

position). Further, assume that

E[ΛT ] = 1. (14)

9

Remark 1 Using stopping time localization of Doleans-Dade exponentials,one obtains e.g. the following sufficient conditions for (14):

sup0≤t≤T

E[exp(6

∫ t

0‖h(s,Xs)‖2 ds (15)

+4

∫ t

0

∫Rm0

(1− λ−1(s,Xs, ς))λ(s,Xs, ς)dsν(dς)

−∫ t

0

∫Rm0

(1− λ−4(s,Xs, ς))λ(s,Xs, ς)dsν(dς)]

< ∞

E[

∫ T

0

∫Rm0

∣∣(λ−4(s,Xs, ς)− 1)λ(s,Xs, ς)∣∣ ν(dς)ds] (16)

+E[

∫ T

0(

∫Rm0

|(λ(s,Xs, ς)− 1)| ν(dς))2ds]< ∞

E[

∫ T

0

∫Rm0

|λ(s,Xs, ς) log λ(s,Xs, ς)| dsν(dς)] < ∞ (17)

An example which satisfies the conditions (15), (16) and (17) in the casem = 1 is given by

ν(dς) = ϕ(ς)dς, (18)

where

ϕ(ς) =

{1

|ς|1+α , if |ς| ≤ 1

0 else

for α ∈ (0, 1) as well as h is a bounded Borel measurable function and

λ(s, x, ς) = exp(Ψ(x) |ς|) (19)

for a bounded and continuous function Ψ : R −→ R.

Define now the probability measure π with Radon-Nikodym derivativeon (Ω,Ft ) given by

dπ

dμ

∣∣∣∣Ft

= Λt.

and require that ∫Rd0

‖z‖ ν(dz) < ∞ (20)

10

Then by Girsanov’s theorem and the uniqueness of semimartigale charac-teristics (see e.g. [19]), the observation process Yt, 0 ≤ t ≤ T becomes aLevy process being independent of the signal process under the new prob-ability measure π. More precisely, the system (8), (9) has the followingrepresentation under π :

dXt = b(Xt)dt+ σ(Xt)dBXt

dYt = dBt + dLt, (21)

where Y· is a Levy process independent of X· with

Bt := BYt −

∫ t

0(−h(s,Xs))ds, 0 ≤ t ≤ T

the Gaussian part and

Lt =

∫ t

0

∫Rm0

ςN(ds, dς)

the jump component with respect the Poisson random measure N(ds, dς) :=Nλ(ds, dς) with compensator dsν(dς).

Since Y· is a Levy process under π, we also observe that the (augmented)filtration FY

t , 0 ≤ t ≤ T is right-continuous.

The so called unnormalized filter 〈Ψt, ·〉 , 0 ≤ t ≤ T is a stochastic processtaking values in the space of finite Borel measures on R

n, and is given bythe Kallianpur-Striebel-formula, which is a consequence of Bayes’ rule:

Theorem 2 The optimal filter πt has the representation

〈πt, f〉 = 〈Ψt, f〉〈Ψt, 1〉

with〈Ψt, f〉 := Eπ[Ztf(Xt)

∣∣FYt

]for all f ∈ Cb(R

n), where Eπ denotes the espectation with respect to π and

11

where

Zt : = Λ−1t (22)

= exp{m∑i=1

∫ t

0hi(s,Xs)dB

is −

1

2

∫ t

0‖h(s,Xs)‖2 ds

+

∫ t

0

∫Rm0

log λ(s,Xs, ς)N(ds, dς)

+

∫ t

0

∫Rm0

(1− λ(s,Xs, ς))dsν(dς)},

0 ≤ t ≤ T

under π.

Remark 3 We mention the fact that

Eπ[ξ∣∣FY

t

]= Eπ[ξ |A]

for all Ft−measurable ξ with Eπ[|ξ|] < ∞, where

A :=∨

0≤t≤T

FYt .

See Proposition 3.15 in [1].

We also need the following Lemmata for the derivation of the Zakaiequation:

Lemma 4 Let f ∈ C∞b (Rn) (space of smooth functions on R

n with boundedpartial derivatives). Assume that the coefficients b, σ in (11), (12) arebounded and that

E[exp(496

∫ T

0‖h(s,Xs)‖2 ds+

∫ T

0

∣∣∣∣∣∫Rm0

(1− λ32(s,Xs(θ), ς))ν(dς)

∣∣∣∣∣ ds(23)

+ 32

∫ T

0

∣∣∣∣∣∫Rm0

(1− λ(s,Xs(θ), ς))ν(dς)

∣∣∣∣∣ ds] < ∞,

12

E[(

∫ T

0(

∫Rm0

|log λ(r,Xr(θ), ς)|j ν(dς))kdr)4] (24)

< ∞, j = 1, 2, 4, 8, k = 1, 2, 3

and

E[

∫ T

0

∫Rm0

∣∣1− λ32(s,Xs(θ), ς)∣∣ ν(dς)ds] (25)

+E[(

∫ T

0(

∫Rm0

(1− λ(r,Xr(θ), ς))ν(dς))2ds)4]

< ∞.

Then there exists a cadlag modification of the unnormalized filter 〈Ψ·, f〉 .

Proof. See Appendix.

Remark 5 An example satisfying the assumptions (23)-(25) in Lemma 4is given by Remark 1.

Lemma 6 Consider F−predictable processes αt, βt, γt(·), 0 ≤ t ≤ T suchthat

Eπ[

∫ T

0(|αs|+ |βs|2)ds] < ∞,

Eπ[

∫ T

0

∫Rm0

|γs(ς)|2 dsν(dς)] < ∞.

Then

Eπ[

∫ t

0αsds

∣∣FYt

]=

∫ t

0Eπ[αs

∣∣FYs

]ds,

Eπ[

∫ t

0βsdBs

∣∣FYt

]=

∫ t

0Eπ[βs

∣∣FYs

]dBs

Eπ[

∫ T

0

∫Rm0

γs(ς)N(ds, dς)∣∣FY

t

]=

∫ t

0

∫Rm0

Eπ[γs(ς)∣∣FY

s

]N(ds, dς)

and

Eπ[

∫ t

0βsdB

Xs

∣∣FYt

]= 0.

13

Proof. The proof is essentially based on the independence of the incre-ments of the process Yt, 0 ≤ t ≤ T under π and can be e.g. found in [2] or[38] in the case of Brownian motion.

Using the latter auxiliary result, we obtain the following Zakai equationfor the unnormalized filter of the non-linear filtering problem (8), (9):

Theorem 7 Assume the conditions of Lemma 4 and require that

sup0≤s≤T

Eπ[|Zs(λ(s,Xs, ς)− 1)|p] < ∞ (26)

for all ς and some p > 1. Then the unnormalized filter 〈Ψt, ·〉 , 0 ≤ t ≤ T isa cadlag FY

t −adapted solution to the Zakai equation, that is to the SPDE

〈Ψt, f〉 = 〈Ψ0, f〉+∫ t

0〈Ψs,Lf〉 ds+

∫ t

0〈Ψs, f · h∗(s, ·)〉 dBs (27)

+

∫ t

0〈Ψs−, f · (λ(s, ·, ς)− 1)〉 N(ds, dς)

for all f ∈ D ⊂ C∞c ((Rn) (space of compactly supported infinitely often

differentiable functions of Rn), where D is a (countable) dense subset of

L2(Rn) and L the generator of the diffusion process X· given by

Lf(x) = 1

2

n∑i,j=1

σij(x)∂2

∂xi∂xjf(x) +

n∑i=1

bi(x)∂

∂xif(x) (28)

with σ(x) = (σij(x))1≤i,j≤n and b(x) = (b1(x), ..., bn(x))∗ and where N(ds, dς)

is the compensated Poisson random measure associated with the Levy processYt, 0 ≤ t ≤ T under π.

Proof. It follows from Ito’s Lemma for f ∈ C∞c ((Rn) that

f(Xt) = f(X0) +

∫ t

0Lf(Xs)ds+

∫ t

0∇∗f(Xs)σ(Xs)dB

Xs ,

where ∇∗ denotes the transposed gradient. On the other hand we know thatthe process Zt, 0 ≤ t ≤ T in Theorem 2 satisfies the SDE

Zt = 1+m∑i=1

∫ t

0Zshi(s,Xs)dB

is+

∫ t

0

∫Rm0

Zs−(λ(s,Xs, ς)−1)N(ds, dς). (29)

14

So using integration by parts, we obtain that

Ztf(Xt) = f(X0) +

∫ t

0ZsLf(Xs)ds+

∫ t

0Zs∇∗f(Xs)σ(Xs)dB

Xs

+m∑i=1

∫ t

0Zsf(Xs)hi(s,Xs)dB

is

+

∫ t

0

∫Rm0

Zs−f(Xs)(λ(s,Xs, ς)− 1)N(ds, dς).

The conditional expectation with respect to FYt applied to the latter equa-

tion combined with Lemma 6 then gives

〈Ψt, f〉 = 〈Ψ0, f〉+ Eπ[

∫ t

0ZsLf(Xs)ds

∣∣FYt

]+Eπ[

∫ t

0Zs∇∗f(Xs)σ(Xs)dB

Xs

∣∣FYt

]+

m∑i=1

Eπ[

∫ t

0Zsf(Xs)hi(s,Xs)dB

is

∣∣FYt

]+Eπ[

∫ t

0

∫Rm0

Zs−f(Xs)(λ(s,Xs, ς)− 1)N(ds, dς)∣∣FY

t

]= 〈Ψ0, f〉+

∫ t

0〈Ψs,Lf〉 ds+

m∑i=1

∫ t

0〈Ψt, f · hi(s, ·)〉 dBi

s

+

∫ t

0

∫Rm0

〈Ψs−, f · (λ(s, ·, ς)− 1)〉 N(ds, dς),

where we used Lemma 4, Remark 3, the continuity of the paths of Xt, 0 ≤t ≤ T, the continuity of λ and (26) in connection with uniform integrabilityunder the measure π.

Remark 8 The condition (26) in Theorem 7 holds, if e.g.

sup0≤s≤T

Eπ[Zprs ] < ∞

andsup

0≤s≤TEπ[|λ(s,Xs, ς)− 1|pq] < C

for all ς, some constant C with pr < 2, 1r +

1q = 1, r, q > 1 are satisfied. The

latter conditions are e.g. covered by the conditions B1 − B6 in the paperlater on.

15

In addition to the conditions (11), (12) let us from now on also requirethat the drift coefficient b is bounded and σ = Id (identity).

Using the independence of the increments of the observation process Y·under π and the probability density of the signal process Xt, which ex-ists in this case, our assumptions on b, h, λ and ν imply that there is anFYt −adapted process Φ(t, ·), 0 ≤ t ≤ T , called unnormalized conditional

density, such that

〈Ψt, f〉 =∫Rn

f(x)Φ(t, x)dx, 0 ≤ t ≤ T

for all f ∈ Cb(Rn). Hence we can recast the Zakai equation (27) in terms

of the unnormalized density and find that Φ(t, ·), 0 ≤ t ≤ T satisfies astochastic Fokker-Planck-equation, that is the SPDE

dtΦ(t, x) = L∗Φ(s, x)dt+ (30)

Φ(t, x)h∗(s, x)dBt +

∫Rm0

Φ(t−, x)(λ(s, x, ς)− 1)N(dt, dς)

Φ(0, x) = p0(x),

where L∗ is the adjoint operator of the generator L of Xt and where p0(x)is the probability density of X0, in a weak sense, that is Φ ∈ L2

loc([0, T ] ×Rn;L2(Ω)) is FY

t −adapted process, which solves the equation∫Rn

Φ(t, x)f(x)dx (31)

=

∫Rn

p0(x)f(x)dx+

∫ t

0

∫Rn

Φ(s, x)Lf(x)dxds

+

∫ t

0

∫Rn

Φ(s, x)h∗(s, x)f(x)dxdBs

+

∫ t

0

∫Rm0

∫Rn

Φ(s−, x)(λ(s, x, ς)− 1)f(x)dxN(ds, dς), 0 ≤ t ≤ T

for all f ∈ C∞c ((Rn).

In fact, it was shown in [30] that the Zakai equation for the unnormalizeddensity (27) has a unique strong solution Φ(t, x) to (30) in Lp(μ), p ≥ 1,

16

which is twice continuously differentiable in x, under the following conditions

A1 : The Levy measure ν is bounded.

A2 : The drift coefficient b is contained in C2+βb (Rn).

A3 : The initial condition p0 in (30) is positive and belongs to

C2+βb (Rn).

A4 : The intensity rate λ is strictly positive and λ(·, ·, ς) ∈C1,2b (R+ × R

m) ∩ C2+β(R+ × Rm) uniformly in ς.

A5 :

n∑i=1

∂

∂xibi ∈ C2

b (Rn) ∩ C2+β(Rn).

A6 : The observation function h is contained in

C1,2b (R+ × R

n) ∩ C2+β(R+ × Rn).

A7 : Λt, 0 ≤ t ≤ T in (13) is a martingale,

where C l,rb (R+ × R

d) is the space of l-times in t ∈ (0,∞) and r-times inx ∈ R

d continuously differentiable, whose partial derivatives are boundedand have continuous extensions to R+ × R

d (R+ := [0,∞)). The spaceCr+β(U) denotes the space of functions in Cr(U) with all partial derivativesup to order r being Holder continuous of order β ∈ (0, 1).

Moreover, the strong solution Φ to (30) has the following explicit repre-sentation:

Φ(t, x, ω) (32)

= Exϑ[p0(X

∗t (θ)) exp(−

n∑i=1

∫ t

0

∂

∂xibi(X

∗s (θ))ds)

exp{∫ T

T−th∗(s,X∗

s−(T−t)(θ))dBs(ω)− 1

2

∫ T

T−t

∥∥∥h(s,X∗s−(T−t)(θ))

∥∥∥2 ds+

∫ T

T−t

∫Rm0

log(λ(s,X∗s−(T−t)(θ), ς))N(ds, dς, ω)

+

∫ T

T−t

∫Rm0

(log(λ(s,X∗s−(T−t)(θ), ς))− (λ(s,X∗

s−(T−t)(θ), ς)− 1))dsν(dς)}]

where X∗s (θ) = X∗,x

s (θ), 0 ≤ s ≤ T, is the solution to the time-homogeneousSDE

dX∗t = −b(X∗

t )dt+ dB∗t , X

∗0 = x ∈ R

d (33)

for a Brownian motionB∗· , defined on an auxiliary probability space (Θ,K, ϑ).

17

In order to capture ”regime switching effects” in the framework of ourmodel for the stochastic transition rates μik(t, x), 0 ≤ t ≤ T in (3) and inview of Monte Carlo simulation techniques with respect to such transitionrates, we now want to extend the representation of Φ (32) under the condi-tions A1 − A7 to the case, when the drift coefficent b of the signal processis merely bounded and measurable. In addition, we aim at relaxing thecondition A1 of compound Poisson Levy measures ν in (32) to that of finite-variation Levy measures ν satisfying (20). Furthermore, we will show thatsuch a Φ solves the Zakai equation (30) in the weak sense.

To this end we need to recall the concept of stochastic integration overthe plane with respect to Brownian local time. See [12]:

Consider elementary functions fΔ : [0, 1]× R −→R given by

fΔ(s, x) =∑

(sj ,xi)∈Δfijχ(sj ,sj+1](s).χ(xi,xi+1](x) , (34)

where (xi)1≤i≤n , (fij)1≤i≤n,1≤j≤m are finite sequences of real numbers, (sj)1≤j≤ma partition of [0, 1] and Δ = {(sj , xi), 1 ≤ i ≤ n, 1 ≤ j ≤ m}. Denote by{L(t, x)}0≤t≤1,x∈R the local time of a 1−dimensional Brownian motion B.Then the integral of integration of fΔ with respect to L is defined as∫ 1

0

∫R

fΔ(s, x)L(ds, dx) (35)

=∑

(sj ,xi)∈Δfij(L(sj+1, xi+1)− L(sj , xi+1)− L(sj+1, xi) + L(sj , xi)).

The latter integral can be generalized to integrands of the Banach space(H, ‖·‖) of measurable functions f endowed with the norm

‖f‖ = 2

(∫ 1

0

∫R

(f(s, x))2 exp(−x2

2s)dsdx√2πs

)1/2

(36)

+

∫ 1

0

∫R

|xf(s, x)| exp(−x2

2s)dsdx

s√2πs

.

If f is such that f(t, ·) is locally square integrable and f(t, ·) continuousin t as a map from [0, T ] to L2

loc(R), then f ∈ H and∫ t

0

∫R

f(s, x)L(ds, dx), 0 ≤ t ≤ T

18

exists as well as

E

[∣∣∣∣∫ t

0

∫R

f(s, x)L(ds, dx)

∣∣∣∣] ≤ ‖f‖

for 0 ≤ t ≤ T. Further, if f(t, x) is differentiable in x, then∫ t

0

∫R

f(s, x)L(ds, dx) = −∫ t

0f ′(s,Bs) ds , 0 ≤ t ≤ T ,

where f ′(s, x) denotes the derivative in x. See [12].

Assume now that Bt =(B

(1)t , ..., B

(n)t

), 0 ≤ t ≤ T is a Brownian motion,

whose components B(i)t are defined on probability spaces (Ωi,Fi, μi), i =

1, ..., n. In what follows we denote by

Bt := (B(1)t , ..., B

(d)t ) := BT−t , 0 ≤ t ≤ T , (37)

the time-reversed Brownian motion. The process B(i)t satisfies for each i =

1, ..., d the equation

B(i)t = B

(i)1 + W

(i)t −

∫ t

0

B(i)s

T − sds , 0 ≤ t ≤ T , a.e., (38)

where W(i)t , 0 ≤ t ≤ T are independent μi-Brownian motions with respect

to the filtrations F B(i)

t generated by B(i)· , i = 1, ..., n. See [12].

Using the relation (38) one obtains the following decomposition of localtime-space integrals (see [12]):∫ t

0

∫R

fi(s, x)Li(ds, dx) (39)

=

∫ t

0fi(s, B

(i)s )dB(i)

s +

∫ T

T−tfi(T − s, B(i)

s )dW (i)s

−∫ T

T−tfi(T − s, B(i)

s )B

(i)s

T − sds,

0 ≤ t ≤ T, a.e. for fi ∈ H, i = 1, ..., n. Here Li(t, x) is the local time of B(i)·

on (Ωi, μi), i = 1, ..., n.

In the sequel we also need the following auxiliary result (see also [31]):

19

Lemma 9 Let Bt, 0 ≤ t ≤ T be a 1−dimensional Brownian motion. Then

E[exp(k

∫ T

0

|Bt|t

dt)] < ∞

for all k ≥ 0.

Proof. See Appendix.

’Let us now assume that the following conditions are satisfied

B1 : The Levy measure ν fulfills condition (20).

B2 : The drift coefficient b is Borel measurable and bounded.

B3 : The initial condition p0 in (30) is positive and belongs to

C2+βb (Rn).

B4 : The intensity rate λ is strictly positive and λ(·, ·, ς) ∈C1,2b (R+ × R

n) ∩ C2+β(R+ × Rn) uniformly in ς.

B5 : The observation function h is contained in

C1,2b (R+ × R

n) ∩ C2+β(R+ × Rn).

B6 : λ satisfies (15)-(17), (23)-(25)

and the following integrability conditions

supx∈U

E[exp(200{∫ T

0

∫Rm0

∣∣(λ(s, Bxs , ς)− 1

∣∣ (40)

+100maxn=1

∫ T

0

∫Rm0

∣∣(λ2n(s, Bxs , ς)− 1

∣∣ dsν(dς)})]< ∞

and

supx∈U

E[(

∫ T

0

∫Rm0

∣∣log λ(s, Bxs , ς)

∣∣i dsν(dς))8] (41)

< ∞, i = 1, 2, 4, 8

for all bounded U ⊂ Rn.

We mention that condition B2 implies the the existence of a uniquestrong solution X∗· to the the SDE (33). See e.g. [40].

20

We obtain the following existence result for weak solutions of a singularstochastic Fokker-Planck equation driven by Levy noise, that is the SPDE(30) with the adjoint operator L∗ of the generator L of Xt for merelybounded and measurable drift coefficients b : Rd −→ R

d:

Theorem 10 Suppose that the conditions B1−B6 hold. Then there exists aweak solution Φ to the SPDE (30), which is given in law by the unnormalizeddensity and takes the explicit form

Φ(t, x, ω) (42)

= Eϑ[p0(Bxt (θ)) exp(

n∑i=1

{∫ t

0bi(B

xs (θ))dB

(i)s +

∫ T

T−tbi(B

xs (θ))dW

(i)s

−∫ T

T−tbi(B

xs (θ))

B(i)s

T − sds})

exp{∫ T

T−th∗(s, Bx


2

∫ T

T−t

∥∥∥h(s, Bxs−(T−t)(θ))

∥∥∥2 ds+

∫ T

T−t

∫Rm0

log(λ(s, Bxs−(T−t)(θ), ς))N(ds, dς, ω)

+

∫ T

T−t

∫Rm0

(log(λ(s, Bxs−(T−t)(θ), ς))

−(λ(s, Bxs−(T−t)(θ), ς)− 1))dsν(dς)}E(

∫ T

0−b∗(Bx

s (θ))dBs)],

where Eϑ denotes the expectation with respect to the product measure ϑ =

μ1×...×μn with B(i)· is a Brownian motion on (Ωi, μi), i = 1, ..., n, Bx

t (θ) :=x+ Bt(θ) and Bx

t (θ) := x+ Bt(θ). Further,

E(∫ t

0−b∗(Bx

s (θ)dBs)

= exp(

∫ t

0−b∗(Bx

s (θ))dBs(θ)− 1

2

∫ t

0

∥∥b(Bxs (θ)

∥∥2 ds), 0 ≤ t ≤ T

is the Doleans-Dade exponential.

Proof. The proof is based on the explicit representation for the unnor-

malized density Φ in (32) and an approximation argument with respect tothe function b and the Levy measure ν.

21

Consider a sequence of Borel sets Ur, r ≥ 1 of Rm0 with Ur ↗ R

m0 such

that ν(Ur) < ∞ for all r. Define the compound Poisson Levy measures νrby

νr(B) =

∫B1Ur(ς)ν(dς),

where 1A is the indicator function of a set A. In the sequel we denote byNr(ds, dς) the Poisson random measure associated with the Levy measureνr, r ≥ 1.

Let us also choose functions br ∈ C∞c (Rn), r ≥ 1 such that

‖br(x)‖ ≤ M < ∞

for a constant M and all x, r as well as

br(x) −→ b(x) a.e.

for r −→ ∞.In the following let us denote by Φr the unique (strong) solution to the

SPDE (30) with respect to the drift coefficent br and by X∗,rt , 0 ≤ t ≤ T

the strong solution to the SDE

dX∗,rt = −br(X∗,r

t )dt+ dB∗t , X

∗,r0 = x ∈ R

n (43)

for all r.In what follows let x ∈ U for a bounded set U ⊂ R

n.Using Girsanov’s theorem and the explicit representation of Φr in (32)

for b = br and ν = νr based on the condition B6 we find that

Φr(t, x, ω) (44)


n∑i=1

{∫ t

0

∂

∂xibri (B

xs (θ))ds)

exp{∫ T

T−th∗(s, Bx


2

∫ T

T−t

∥∥∥h(s, Bxs−(T−t)(θ))

∥∥∥2 ds+

∫ T

T−t

∫Rm0

log(λ(s, Bxs−(T−t)(θ), ς))Nr(ds, dς, ω)

+

∫ T

T−t

∫Rm0


−(λ(s, Bxs−(T−t)(θ), ς)− 1))dsνr(dς)}E(

∫ T

0−(br(Bx

s (θ)))∗dBs)].

22

If we apply (39) to B(i)· on (Ωi, μi) for

fi(s, z)

= bri (x1 + B(1)s (ω1), ...,

xi−1 + B(i−1)s (ωi−1), z, xi + B(i+1)

s (ωi+1), ..., xn + B(n)s (ωn))

then we get that

Φr(t, x, ω) (45)


n∑i=1

{∫ t

0bri (B

xs (θ))dB

(i)s +

∫ T

T−tbri (B

xs (θ))dW

(i)s

−∫ T

T−tbri (B

xs (θ))

B(i)s

T − sds})

exp{∫ T

T−th∗(s, Bx


2

∫ T

T−t

∥∥∥h(s, Bxs−(T−t)(θ))

∥∥∥2 ds+

∫ T

T−t

∫Rm0


+

∫ T

T−t

∫Rm0


−(λ(s, Bxs−(T−t)(θ), ς)− 1))dsνr(dς)}E(

∫ T

0−(br(Bx

s (θ)))∗dBs)].

Then it follows from the mean value theorem that

E[(Φr(t, x, ω)− Φ(t, x, ω))2]

= E[(p0(Bxt (θ)))

2(Ir1 + Ir2 + Ir3)2

(

∫ 1

0exp(Ir0 + τ(Ir1 + Ir2 + Ir3))dτ)

2],

where E is an expectation with respect to a probability measure under whichY· associated with the bounded and measurable drift coefficient b is the Levyprocess of the type in (21) and where

Ir0 := Ir0,1 + Ir0,2 + Ir0,3

23

with

Ir0,1

: =n∑

i=1

{∫ t

0bri (B

xs (θ))dB

(i)s +

∫ T

T−tbri (B

xs (θ))dW

(i)s

−∫ T

T−tbri (B

xs (θ))

B(i)s

T − sds},

Ir0,2

: =

∫ T

T−th∗(s, Bx


2

∫ T

T−t

∥∥∥h(s, Bxs−(T−t)(θ))

∥∥∥2 ds+

∫ T

T−t

∫Rm0


+

∫ T

T−t

∫Rm0


−(λ(s, Bxs−(T−t)(θ), ς)− 1))dsνr(dς)

=

∫ T

T−th∗(s, Bx


2

∫ T

T−t

∥∥∥h(s, Bxs−(T−t)(θ))

∥∥∥2 ds+

∫ T

T−t

∫Rm0

log(λ(s, Bxs−(T−t)(θ), ς))1Ur(ς)N(ds, dς, ω)

+

∫ T

T−t

∫Rm0


−(λ(s, Bxs−(T−t)(θ), ς)− 1))1Ur(ς)dsν(dς)

Ir0,3 :=

∫ T

0−(br(Bx

s (θ)))∗dBs − 1

2

∫ T

0

∥∥br(Bxs (θ))

∥∥2 dsand

Ir1

: =n∑

i=1

{∫ t

0(bri (B

xs (θ))− bi(B

xs (θ)))dB

(i)s +

∫ T

T−t(bri (B

xs (θ))− bi(B

xs (θ)))dW

(i)s

−∫ T

T−t(bri (B

xs (θ))− bi(B

xs (θ)))

B(i)s

T − sds},

24

Ir2

: =

∫ T

T−t

∫Rm0

log(λ(s, Bxs−(T−t)(θ), ς))(Nr(ds, dς, ω)− N(ds, dς, ω))

+

∫ T

T−t

∫Rm0


−(λ(s, Bxs−(T−t)(θ), ς)− 1))ds(νr(dς)− ν(dς))

=

∫ T

T−t

∫Rm0

log(λ(s, Bxs−(T−t)(θ), ς))(1Ur(ς)− 1)N(ds, dς, ω)

+

∫ T

T−t

∫Rm0


−(λ(s, Bxs−(T−t)(θ), ς)− 1))(1Ur(ς)− 1)dsν(dς),

Ir3 : =

∫ T

0−((br(Bx

s (θ)))∗ − (b(Bx

s (θ)))∗)dBs

−1

2

∫ T

0(∥∥br(Bx

s (θ))∥∥2 − ∥∥b(Bx

s (θ))∥∥2)ds.

It follows from the boundedness of the probability density p0 and Holder’sinequality that

E[(Φr(t, x, ω)− Φ(t, x, ω))2]

≤ CJr1J

r2

for a constant C < ∞, where

Jr1 := E[(Ir1 + Ir2 + Ir3)

4]1/2,

Jr2 := (

∫ 1

0E[exp(4(Ir0 + τ(Ir1 + Ir2 + Ir3)))]dτ)

1/2.

25

Using Burkholder’s inequality, we find that

Jr1

≤ K1(E[|Ir1 |4]1/2 + E[|Ir2 |4]1/2 + E[|Ir3 |4]1/2)

≤ K2(n∑

i=1

{E[

∫ T

0(bri (B

xs (θ))− bi(B

xs (θ)))

4ds]1/2

+E[

∫ T

T−t(bri (B

xs (θ))− bi(B

xs (θ)))

4ds]1/2

+E[(

∫ T

T−t

∣∣∣bri (Bxs (θ))− bi(B

xs (θ))

∣∣∣ (∣∣∣B(i)s

∣∣∣ /(T − s))ds)4]1/2}

+E[(

∫ T

T−t

∫Rm0

((log(λ(s, Bxs−(T−t)(θ), ς)))

2(1Ur(ς)− 1))2dsν(dς))2]1/2

+E[

∫ T

T−t

∫Rm0

((log(λ(s, Bxs−(T−t)(θ), ς)))

4(1Ur(ς)− 1))4dsν(dς)]1/2

+E[(

∫ T

T−t

∫Rm0

∣∣∣(log(λ(s, Bxs−(T−t)(θ), ς))− (λ(s, Bx

s−(T−t)(θ), ς)− 1))∣∣∣

· |(1Ur(ς)− 1)| dsν(dς))4]1/2

+E[

∫ T

0

∥∥br(Bxs (θ))− b(Bx

s (θ))∥∥4 ds]1/2

+E[

∫ T

0(∥∥br(Bx

s (θ))∥∥2 − ∥∥b(Bx

s (θ))∥∥2)4ds]1/2)

for constants K1,K2 < ∞. Since

E[(

∫ T

T−t

∣∣∣B(i)s

∣∣∣ /(T − s))ds)l] < ∞

for all l ≥ 1 (see Lemma 9) and since by assumption

E[(

∫ T

T−t

∫Rm0

(log(λ(s, Bxs−(T−t)(θ), ς)))

2dsν(dς))2] < ∞,

E[(

∫ T

T−t

∫Rm0

∣∣∣log(λ(s, Bxs−(T−t)(θ), ς))

∣∣∣+∣∣∣λ(s, Bx

s−(T−t)(θ), ς)− 1∣∣∣ dsν(dς))4]

< ∞,

26

it follows from dominated convergence that

Jr1 −→ 0 for r −→ ∞.

Further, we obtain by Holder’s inequality that

Jr2

= (

∫ 1

0E[exp(4(Ir0 + τ(Ir1 + Ir2 + Ir3)))]dτ)

1/2

≤ (

∫ 1

0E[exp(24Ir0,1)]

1/6E[exp(24Ir0,2)]1/6E[exp(24Ir0,3)]

1/6

E[exp(6τIr1)]1/6E[exp(6τIr2)]

1/6E[exp(6τIr3)]1/6dτ)1/2.

Using localization applied to Doleans-Dade exponentials combined with Lemma9 once more, we get that

E[exp(24Ir0,1)]

≤ E[exp(K1

∫ T

0

∥∥br(Bxs (θ))

∥∥2 ds)]2/3·E[exp(K2

∫ T

T−t

∣∣∣B(i)s

∣∣∣ /(T − s))ds)]1/3

≤ K3E[exp(K2

∫ T

T−t

∣∣∣B(i)s

∣∣∣ /(T − s))ds)]1/3

≤ K4 < ∞

for constants Ki, i = 1, ..., 4. On the other hand, repeated use of a localiza-tion argument with respect to Doleans-Dade exponentials yields

E[exp(24Ir0,2)] (46)

≤ E[exp(1128

∫ T

0

∥∥h(s, Bxs )∥∥2 ds

+48

∫ T

0

∫Rm0

∣∣1− λ(s, Bxs , ς)

∣∣ dsν(dς)+

∫ T

0

∫Rm0

∣∣1− λ48(s, Bxs , ς)

∣∣ dsν(dς))]1/2< ∞

27

Similarly to the above estimates we see from our assumptions that

E[exp(24Ir0,3)]

≤ E[exp(K

∫ T

0

∥∥br(Bxs )∥∥2 ds)]1/2

≤ M < ∞,

E[exp(6τIr1)]

≤ E[exp(K1

∫ T

0

∥∥br(Bxs (θ))− b(Bx

s (θ))∥∥2 ds)]2/3

·E[exp(K2

∫ T

T−t

∣∣∣B(i)s

∣∣∣ /(T − s))ds)]1/3

≤ C < ∞,

E[exp(6τIr2)] (47)

≤ E[exp(24

∫ T

0

∫Rm0

∣∣1− λ(s, Bxs , ς)

∣∣ dsν(dς)+

∫ T

0

∫Rm0

∣∣1− λ12(s, Bxs , ς)

∣∣ dsν(dς))]1/2< R < ∞

and

E[exp(6τIr3)]

≤ E[exp(K

∫ T

0(∥∥br(Bx

s

∥∥2 + ∥∥b(Bxs )∥∥2)ds)]1/2

≤ H < ∞.

Altogether, we obtain that for all bounded sets U ⊂ Rn :

sup0≤t≤T,x∈U

E[(Φr(t, x)− Φ(t, x))2] −→ 0 (48)

for r −→ ∞.Denote by Lr the differential operator in (28) for b = br. Then, using

the Ito-isometry, relation (48) implies that∫Rn

Φr(t, x)f(x)dx −→∫Rn

Φ(t, x)f(x)dx,

28

∫ t

0

∫Rn

Φr(s, x)Lrf(x)dxds −→∫ t

0

∫Rn

Φ(s, x)Lf(x)dxds,∫ t

0

∫Rn

Φr(s, x)h∗(s, x)f(x)dxdBs −→

∫ t

0

∫Rn

Φ(s, x)h∗(s, x)f(x)dxdBs

and ∫ t

0

∫Rm0

∫Rn

Φm(s−, x)(λ(s, x, ς)− 1)f(x)dxNr(ds, dς)

=

∫ t

0

∫Rm0

∫Rn

Φm(s−, x)(λ(s, x, ς)− 1)f(x)1Ur(ς)dxN(ds, dς)

−→∫ t

0

∫Rm0

∫Rn

Φ(s−, x)(λ(s, x, ς)− 1)f(x)dxN(ds, dς)

for r −→ ∞ in L2(Ω) uniformly in t for all f ∈ C∞c (Rn). Thus Φ ∈

L2loc([0, T ]× R

n;L2(Ω)) is an adapted process, which solves the SPDE (30)in a weak sense.

Consider now the unique strong solutions Xrt , 0 ≤ t ≤ T , r ≥ 1 of the

SDE for the signal process (8)

dXrt = br(Xr

t )dt+ dBXt , Xr

0 = x

and denote by Y rt , 0 ≤ t ≤ T the corresponding oberservation process.

It is known thatXr

t −→ Xt for r −→ ∞in L2(Ω) for all t. See e.g. [29].

Denote by Zr· , πr the Doleans-Dade exponentials and probability mea-sures in (22) with respect to br and νr, r ≥ 0, where we set b0 := b, ν0 := ν,Z· := Z0· , πr := π0. Since Xr· is independent of Y r· under πr, we find bymeans of Girsanov’s theorem applied to the signal process that

Eπr [Zrt f(X

rt )

∣∣FY r

t

](ω)

= Eϑ[exp{m∑i=1

∫ t

0hi(s, B

xs (θ))dB

i,rs (ω)− 1

2

∫ t

0

∥∥h(s, Bxs (θ))

∥∥2 ds+

∫ t

0

∫Rm0

log λ(s, Bxs (θ), ς)1Ur(ς)N

r(ds, dς, ω)

+

∫ t

0

∫Rm0

(1− λ(s, Bxs (θ), ς))1Ur(ς)dsν(dς)}

f(Bxt (θ))E(

∫ T

0(br(Bx

s (θ)))∗dBs)]

29

πr−a.e. and therefore μ−a.e., where Br· and N r are the Brownian motionand Poisson random measure under πr, respectively.

Let g be a bounded Lipschitz function on R. Then

Eμ[(g(Eπr [Zrt f(X

rt )

∣∣FY r

t

])]

= Eπ∗ [g(Eϑ[exp{m∑i=1

∫ t

0hi(s, B

xs (θ))dB

is −

1

2

∫ t

0


∥∥2 ds+

∫ t

0

∫Rm0

log λ(s, Bxs (θ), ς)1Ur(ς)N(ds, dς)

+

∫ t

0

∫Rm0

(1− λ(s, Bxs (θ), ς))1Ur(ς)dsν(dς)}

f(Bxt (θ))E(

∫ T

0(br(Bx

s (θ)))∗dBs)])]

where π∗ is a probability measure under which B· is a Brownian motion inde-pendent of a Poisson random measure N associated with the Levy measureν. By using the same reasoning as above, one sees that

Eμ[(g(Eπr [Zrt f(X

rt )

∣∣FY r

t

])] −→ Eμ[(g(Eπ[Ztf(Xt)

∣∣FYt

])]

for r −→ ∞. Hence

Eπr [Zrt f(X

rt )

∣∣FY r

t

] −→ Eπ[Ztf(Xt)∣∣FY

t

]for r −→ ∞ in distribution. Similarly, by employing the representation (44),we have that

Eπr [Zrt f(X

rt )

∣∣FY r

t

] −→ ∫Rn

f(x)Φ(t, x)dx

for r −→ ∞ in distribution for all f ∈ C∞c (Rn).

On the other hand, since Xt possesses a probability density, we alsoknow that there exists an unnormalized density Φ of the corresponding filterproblem. So we obtain that

Eμ[(g(Eπ[Ztf(Xt)∣∣FY

t

])]

= Eμ[(g(

∫Rn

f(x)Φ(t, x)dx)] = Eμ[(g(

∫Rn

f(x)Φ(t, x)dx)]

for all bounded Lipschitz functions g and f ∈ C∞c (Rn). If we now choose f

such that

f(x) =1

εnη(

y − x

ε)

30

for a standard mollifier and ε > 0, then we find for ε ↘ 0 that

Eμ[(g(Φ(t, x))] = Eμ[(g(Φ(t, x))]

x−a.e. Hence, separability implies that for all t

Φ(t, x)law= Φ(t, x)

x−a.e.

Our next result, which pertains to the regularity of solutions Φ given by(42) in the case of discontinuous drift coefficients b : R −→ R, requires thefollowing additional condition:

Eϑ[

∫ T

0

∥∥h(s, Bx1s (θ))− h(s, Bx2

s (θ))∥∥16 ds (49)

+

∫ T

0

∫Rm0

∣∣log(λ(s, Bx1s (θ), ς))− log(λ(s, Bx2

s (θ), ς))∣∣8 ν(dς)ds

+(

∫ T

0

∫Rm0


s (θ), ς))∣∣4 ν(dς)ds)2∫ T

0

∫Rm0


s (θ), ς))∣∣2 ν(dς)ds)4

+(

∫ T

0

∫Rm0

∣∣∣log(λ(s, Bx1s (θ), ς))− log(λ(s, Bx2

s−(T−t)(θ), ς))∣∣∣ dsν(dς))8

(

∫ T

0

∫Rm0

∣∣λ(s, Bx1s (θ), ς)− λ(s, Bx2

s (θ), ς)∣∣ dsν(dς))8]

≤ C(|x1 − x2|16 + |x1 − x2|8)

for all x1, x2 ∈ R, where C is a constant.

We obtain the following regularity result:

Theorem 11 Retain the conditions of Theorem 10 and suppose that (49)holds. Further, assume that the drift coefficient b in (8) is a step functionof the form

b(x) =r∑

i=1

ξi1(ai,bi](x), x ∈ R,

31

where ξi, ai, bi ∈ R, i = 1, ..., r.Then for all t a modification of the weak solution Φ(t, ·) to the SPDE (30) inTheorem 10 is locally Holder continuous with exponent α for all α ∈ (0, 1/4).

Proof. Using relation (39), we see that Φ can be written as

Φ(t, x, ω) = Eϑ[p0(Bxt (θ))I(x)],

where

I(x) := exp{∫ t

0

∫R

b(y)LBx(ds, dy)∫ T

T−th∗(s, Bx


2

∫ T

T−t

∥∥∥h(s, Bxs−(T−t)(θ))

∥∥∥2 ds+

∫ T

T−t

∫Rm0


+

∫ T

T−t

∫Rm0


−(λ(s, Bxs−(T−t)(θ), ς)− 1))dsν(dς)}E(

∫ T

0−b(Bx

s (θ))dBs).

Hence,

E[(Φ(t, x1, ω)− Φ(t, x2, ω))4] (50)

≤ CE[Eϑ[(p0(Bx1t (θ))− p0(B

x2t (θ)))4(I(x1))

4

+(I(x1)− I(x2))4]]

for a constant C. On the other hand, it follows from the mean value theoremand Holder’s inequality that

E[Eϑ[(I(x1)− I(x2))4]]

≤ CE[Eϑ[((I1(x1)− I1(x2))8 + (I2(x1)− I2(x2))

8 + (I3(x1)− I3(x2))8)]]

12

E[Eϑ[

∫ 1

0exp(8(I1(x1) + I2(x1) + I3(x1)

+τ(I1(x1)− I1(x2) + I2(x1)− I2(x2) + I3(x1)− I3(x2))))dτ)]]12 ,

where

I1(x) :=

∫ t

0

∫R

b(y)LBx(ds, dy),

32

I2(x)

: =

∫ T

T−th∗(s, Bx


2

∫ T

T−t

∥∥∥h(s, Bxs−(T−t)(θ))

∥∥∥2 ds+

∫ T

T−t

∫Rm0


+

∫ T

T−t

∫Rm0

log(λ(s, Bxs−(T−t)(θ), ς))− (λ(s, Bx

s−(T−t)(θ), ς)− 1))dsν(dς)

and

I3(x) :=

∫ T

0−b(Bx

s (θ))dBs − 1

2

∫ T

0(b(Bx

s (θ)))2ds.

Using the above notation, we have that

E[Eϑ[((I3(x1)− I3(x2))8]]

≤ C{Eϑ[(

∫ T

0b(Bx1

s (θ))dBs −∫ T

0b(Bx2

s (θ))dBs)8]

+Eϑ[(

∫ T

0(b(Bx1

s (θ)))2ds−∫ T

0(b(Bx2

s (θ)))2ds)8].

Further , it follows from the Tanaka formula and our assumptions on thedrift coefficient b that∫ T

0b(Bx

s (θ))dBs

=r∑

i=1

ξi(L(T, bi − x)− L(T, ai − x))

−r∑

i=1

ξi{(Bxs − bi)

− − (Bxs − ai)

−} a.e.

where (a)− := min(0, a), a ∈ R.So

Eϑ[(

∫ T

0b(Bx1

s (θ))dBs −∫ T

0b(Bx2

s (θ))dBs)8]

≤ C

r∑i=1

|ξi|8 {Eϑ[(L(T, bi − x1)− L(T, bi − x2))8]

+Eϑ[(L(T, ai − x1)− L(T, ai − x2))8]

+Eϑ[((Bx1s − bi)

− − (Bx2s − bi)

−)8]+Eϑ[((B

x1s − ai)

− − (Bx2s − ai)

−)8]}.

33

On the other hand, it is well known that

Eϑ[(L(t1, z1)− L(t2, z2))2l] ≤ Cn,T {|t1 − t2|l + (|z1 − z2|l} (51)

for a constant Cn,T . See e.g. [22]. Thus

Eϑ[(

∫ T

0b(Bx1

s (θ))dBs −∫ T

0b(Bx2

s (θ))dBs)8]

12

≤ C(

r∑i=1

|ξi|4)(|x1 − x2|2 + |x1 − x2|4).

Further, we also see from the occupation time formula that

Eϑ[(

∫ T

0(b(Bx1

s (θ)))2ds−∫ T

0(b(Bx2

s (θ)))2ds)8]

= Eϑ[(

∫R

(b(y))2L(T, y − x1)dy −∫R

(b(y))2L(T, y − x2)dy)8]

= Eϑ[(

∫R

(b(y))2(L(T, y − x1)− L(T, y − x2))dy)8]

CbEϑ[

∫R

(b(y))16(L(T, y − x1)− L(T, y − x2))8dy]

for a constant Cb depending on the compact support of the function b.Therefore, we see from (51) that

Eϑ[(

∫ T

0(b(Bx1

s (θ)))2ds−∫ T

0(b(Bx2

s (θ)))2ds)4]12

≤ Cb(

∫R

(b(y))16)Eϑ[(L(T, y − x1)− L(T, y − x2))8]dy)

12

≤ C(

∫R

(b(y))16dy)12 |x1 − x2|4 .

The latter yields

E[Eϑ[((I3(x1)− I3(x2))8]]

12

≤ C{|x1 − x2|2 + |x1 − x2|4}.Since

I1(x)

=

∫ t

0

∫R

b(y)LBx(ds, dy)

=

r∑i=1

ξi(L(t, bi − x)− L(t, ai − x))

34

by definition (see 35), we can employ the same reasoning as above and obtainthat

E[Eϑ[((I1(x1)− I1(x2))8]]

12

≤ C |x1 − x2|2 .Further, it follows from Burkholder’s inequality in connection with the as-sumptions (41), (49) and the inequality (51) that

E[Eϑ[(I2(x1)− I2(x2))8]]

≤ CEϑ[

∫ T

0

∥∥h(s, Bx1s (θ))− h(s, Bx2

s (θ))∥∥16 ds

+

∫ T

0

∫Rm0


s (θ), ς))∣∣8 ν(dς)ds

+(

∫ T

0

∫Rm0


s (θ), ς))∣∣4 ν(dς)ds)2∫ T

0

∫Rm0


s (θ), ς))∣∣2 ν(dς)ds)4

+(

∫ T

0

∫Rm0

∣∣∣log(λ(s, Bx1s (θ), ς))− log(λ(s, Bx2

s−(T−t)(θ), ς))∣∣∣ dsν(dς))8

(

∫ T

0

∫Rm0

∣∣λ(s, Bx1s (θ), ς)− λ(s, Bx2

s (θ), ς)∣∣ dsν(dς))8

+

∫ T

0

∣∣b(Bx1s (θ))− b(Bx2

s (θ))∣∣16 ds]

≤ C(|x1 − x2|16 + |x1 − x2|8)for a constant C.

Finally, using the same arguments as in the proof of Theorem 10, ourassumptions imply that

supx1,x2∈U×U

E[Eϑ[

∫ 1

0exp(8(I1(x1) + I2(x1) + I3(x1)

+τ(I1(x1)− I1(x2) + I2(x1)− I2(x2) + I3(x1)− I3(x2))))dτ)]]12

≤ M < ∞for bounded sets U ⊂ R, where M is a constant depending on U .

Altogether, we see from the above estimates that

E[(Φ(t, x1, ω)− Φ(t, x2, ω))4] ≤ C |x1 − x2|2

35

for all x1, x2 on bounded intervals U ⊂ R with a constant C depending onU .

So it follows from Kolmogorov’s Lemma that for all t there is a continuousmodification of Φ(t, ·), which is locally Holder continuous with exponent αfor all α ∈ (0, 1/4).

Remark 12 An example which fulfills the assumptions of Theorem 11 isgiven by (18), (19) in the case of a truncated α−stable Levy process withα ∈ (0, 1), when Ψ ∈ C3

b (R) and h = 0.

3 The Model

As mentioned in the introduction of this paper, we aim at modeling stochas-tic transition rates μik(t, x), 0 ≤ t ≤ T for states i, k ∈ S of the insured bythe following stochastic Gompertz-Makeham model GM(r, s) given by

μik(t, x) = h1,rik (t, x) + exp(h2,sik (t, x)), (52)

where h1,rik (t, x), h2,sik (t, x) are time-dependent stochastic polynomials of de-gree r and s, respectively, that is

h1,rik (t, x) =r∑

l=0

Y(l)t xl

and

h2,sik (t, x) =s∑

l=0

Y(r+1+l)t xl

for all i, k ∈ S.Here the coefficients of the polynomials

Yt = (Y(0)t , ..., Y

(r)t , Y

(r+1)t , ..., Y

(r+s)t )∗, 0 ≤ t ≤ T (53)

are described by a generalized Cox process given by


∫Rm

ςNλ(dt, dς), (54)

36

where m = r + s+ 2 and where the integer valued random measure Nλ hasa F−predictable compensator of the form

μ(dt, dς, ω) = λ(t,Xt, ς)dtν(dς)

for a Levy measure ν associated with a Levy process Lt ∈ Rm and a Borel

function λ. Further, the process Xt, 0 ≤ t ≤ T is the strong solution to theSDE


where the Brownian motion BYt ∈ R

n is independent of the Brownian motionBX

t ∈ Rm and the integer valued random measure Nλ. Here the initial value

X0 is supposed to be square integrable and to be independent of BX· , BY· ,Nλ.

An important feature of our model is the unknown ”parametrization”process Xt, 0 ≤ t ≤ T , which we use to describe the occurrence of changingtypes of jumps or types of heavy-tailedness of distributions of real data.The phenomenon of (semi-) heavy tailedness- as mentioned- may arise fromdata with cohort effects in mortality modeling or short term catastrophicevents as e.g. earthquakes. However, the ”regime switch” itself betweendifferent types of jumps, which may be of long-term nature and due toregulatory changes in the insurance branch or political decisions with a long-term impact on the economy, is modeled by the process Xt, 0 ≤ t ≤ T. Inorder to capture the ”regime switching” effects of data, we may assume thatXt, 0 ≤ t ≤ T is the strong solution to a SDE with singular drift coefficientgiven by

dXt =

⎧⎪⎨⎪⎩dXt = b(Xt)dt+ dBX

t

dYt = 0 , Y0 = a1dZt = 0 , Z0 = a2

,

X0 = (X0, Y0, Z0)∗ ∈ R

3l

where

b(x) = (56)

b(x1, ..., xl, a1, a2) =

{a1 , if ‖(x1, ..., xl)∗‖ ≥ τa2 else

for Xt := (X(1)t , ..., X

(l)t )∗ and a ”critical” threshold τ > 0. Here, the vectors

a1, a2 ∈ Rl can be interpreted as the different ”regime switching” states of

37

the jump intensity of the generalized Cox process Yt, 0 ≤ t ≤ T in (54),

depending on whether∥∥∥Xt

∥∥∥ ≥ τ or not.

A natural generalization of the model (56) to the case of multiple ”regimeswitching” states a1, ..., ar ∈ R

l is the following

b(x) = b(x1, ..., xl, a1, ..., ar) =r∑

i=1

ai1Γi(x1, ..., xl), (57)

where {Γi}i=1,...,r is a partition of Rl.An alternative model to the above ones, which is able capture long-term

effects of data, is given by the following ”regime switching” mean-reversionmodel

dXt =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

dX(1)t = b(Xt)dt+ dBX

t

dX(2)t = 0, X

(2)0 = a

dX(3)t = 0, X

(3)0 = b1

dX(4)t = 0, X

(4)0 = b2

dX(5)t = 0, X

(5)0 = τ

, (58)

where

b(x) =

{a(b1 − x) , if x ≥ τa(b2 − x) else

.

for a mean reversion coefficient a ≥ 0 and long-run average levels b1, b2 ≥ 0,depending on a ”critical” threshold τ > 0.

We mention that a unique strong solution to (58) exists. See also [3],where the authors consider the latter model in connection with a regimeswitching short rate model in finance.

In our model (52), (53) we may e.g. choose the observation function h in(54) to be a constant. In this case the process Yt, 0 ≤ t ≤ T can be regardedas a Levy process with a Levy measure ν ”parametrized” by the processXt, 0 ≤ t ≤ T.

The unknown process Xt, 0 ≤ t ≤ T or more generally f(Xt), 0 ≤ t ≤ Tfor Borel functions f can be estimated from (indirectly and under optimiza-tion constraints unique) observed insurance data Yt, 0 ≤ t ≤ T by means ofthe optimal filter

〈πt, f〉 = E[f(Xt)∣∣FY

t

], 0 ≤ t ≤ T. (59)

Using the Kallianpur-Striebel formula in Theorem 2 under the condition(14) , we may also write (59) as

〈πt, f〉 = 〈Ψt, f〉〈Ψt, 1〉 (60)

38

with〈Ψt, f〉 := Eπ[Ztf(Xt)

∣∣FYt

]for all f ∈ Cb(R

n), where Zt, 0 ≤ t ≤ T is the Doleans-Dade exponential(22).

Principally, we could now use Monte-Carlo techniques in connection withTheorem 10 to simulate the unnormalized conditional density (42) and tocompute the optimal filter by means of (60).

An alternative method to the latter one, which we want to discuss in thisSection, is the Monte-Carlo method directly applied to the unnormalizedfilter 〈Ψt, ·〉 , 0 ≤ t ≤ T .

In fact, we have the following result in the case of Lipschitz continuouscoefficients b, σ, h and λ :

Proposition 13 Assume that the functions b, σ, h and λ are bounded andsatisfy the conditions (11), (12). In addition, require (15), (16) and (17)hold. Let Xi

t , 0 ≤ t ≤ T, i ≥ 1 be a sequence of i.i.d. copies of the solutionXt, 0 ≤ t ≤ T to (55) on our probability space, being independent of Yt, 0 ≤t ≤ T, and denote by Zi

t ,0 ≤ t ≤ T the stochastic exponential in (22) basedon X i

t , 0 ≤ t ≤ T for all i ≥ 1. Let f ∈ Cb(Rn). Then

M l(f) :=1

l

l∑i=1

Zitf(X

it) −→

l−→∞〈Ψt, f〉 = Eπ[Ztf(Xt)

∣∣FYt

]a.e. (61)

for all t. Moreover, for all t there exists a constant C < ∞ such that

Eπ[(Ml(f)− 〈Ψt, f〉)2] ≤ 1

lC ‖f‖2∞ (62)

for all l ≥ 1.

Proof. Since X· is independent of Y· under π, we can represent 〈Ψt, f〉as

〈Ψt, f〉 (ω) (63)

= Eϑ[exp{m∑i=1

∫ t

0hi(s,Xs(θ))dB

is(ω)−

1

2

∫ t

0‖h(s,Xs)‖2 ds

+

∫ t

0

∫Rm0

log λ(s,Xs(θ), ς)N(ds, dς, ω)

+

∫ t

0

∫Rm0

(1− λ(s,Xs(θ), ς))dsν(dς)}f(Xt(θ))],

39

where Eϑ denotes the expectation with respect to Xs(θ), 0 ≤ s ≤ T on a sep-arate probability space. Using the latter in connection with an expectationEω in the direction of the other probability space, we get that

Eπ[(Ml(f)− 〈Ψt, f〉)2]

= Eω[Eϑ[(Ml(f)− 〈Ψt, f〉)2]]

= Eω[1

l2Eϑ[(

l∑i=1

(Zitf(X

it)−

⟨Ψi

t, f⟩))2]

= Eω[1

l2

l∑i=1

Eϑ[((Zitf(X

it)−

⟨Ψi

t, f⟩))2]

=1

lEω[Eϑ[((Z

1t f(X

1t )−

⟨Ψ1

t , f⟩))2]

≤ 4 ‖f‖2∞l

Eπ[(Zt)2].

It follows from the conditions (15), (16)) and (17) that there is a constantdepending on the sizes of h and λ such that

Eπ[(Zt)2] ≤ C.

Relation (61) is a consequence of the strong law of large numbers applied to(63).

Since we are interested to apply Proposition 13 to our model for stochas-tic transition rates in the case of discontinuous coefficients b in (56), (57) or(58) and σ = Id, we may approximate the drift coefficient b by a boundedLipschitz continuous function b. To be more precise, using the notation ofthe previous Section, we mention that under the conditions (40)-(41) we getfor bounded coefficients b that

Eπ[Ztf(Xt)∣∣FY

t

](ω)

= Eϑ[exp{m∑i=1

∫ t

0hi(s, B

xs (θ))dB

is(ω)−

1

2

∫ t

0


∥∥2 ds+

∫ t

0

∫Rm0

log λ(s, Bxs (θ), ς)N(ds, dς, ω)

+

∫ t

0

∫Rm0

(1− λ(s, Bxs (θ), ς))dsν(dς)}

f(Bxt (θ))E(

∫ T

0(b(Bx

s (θ)))∗dBs)].

40

Denote by⟨Ψb

t , ·⟩, 0 ≤ t ≤ T the unnormalized filter associated with b. Then,

using the mean value theorem, we obtain just as in the proof of Theorem 10that

sup0≤t≤T

Eπ[(⟨Ψb

t , ·⟩−

⟨Ψ˜bt , ·

⟩)2] (64)

≤ CEπ[

∫ T

0

∥∥∥b(Bxs )− b(Bx

s )∥∥∥2 ds

+(

∫ T

0

∥∥b(Bxs )∥∥2 − ∥∥∥b(Bx

s )∥∥∥2 ds)2] 12

for a constant C depending on the sizes of b, h and λ. So, if we approximateb by b in the sense that∫ T

0

∫Rn

∥∥∥b(y)− b(y)∥∥∥4 1

(2πt)n2

exp(−‖y − x‖2 /2t)dydt < 1

r2

for r ≥ 1 sufficiently large, then we see from (62) and (64) that

Eπ[(Ml,˜b(f)− 〈Ψt, f〉)2] ≤ C(

1

r+

1

l) (65)

for all l ≥ 1 and a constant C, where M l,˜b(f) is the sum in (61) with respectto b.

We remark that the estimate (64) can also be established in the caseof coefficients b in (58), since in this case one can still apply Girsanov’stheorem.

Finally, we mention that the paths of Zit , 0 ≤ t ≤ T and X i

t , 0 ≤ t ≤ Twith respect to b can be simulated by using Euler-approximation schemeapplied to the SDE’s (29) and (55).

4 Appendix

We collect in this Section some results and proofs which we need in themaintext of the article.

Lemma 14 Let Mt, 0 ≤ t ≤ T be stochastically continuous process on(Ω,F , μ). Suppose that

E[|Mt −Ms|a |Ms −Mu|b] ≤ C |t− u|1+γ

for some constants a, b, C, γ > 0 and all 0 ≤ u ≤ s ≤ t ≤ T . Then Mt, 0 ≤t ≤ T has a cadlag modification.

41

Proof. See e.g. Theorem 6.4.1 in [6] for a proof.

Proof of Lemma 4. Without loss of generality, we consider the case,when σ = Id.

Since X· is independent of Y· under π, we can represent 〈Ψt, f〉 as

〈Ψt, f〉 (ω) (66)

= Eϑ[exp{n∑

i=1

∫ t

0hi(s,Xs(θ))dB

is(ω)−

1

2

∫ t

0‖h(s,Xs)‖2 ds

+

∫ t

0

∫Rm0


+

∫ t

0

∫Rm0

(1− λ(s,Xs(θ), ς))dsν(dς)}f(Xt(θ))],

where Eϑ denotes the expectation with respect to Xs(θ), 0 ≤ s ≤ T on aseparate probability space.

Set

It =n∑

i=1

∫ t

0hi(s,Xs(θ))dB

is(ω)−

1

2

∫ t

0‖h(s,Xs)‖2 ds∫ t

0

∫Rm0


+

∫ t

0

∫Rm0

log λ(s,Xs(θ), ς)dsν(dς)

+

∫ t

0

∫Rm0

(1− λ(s,Xs(θ), ς))dsν(dς).

Using Holder’s inequality we see that

E[|〈Ψt, f〉 − 〈Ψs, f〉|2 |〈Ψs, f〉 − 〈Ψu, f〉|2]= E[Eϑ[|exp(It)(f(Xt(θ))− f(Xs(θ))) + f(Xs(θ))(exp(It)− exp(Is))|2

|exp(Is)(f(Xs(θ))− f(Xu(θ))) + f(Xu(θ))(exp(Is)− exp(Iu))|2]]≤ C(J1 + J2 + J3 + J4),

42

where

J1 : = E[Eϑ[exp(2It) exp(2Is)

(f(Xt(θ))− f(Xs(θ)))2(f(Xs(θ))− f(Xu(θ)))

2]],

J2 : = E[Eϑ[exp(2It)f(Xu(θ))2

(f(Xt(θ))− f(Xs(θ)))2(exp(Is)− exp(Iu))

2]],

J3 : = E[Eϑ[f(Xs(θ))2 exp(2Is)

(f(Xs(θ))− f(Xu(θ)))2(exp(It)− exp(Is))

2]],

J4 : = E[Eϑ[f(Xs(θ))2f(Xu(θ))

2

(exp(It)− exp(Is))2(exp(Is)− exp(Iu))

2]].

Using our assumptions we obtain by Holder’s inequality in connectionwith the independence of Brownian increments that

J1 ≤ CEϑ[(Xt(θ)−Xs(θ))4(Xs(θ)−Xu(θ))

4]1/2

≤ C |t− u|2 .

On the other hand, using the mean value theorem and Burkholder’s inequal-ity we see that

E[(exp(Is)− exp(Iu))4]

= E[(

∫ 1

0(Is − Iu) exp(Is + θ(Is − Iu))dθ)

4]

≤ CE[(Is − Iu)8]1/2

≤ C{E[(

∫ s

u‖h(r,Xr)‖2 dr)4]1/2 + E[(

∫ s

u

∫Rm0

log λ(r,Xr(θ), ς)ν(dς)dr)8]1/2

+E[(

∫ s

u

∫Rm0

(1− λ(r,Xr(θ), ς))ν(dς)ds)8]1/2

+

n∑i=1

E[(

∫ s

u(hi(r,Xr(θ)))

2dr)4]1/2

+E[

∫ s

u

∫Rm0

(log λ(r,Xr(θ), ς))8ν(dς)dr]1/2

+E[(

∫ s

u

∫Rm0

(log λ(r,Xr(θ), ς))4ν(dς)dr)2]1/2

+E[(

∫ s

u

∫Rm0

(log λ(r,Xr(θ), ς))2ν(dς)dr)4]1/2}.

43

So

E[(exp(Is)− exp(Iu))4]

≤ C{|s− u|4 + |s− u|2E[(

∫ T

0(

∫Rm0

log λ(r,Xr(θ), ς)ν(dς))2dr)4]1/2

+ |s− u|2E[(

∫ T

0(

∫Rm0

(1− λ(r,Xr(θ), ς))ν(dς))2ds)4]1/2

+ |s− u|2 + |s− u| 14 E[(

∫ T

0(

∫Rm0

(log λ(r,Xr(θ), ς))8ν(dς))2dr)1/2]1/2

+ |s− u| 12 E[(

∫ T

0(

∫Rm0

(log λ(r,Xr(θ), ς))4ν(dς))2dr)]1/2

+ |s− u|E[(

∫ T

0(

∫Rm0

(log λ(r,Xr(θ), ς))2ν(dς))2dr)2]1/2}

≤ C |t− u| 14 .

We also see by using Lipschitzianity that

Eϑ[(f(Xt(θ))− f(Xs(θ)))8]1/4 ≤ C |t− u| .

Hence by our assumptions we get that

J2 ≤ C |t− u|1+ 18 .

Similarly, we see that

J3 ≤ C |t− u|1+ 18 .

Because of the mean value theorem and the independence of incrementsof Levy processes we obtain from our assumptions that

E[Eϑ[(exp(It)− exp(Is))2(exp(Is)− exp(Iu))

2]]

= E[Eϑ[(

∫ 1

0(It − Is) exp(Is + θ1(Is − Iu))dθ1)

2

(

∫ 1

0(Is − Iu) exp(Iu + θ1(Is − Iu))dθ1)

2]]

≤ CEϑ[E[(It − Is)4(Is − Iu)

4]] = CEϑ[E[(It − Is)4]E[(Is − Iu)

4]].

44

Further, Holder’s and Burkholder’s inequality implies that

E[(Is − Iu)4]

≤ C{E[(

∫ s

u‖h(r,Xr)‖2 dr)2] + E[(

∫ s

u

∫Rm0

log λ(r,Xr(θ), ς)ν(dς)dr)4]

+E[(

∫ s

u

∫Rm0

(1− λ(r,Xr(θ), ς))ν(dς)ds)4]

+

n∑i=1

E[(

∫ s

u(hi(r,Xr(θ)))

2dr)2]

+E[

∫ s

u

∫Rm0

(log λ(r,Xr(θ), ς))4ν(dς)dr]

+E[(

∫ s

u

∫Rm0

(log λ(r,Xr(θ), ς))2ν(dς)dr)2]}

≤ C{|s− u|2 + |s− u|2E[(

∫ T

0(

∫Rm0

log λ(r,Xr(θ), ς)ν(dς))2dr)2]

+ |s− u|2E[(

∫ T

0(

∫Rm0

(1− λ(r,Xr(θ), ς))ν(dς))2ds)2] + |s− u|2

+ |s− u| 23 E[(

∫ T

0(

∫Rm0

(log λ(r,Xr(θ), ς))4ν(dς))3dr)1/3]

+ |s− u|E[(

∫ T

0(

∫Rm0

(log λ(r,Xr(θ), ς))2ν(dς))2dr)]}

≤ |t− u| 23 (K + CL(θ))

θ−a.e., where

L(θ) : = E[(

∫ T

0(

∫Rm0

log λ(r,Xr(θ), ς)ν(dς))2dr)2]

+E[(

∫ T

0(

∫Rm0

(1− λ(r,Xr(θ), ς))ν(dς))2ds)2]

+E[(

∫ T

0(

∫Rm0

(log λ(r,Xr(θ), ς))4ν(dς))3dr)1/3]

+E[(

∫ T

0(

∫Rm0

(log λ(r,Xr(θ), ς))2ν(dς))2dr)].

45

Hence

Eϑ[E[(It − Is)4]E[(Is − Iu)

4]]

≤ |t− u| 43 Eϑ[(K + CL(θ))2]

≤ C |t− u| 43 ·

·(K + Eϑ[E[(

∫ T

0(

∫Rm0

log λ(r,Xr(θ), ς)ν(dς))2dr)4]]

+Eϑ[E[(

∫ T

0(

∫Rm0

(1− λ(r,Xr(θ), ς))ν(dς))2ds)4]]

+Eϑ[E[(

∫ T

0(

∫Rm0

(log λ(r,Xr(θ), ς))4ν(dς))3dr)2/3]]

+Eϑ[E[(

∫ T

0(

∫Rm0

(log λ(r,Xr(θ), ς))2ν(dς))2dr)2]]).

Altogether, it follows that

E[|〈Ψt, f〉 − 〈Ψs, f〉|2 |〈Ψs, f〉 − 〈Ψu, f〉|2] ≤ C |t− u|1+ 18

for a constant C < ∞ depending on f , which gives the proof in connectionwith Lemma 14.

Proof of Lemma 9. It follows from (39) that∫ t

0

∫R

sgn(x)L(ds, dx)

=

∫ t

0sgn(Bs)dBs +

∫ T

T−tsgn(Bs)dWs −

∫ T

T−tsgn(Bs)

Bs

T − sds

=

∫ t

0sgn(Bs)dBs +

∫ T

T−tsgn(Bs)dWs −

∫ T

T−t

∣∣∣Bs

∣∣∣T − s

ds,

where

sgn(x) :=

{1 , if x > 0−1 else

.

Since ∫ t

0

∫R

sgn(x)L(ds, dx) = −2 |Bt|+ 2

∫ t

0sgn(Bs)dBs

46

by means of Tanaka’s formula, we find that

∫ t

0

|Bs|s

ds =

∫ T

T−t

∣∣∣Bs

∣∣∣T − s

ds

= −∫ t

0sgn(Bs)dBs +

∫ T

T−tsgn(Bs)dWs + 2 |Bt| .

Using the latter combined with the supermartingale property of Doleans-Dade exponentials and Holder’s inequality, we get that

E[exp(k

∫ T

0

|Bt|t

dt)]

≤ E[exp(−3k

∫ t

0sgn(Bs)dBs)]

1/3E[exp(3k

∫ T

T−tsgn(Bs)dWs)]

1/3 ·

·E[exp(6k |Bt|)]1/3≤ CT,k < ∞

for a constant CT,k depending on T and k.

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