In Copyright - Non-Commercial Use Permitted Rights / License: …23396/eth-23396-02.pdf · Stochastic Equations in Infinite Dimensions 9 1.1. Infinite DimensionalBrownianMotion 9

Research Collection

Doctoral Thesis

Consistency problems for HJM interest rate models

Author(s): Filipović, Damir

Publication Date: 2000

Permanent Link: https://doi.org/10.3929/ethz-a-003882242

Rights / License: In Copyright - Non-Commercial Use Permitted

This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.

ETH Library

https://doi.org/10.3929/ethz-a-003882242

http://rightsstatements.org/page/InC-NC/1.0/

https://www.research-collection.ethz.ch

https://www.research-collection.ethz.ch/terms-of-use

Diss. ETH No. 13603

Consistency Problems

for

HJM Interest Rate Models

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY

ZURICH

for the degree of

Dr. sc. math.

presented byDAMIR FILIPOVIC

Dipl. Math. ETH

born March 26, 1970

citizen of Schmiedrued AG

accepted on the recommendation of

Prof. Dr. F. Delbaen, examiner

Prof. Dr. P. Embrechts, co-examiner

2000

I

Meinen Eltern gewidmet

Contents

Abstract iii

Zusammenfassung v

Introduction 1

Part 1. The HJM Methodology 7

Chapter 1. Stochastic Equations in Infinite Dimensions 9

1.1. Infinite Dimensional Brownian Motion 9

1.2. The Stochastic Integral 11

1.3. Fundamental Tools 13

1.4. Stochastic Equations 16

Chapter 2. The HJM Methodology Revisited 19

2.1. The Bond Market 19

2.2. The Musiela Parametrization 20

2.3. Arbitrage Free Term Structure Movements 24

2.4. Contingent Claim Valuation 28

A Summary of Conditions 32

2.5. What Is a Model? 32

Chapter 3. The Forward Curve Spaces Hw 35

3.1. Definition of Hw 35

3.2. Volatility Specification 40

3.3. The Yield Curve 41

3.4. Local State Dependent Volatility 42

3.5. Functional Dependent Volatility 45

3.6. The BGM Model 46

Chapter 4. Consistent HJM Models 51

4.1. Consistency Problems 51

4.2. A Simple Criterion for Regularity of G 53

4.3. Regular Exponential-Polynomial Families 54

4.4. Affine Term Structure 58

Part 2. Publications 63

Chapter 5. A Note on the Nelson-Siegel Family 65

5.1. Introduction 65

5.2. The Interest Rate Model 66

ii CONTENTS

5.3. Consistent State Space Processes 67

5.4. The Class of Consistent Itô Processes 69

5.5. E-Consistent Itô Processes 73

Chapter 6. Exponential-Polynomial Families and the Term Structure of

Interest Rates 75


6.2. Consistent Itô Processes 77

6.3. Exponential-Polynomial Families 78

6.4. Auxiliary Results 81

6.5. The Case BEP(l,n) 84

6.6. The General Case BEP{K, n) 86

6.7. E-Consistent Itô Processes 92

6.8. The Diffusion Case 93

6.9. Applications 95

6.10. Conclusions 97

Chapter 7. Invariant Manifolds for Weak Solutions to Stochastic Equations 99


7.2. Preliminaries on Stochastic Equations 101

7.3. Invariant Manifolds 105

7.4. Proof of Theorem 7.6 106

7.5. Proof of Theorems 7.7, 7.9 and 7.10 110

7.6. Consistency Conditions in Local Coordinates 113

Appendix: Finite Dimensional Submanifolds in Banach Spaces 115

Bibliography 121

Curriculum Vitae 123

Abstract

This thesis brings together estimation methods and stochastic factor models

for the term structure of interest rates within the HJM framework. It is based on

the complex of consistency problems introduced by Björk and Christensen [7].There exist commonly used methods for fitting the current term structure.

Any curve fitting method can be represented as a parametrized family of smooth

curves G = {G(-,z) \ z £ Z} with finite dimensional parameter set Z. A lot

of cross-sectional data z is available, and one may ask for a suitable stochastic

model for z which provides accurate bond option prices. However, this requires

the absence of arbitrage. We characterize all consistent i?-valued state space Itô

processes Z which, by definition, provide an arbitrage-free model when representingthe parameter z. Obviously, G{T — t,Zt) has to satisfy the HJM drift condition.

It turns out that selected common curve fitting methods do not go well with the

HJM framework.

We then consider the preceding consistency problem from a geometric point of

view, as proposed by Björk and Christensen [7]. We extend the HJM framework to

incorporate an infinite dimensional driving Brownian motion. We then perform the

change of parametrization due to Musiela [37] and arrive at a stochastic equation in

a Hilbert space H, describing the arbitrage-free evolution of the forward curve. The

family Q can be treated as a subset of H and the above consistency considerations

yield a stochastic invariance problem for the previously derived stochastic equation.Under the assumption that G is a regular submanifold of H, we derive sufficient

and necessary conditions for its invariance. Expressed in local coordinates theyturn out to equal the HJM drift condition.

Classical models, such as the Vasicek [48] and CIR [17] short rate model, and

the popular BGM [12] LIBOR rate model, are shown to fit well into that framework.

By their very definition, affine HJM models are consistent with finite dimensional

linear submanifolds of H. A straight application of our results yields a completecharacterization of them, as obtained by Duffie and Kan [21].

In conclusion, we provide a general tool for exploiting the interplay between

curve fitting methods and HJM factor models.

Zusammenfassung

Die vorliegende Arbeit behandelt den Zusammenhang zwischen Schätzmetho¬

den und HJM-Faktormodellen für die Zinsstrukturkurve. Sie ist motiviert durch den

Themenkomplex von Konsistenzproblemen, welcher von Björk und Christensen [7]eingeführt wurde.

Es gibt geläufige Methoden, um die aktuelle Zinsstrukturkurve zu schätzen.

Jede dieser Methoden führt zu einer parametrisierten Familie von glatten Kur¬

ven G = {G{-,z) | z e Z} mit einer endlichdimensionalen Parametermenge Z.

Eine Fülle von täglichen Schätzdaten für z ist erhältlich und man mag daher nach

einem geeigneten stochastischen Modell für den Parameter z verlangen, welches

entsprechend genaue Bondoptionspreise liefert. Dies setzt allerdings Arbitrage¬freiheit voraus. Wir charakterisieren alle konsistenten Z-wertigen Zustandsraum-

Itôprozesse Z, dadurch definiert, dass GiT—t, Zt) ein arbitragefreies Modell erzeugt.

Notwendigerweise hat G(T — t, Zt) die HJM-Driftbedingung zu erfüllen. Es stellt

sich heraus, dass sich häufig verwendete Schätzmethoden schlecht mit diesen Rah¬

menbedingungen vertragen.

Wir betrachten anschliessend das vorhergehende Konsistenzproblem aus geo¬

metrischer Sicht, wie von Björk und Christensen [7] vorgeschlagen. Wir erweit¬

ern den HJM-Rahmen durch Einbeziehung einer unendlichdimensionalen zugrun¬

deliegenden Brownschen Bewegung. Wir führen Musielas [37] Parameterwechsel

durch und leiten eine stochastische Gleichung in einem Hilbertraum H her, welche

die arbitragefreie Evolution der Zinskurve beschreibt. Die Familie G kann als Un¬

termenge von H aufgefasst werden, wodurch obige Konsistenzbetrachtungen ein

stochastisches Invarianzproblem für die eben hergeleitete stochastische Gleichunghervorrufen. Unter der Annahme, dass G eine reguläre Untermannigfaltigkeit von

H ist, leiten wir hinreichende und notwendige Bedingungen für ihre Invarianz

her. Ausgedrückt in lokalen Koordinaten stellen sich letztere gerade als HJM-

Driftbedingung heraus.

Klassische Modelle wie diejenigen von Vasicek [48] und CIR [17] oder das

bekannte BGM [12] LIBOR-Modell sind im vorliegenden Rahmen enthalten. Affine

Modelle sind definitionsgemäss konsistent mit endlichdimensionalen linearen Un¬

termannigfaltigkeiten von H. Unsere Resultate können daher direkt angewendetwerden und ergeben eine vollständige Charakterisierung der affinen Modelle nach

Duffie und Kan [21].Zusammengefasst vermitteln wir eine allgemeine Methode zur Untersuchung

des Zusammenspiels zwischen Kurvenschätzverfahren und HJM-Faktormodellen.

Introduction

Bond markets differ in one fundamental aspect from standard stock markets.

While the latter are built up by a finite number of traded assets, the underlying

basis of a bond market is the entire term structure of interest rates: an infinite

dimensional variable which is not directly observable. On the empirical side this

necessitates curve fitting methods for the daily estimation of the term structure.

Pricing models on the other hand, are usually built upon stochastic factors repre¬

senting the term structure in a finite dimensional state space.

The aim of this thesis is threefold: to bring together estimation methods and

factor models for interest rates, to provide appropriate consistency conditions and

to explore some important examples.

By a bond with maturity T we mean a default-free zero coupon bond with

nominal value 1. Its price at time t is denoted by P(t,T). There is a one to one

relation between the time t term structure of bond prices and the time t term

structure of interest rates or forward curve {f(t,T) \ T > t} given by

P(t,T) = exp(- J f(t,s)ds).

Accordingly, f(t,T) is the continuously compounded instantaneous forward rate

for date T prevailing at time t. The forward curve contains all the necessary

information for pricing bonds, swaps and forward rate agreements of all maturities.

Furthermore, it is a vital indicator for central banks in setting monetary policy.Several algorithms for constructing the current term structure of interest rates

from market data are applied in practice. A recent document from the Bank for

International Settlements (BIS [6]) provides an overview of all estimation methods

used by some of the most important central banks, see Table 1 on page 76. Promi¬

nent among them are the curve fitting procedures by Nelson and Siegel [39] and

Svensson [44] and smoothing splines. Any common method can be represented as

a parametrized family G = {G(-,z) | z 2} of smooth curves where Z C Mm is a

finite dimensional parameter set (m £ N). By appropriate choice of the parameter

z g Z, an optimal fit of the forward curve x ^ G(x,z) to the observed data is

achieved. Here x > 0 denotes time to maturity. In that sense z represents the

current state of the economy, taking values in the state space Z.

According to [6] a lot of cross-sectional data, i.e. daily estimations of z, is

available for selected estimation methods G- In addition there exists an exten¬

sive literature on statistical inference for diffusion processes based on discrete time

observations, see e.g. [5]. It therefore seems promising to explore the stochastic

evolution of the parameter z continuously over time.

l

2 INTRODUCTION

Henceforth we are given a complete filtered probability space (Q, !F, (Js't)teM+, P)satisfying the usual conditions. Let Z = {Zt)tçu+ be a diffusion model for the above

parameter z. One would think that now f(t, T) = G(T — t, Zt) provides an accurate

factor model for the interest rates. However, there are constraints on the dynamicsof Z, such as the requirement of absence of arbitrage. By this we mean the existence

of a probability measure Q ~ P under which discounted bond price processes follow

local martingales (Q is called an equivalent local martingale measure or risk neutral

measure). Necessary conditions can be formulated in terms of the generator of Z

applied to G. These conditions turn out to be very restrictive. For G fixed we arrive

at a type of inverse problem for the generator of Z. State space processes Z which

comply with these constraints, are called consistent with Q. It can be shown for

example that the Nelson-Siegel family admits no non-trivial consistent diffusion

Z. We achieve these negative results even for generic state space Itô processes

Z (thereby the inverse problem applies to the characteristics of Z instead of the

generator as for diffusions).

An application of Itô's formula shows that the preceding approach is part of

the framework introduced by Heath, Jarrow and Morton (henceforth HJM) [28]which basically unifies all continuous interest rate models. For arbitrary but fixed

T £ M+, HJM let /(•, T) evolve as an Itô process

/(t,T) = /(0,T)+ f a{s,T)+ [ a(s,T)dWs, t £ [0,T].Jo Jo

(1)

Originally in [28], W is a finite dimensional Brownian motion. We will however ex¬

tend this approach at once by considering an infinite dimensional Brownian motion

W. This generalization is straightforward but one has to take care of the definition

of the stochastic integral.The no-arbitrage requirement cumulates in the well-known HJM drift condi¬

tion: existence of an equivalent local martingale measure Q yields a representationfor the Q-drift of /(, T) in terms of a. Thus, pricing formula for interest rate sensi¬

tive contingent claims do only depend on a. The above mentioned inverse problemfor the characteristics of Z is simply this drift condition.

Note that (1) provides a generic description of the arbitrage-free evolution

rather than a model for the interest rates. By HJM's result it is convenient and

standard to describe the evolution of f(-,T) under the risk neutral measure Q,

involving only a and the Girsanov transform W oi W.

Musiela [37] proposed the reparametrization

rt(x) = f(t,x + t), (2)

taking better into account the nature of the forward curve x \-> rt (x) as an (infinitedimensional) state variable, where x > 0 denotes time to maturity. We ask: what

is the implied stochastic dynamics for (rt(x))teM+tThe semigroup theory for stochastic equations in a Hilbert space provided by

Da Prato and Zabczyk [18] is tailor-made for that approach. We give an axiomatic

scheme for the choice of an admissible Hilbert space H of functions h : M+ -> M. in

which (rt)t£K+ can be realized as a solution to the stochastic equation

j drt = {Art + FHJM(t, n)) dt + a(t, rt) dWt

1 r0=h0.[â)

INTRODUCTION 3

Here A is the generator of the semigroup of right-shifts S(t)h(x) = h(x + t). The

coefficients1 a = <r{b,w, h) and Fhjm = ^hjm(£, w, h) are random mappings. The

latter is fully specified by a according to the HJM drift condition under the measure

<Q> (under P the market price of risk is also involved). Equation (3) with /ig = /(0, )represents the system of infinitely many Itô processes (1) as one dynamical system

in infinite dimensions, now allowing for arbitrary initial curves ro £ H.

The axiomatic exposure of an admissible Hilbert space of forward curves is

new. We propose a particular class of admissible Hilbert spaces, which are both

economically reasonable and appropriate for providing existence and uniqueness

results for (3) in full generality. Any Lipschitz continuous and bounded volatilitycoefficient a provides an HJM model therein. Classical models are shown to fit well

into that framework. These include the short rate models of Vasicek [48] and Cox,

Ingersoll and Ross (henceforth CIR) [17], and also the popular LIBOR model by

Brace, Gatarek and Musicla (henceforth BGM) [12].Equation (3) was introduced and analyzed first in [37] and [13], but only

for deterministic volatility structure. In that Gaussian framework Musiela also

characterized the set of attainable forward curves by the model. This is an issue

which is strongly related to the present discussion. Further aspects of equation (3)have been exploited by several authors. Among them are Björk et al. [7], [8], [10],Zabczyk [50], Vargiolu [47], [46] and Milian [34]. Research is ongoing. A rigorous

exposure of the no-arbitrage property for equation (3) as provided by this thesis

therefore seems to be needed.

In view of the previously mentioned negative consistency results for common

estimation methods, one can proceed in two possible ways. Either one consid¬

ers more general non-continuous (Markovian) state space processes Z, or one stayswithin the continuous (that is HJM) regime and looks for appropriate forward curve

families G- The first approach is not a topic of this thesis and will be exploited

elsewhere, see [26]. The second approach leads naturally to the complex of con¬

sistency problems for HJM models introduced by Björk and Christensen [7]. They

propose the following geometric point of view.

Considering G as a subset of H, we can study the stochastic invariance problemrelated to (3) for G- The set G is called (locally) invariant for (3) if for any space-

time initial point (t0,h0) el+xÇ, the solution r = r(ta'h°ï of (3) stays (locally)in G almost surely. Accordingly, we call the HJM model provided by (3) consistent

with G-

By a solution to (3) we mean a mild or weak solution rather than a strongsolution. These three concepts will be defined below, see Definition 1.19. It is

essential, however, to notice that a mild or a weak solution is not an iî-valued

semimartingale.The existence of a state space diffusion process Z which is (locally) consistent

with G for any initial point (t0,zo) £ M.+ x Z obviously implies consistency of the

induced HJM model G(-,Z) with Q. The converse is not true in general: if the

model (3) is consistent with G it's not clear what the i?-valued coordinate process

Z determined by

G(;Zt)=n (4)

xBy a slight abuse of notation we use the same letter a in (3) as well as (1), although theyrepresent different objects.

4 INTRODUCTION

looks like. We cannot expect it to be an Itô process.

We provide a general regularity result for this problem. Suppose G is a finite

dimensional regular submanifold of H and locally invariant for a stochastic equation

like (3). Then the coordinate process Z in (4) is a Z-valued diffusion. Furthermore,we find Nagumo-type consistency conditions in terms of <x and Fhjm and the

tangent bundle of G- Expressing these consistency conditions in local coordinates we

arrive at exactly the above inverse problem for the characteristics of the coordinate

(i.e. state space) process Z. That way the consistency problem for G is completelysolved.

These results are then applied to the estimation methods provided by [6]. Under

some technical restrictions, we show that exponential-polynomial families, such as

those of Nelson-Siegel or Svensson, are regular submanifolds of H. Thus, the

previously derived negative consistency results can be restated within the present

setup.

A further application yields the characterization of all HJM models which are

consistent with linear submanifolds. These are just the affine HJM models. We

derive the well-known results of Duffie and Kan [21] from our general point of view.

The present consistency results have been basically established by Björk and

Christensen [7]. However, they proceeded under much stronger assumptions which

are not necessarily convenient for the HJM framework, see Remark 2.16 below.

Moreover, they did not address the difference between a regular and an immersed

submanifold, see Figure 1 on page 52. By exploiting the strong structure of the

regular submanifold G we derive global invariance results in contrast to [7] where

only local results are provided. New is also the representation of the consistencyconditions in local coordinates, making them feasible for applications, which work

mainly in one direction: given a curve fitting method G, we can determine the

consistent HJM models. The converse problem of the existence of an invariant finite

dimensional submanifold, given a particular HJM model, is treated in Björk and

Svensson [10], by using the Frobenius theorem. The non-consistency of exponential-

polynomial families has been proved in [7] for the particular case of a deterministic

volatility structure.

Similar stochastic invariance problems have been studied by Zabczyk [50],Jachimiak [30] and for finite dimensional systems by Milian [33] (see also the refer¬

ences therein). Björk et al. [8], [10] discuss further the existence of (minimal) finite

dimensional realizations for (3). In [50] the invariance question for finite dimen¬

sional linear subspaces with respect to Ornstein-Uhlenbeck processes is resolved.

Applications to HJM models, the BGM model and to a second order term struc¬

ture model by Cont [16] are exploited. But the support theorem methods in [50]have not yet been established for the general equation (3). Jachimiak [30] exploitsinvariance for closed sets K C H. However, his methods cannot be applied directlyto (3).

Outline. The thesis is divided into two parts. Part 2 contains a series of

three articles ([25], [23], [24]) developed during my Ph.D. research over the last

three years. All have been accepted for publication by renowned journals. Part 1

represents a synthesis of the three articles cumulating in the central Chapter 4.

INTRODUCTION 5

In Chapter 1 we repeat the relevant material from [18] about stochastic anal¬

ysis in infinite dimensions. We define an infinite dimensional Brownian motion

W and show how it can be realized as Hilbert space valued Wiener process. The

construction of the stochastic integral with respect to W is then sketched and we

provide the main tools from stochastic analysis: Itô's formula, the stochastic Fubini

theorem and Girsanov's theorem. We introduce stochastic equations in a Hilbert

space, define the concepts for a solution and state an existence and uniquenessresult.

Chapter 2 recaptures the HJM [28] methodology and extends it to incorporatean infinite dimensional driving Brownian motion W. We perform the change of

parametrization (2) and arrive at the stochastic equation (3) in a Hilbert space

H which is axiomatically scheduled. The no-arbitrage (or HJM drift) condition is

rigorously exposed. We give a sketch for contingent claim valuation in the precedingbond market and discuss in particular the forward measure, forward LIBOR rates

and caplets. This is introductory for the BGM model to be recaptured in the

subsequent chapter. Finally, we define what we mean by an HJM model.

In Chapter 3 we introduce a class of Hilbert spaces Hw which are both eco¬

nomically reasonable and convenient for analyzing the previously derived stochastic

equation (3). We provide a general existence and uniqueness result for Lipschitzcontinuous and bounded volatility coefficients a. Classical models are shown to fit

well into that framework. So is the classical HJM model which corresponds to lo¬

cally state dependent volatility coefficients a. We also recapture the popular BGM

model and provide further promising examples.

Chapter 4 is central and can be considered as the main chapter. It unifies the

results obtained in [25], [23] and [24] within the previously constructed framework.

We state the main result for regular families G and thus solve completely the consis¬

tency problem. We discuss some pitfalls which arise from the subtle but importantdifference between a regular and an immersed submanifold. A simple criterion

for differentiability in Hw is presented and applied to the class of exponential-polynomial families. We restate the (negative) consistency results for the common

Nelson-Siegel and Svensson families. Finally, we apply our methods for identifyingthe class of affine term structure models which are characterized by possessing lin¬

ear invariant submanifolds. We provide some of the well-known results from [21]and recapture the Vasicek and CIR short rate models.

Chapter 5 is [25] and shows that there exists no non-trivial state space Itô

process which is consistent with the Nelson-Siegel family.Chapter 6 is [23]. Here we extend the methods used in Chapter 5 for char¬

acterizing the class of state space Itô processes which are consistent with generalexponential-polynomial families. We obtain remarkably restrictive results. In par¬

ticular we identify the only non-trivial HJM model which is consistent with the

Svensson family: an extended Vasicek short rate model. Chapters 5 and 6 were

motivated by the examples given in [7], see Remarks 5.3 and 7.1 therein.

Chapter 7 is an extended version (with an additional section and completeproofs) of [24]. It is not directly related to interest rate models and providesgeneral results on stochastic invariance for finite dimensional submanifolds in a

Hilbert space. The main results of [7] are considerably extended.

The appendix includes a comprehensive discussion on finite dimensional sub¬

manifolds in Banach spaces. Their crucial properties are deduced.

6 INTRODUCTION

Remark on Notation. Basically, we follow the notation of [40] and [18].Let G and H be Banach spaces. The norm of H is denoted by || • \\h and we

write B(H) for the cr-algebra generated by its topology. If H is a Hilbert space we

use the notation (•, •)# for its scalar product. The closure of a subset M c H is

denoted by M. We define the open ball of radius R > 0

BR{H) :={h£H\ \\h\\H < R}.

The space of continuous linear operators from G into H is denoted L(G; H)and L(H) = L(H;H) for short. We write D(A) for the domain of an (unbounded)linear operator A: G -> H and A* for its adjoint.

We write C(G; H) and Ck(G; H) for the space of continuous and k times con¬

tinuously differentiable mappings </> : G -» H, respectively. The derivatives are

denoted by D<f> and Dk(ß.For U C Rm, m £ N, the space of continuous, resp. k times continuously

differentiable functions / : U —> M is written shortly C(U), resp. Ck(U), and

Cç{U) consists of those elements of Ck{U) which have compact support in U.

If (fi, J7, P) is a measure space and p > 1, we write Lp(fi, J7, P) for the Banach

space of integrable functions2 X : H —» R equipped with norm

If fi is a Borel subset of M and P the Lebesgue measure, we just write IP (Ü).The space of locally integrable functions3 h : Ü -» R is denoted by 1^(0).

Letters such as (7, (7, -ft', ... represent positive real constants which, if not

otherwise stated, can vary from line to line.

All remaining notation is either standard and self-explanatory or it is intro¬

duced and explained in the text.

Acknowledgements. It is a great pleasure to thank my adviser F. Delbaen

for his guidance throughout my Ph.D. studies. He has given me his time and his

insights in the course of many stimulating conversations. I am very grateful to

him for his generosity and for giving me the opportunity of several educational

excursions abroad.

I have profited from fruitful discussions with many people at several meetingsand seminars. Very special thanks go to T. Björk and J. Zabczyk for their helpfuland motivating suggestions, and their hospitality, which have been essential for

the development of this thesis. I would like to thank also B. J. Christensen and

G. Da Prato for their interest and hospitality. Furthermore, I thank P. Embrechts

for being co-examiner.

Thanks go to Axel Schulze-Halberg, Paul Harpes, Freweini Tewelde, Uwe

Schmock and all my colleagues from ETH for their friendship and support, and

especially to Manuela for her love over the last years.

Financial support from Credit Suisse is gratefully acknowledged.

2In fact, equivalence classes of functions where X ~ Y if X = Y P-a.s.

3That is, equivalence classes (h ~ g if h = g a.s.) of measurable functions h satisfying

/ \h{x)\dx <oo VÄSK+.

Part 1

The HJM Methodology

CHAPTER 1

Stochastic Equations in Infinite Dimensions

In this chapter we provide some introduction to infinite dimensional stochastic

analysis and sketch the relevant material from Da Prato and Zabczyk [18] without

giving detailed proofs. For terminology and notation we refer to [18] and [40].Here and subsequently, (ft, T, (Jrt)teK+, P) stands for a complete filtered prob¬

ability space satisfying the usual conditions1. The predictable u-field is denoted by

V. We will suppress the notational dependence of stochastic variables on cj when

no confusion can arise.

Throughout, H denotes a separable Hilbert space.

Let / C R+. A stochastic process (t,u) >-» Xt(uj) : I x ft -> H will be written

indifferently2 (Xt)tei-> resp. X or (Xt) if there is no ambiguity about the index

set /. Whereas Xt : ft —>• H is a random variable for each t £ I.

Two processes (Xt)tei and (Yt)tei are indistinguishable if

V[Xt = Yt, Vt £ I] = 1.

We do not distinguish between them and write X — Y.

1.1. Infinite Dimensional Brownian Motion

What is an ü-valued "standard Brownian motion" W1 A natural requirementwould be that (W, h)jj were a real Brownian motion with

E[(Wt,h)H(Wt,h')H}=t(h,ti)H, \/h,h' £ H.

If dim H < oo this certainly applies. But in general

E [\\Wt\\2H] = EE [KWi> W2] = £ * = oo

where {hj \ j £ N} is an orthonormal basis in H. Whence Wt does not exist in H,see [49]. Usually one relaxes the condition on Wt belonging to H. Yor [49] gives a

comprehensive overview of the different concepts for defining W.

However, we will use a more direct approach here. For us, an infinite dimen¬

sional Brownian motion is a sequence

W = {ß1)3& (1.1)

of independent real-valued standard (Tt)-Brownian motions. For being consistent

with [18] we show first how W can be realized as a Hilbert space valued Wiener

process.

1 J7 is P-complete, [Ft) is increasing and right continuous, To contains all P-nullsets of T

2Differing notations like (B(t))t6R+ or (F>(*>^'))te[o,T] are self-explanatory.

9

10 1. STOCHASTIC EQUATIONS IN INFINITE DIMENSIONS

For that purpose we introduce the usual Hilbert sequence space

et

^,'CW '

and denote by {gj \ j £ N} the standard orthonormal basis3 in I2. As mentioned

above, W = (ßj)jGn = ^L/j^ß^dj does not exist in (?. In the notation of [18,Section 4.3.1], W is a cylindrical Wiener process in ir. It can be realized in a largerspace as follows.

Let À = (Aj)jgN be a sequence of strictly positive numbers with V. Xj < oo.

We define the weighted sequence Hilbert space

L = (vj)j& 6 KN |M|| := £ A>/ < oo

Clearly, ^2 C ^| with Hilbert-Schmidt embedding, and tj := (Xj) vgj form an

orthonormal basis in t\. Define Q £ L(i\) by Qej := XjCj. Obviously, the operator

Q is strictly positive, self-adjoint and nuclear. Moreover we have Q^((2\ = I2 and

Q~ï : I2 -^ l\ is an isometry since for u,v £ (?

Uj

(u,v)t2 = J2ujvi = J2xi^r.~h. = ^ *U>Q *vh- (L2)

Fix T £ R+.

Definition 1.1. We denote byl-Lj-(H) the space of H-valued continuous mar¬

tingales4 M on [0, T] such that

\\M\\2nm:=E[\\MT\\l] <co. (1.3)

Write My. := sup4e[0T] HMtH^. Doob's inequality [18, Theorem 3.8] says

E[|M^|2] <M\M\\hT(H), iorM£H2T. (1.4)

Using (1.4) it can be shown as in [18, Proposition 3.9] that H^H) is a Hilbert

space for the norm (1.3). Denote by 'H<^2(H) the closed subspace consisting of

M £ U2T(H) with M0 = 0.

We adopt the terminology of [18, Section 4.1].

Definition 1.2. An t\-valued continuous martingale X is called a Q-Wienerprocess if

i) X0 = 0

ii) X has independent increments

iii) Xt — Xs is centered Gaussian0 with covariance operator (t — s) Q, for all

0 < s < t.

Proposition 1.3. The series

W

=

jeN

W

= Y,ßJ9j

converges in Wj. (£\) for all T £ R+ and defines a Q-Wiener process in t\.

3That is, 91 = (1,0,... ), 32 = (0,1,0,...),4see [18, Section 3.4]5see [18, Section 2.3.2]

1.2. THE STOCHASTIC INTEGRAL 11

Observe that this realization of W as a Hilbert space valued Wiener process

depends on the choice of A. Indeed, the same procedure applies for any given

sequence A sharing the above properties. Yet the class of integrable processes does

not depend on A as we shall see.

Proof. Fix T £ R+. We define Wn £ H0/(£\) by

n

3= 1

Obviously, E \\\W% - Wp\\2J =T^=ra+1 A^ -> 0 for m,n -4 0. Hence (Wn)nN

is a Cauchy sequence in H°^2(£2X), converging to W. Since this holds for any T £ R+,we get that W is an l\-valued continuous martingale.

For any v £ £\ the R-valued random variables (W — W,v)p are centered

Gaussian, converging in L2(ft,T, P) to (Wt — Ws,v)p- which is therefore centered

Gaussian too. By the same reasoning

IE (Wt - Ws, ei)e (Wt - Ws, ej)t2 = (t - s)(Qeu ej)e .

Whence Condition iii) of Definition 1.2 holds. Similarly one shows ii) and clearly

W0 = 0. D

REMARK 1.4. We notice that any Hilbert space valued (cylindrical) Wiener pro¬

cess in the terminology of Da Prato and Zabczyk [18] can be considered in the

preceding RN-framework (1.1), see [18, Propositions 4.1 and 4.11].

1.2. The Stochastic Integral

For the definition of the stochastic integral with respect to W, Da Prato and

Zabczyk [18] introduce the space L°(H) of Hilbert Schmidt operators from £2 into

iT.6 It consists of linear mappings $ : I2 —> H with

n*iilg(io:=Eii*^<00' (L6)

where & := $(<?_/)• In the sequel we shall identify $ with (&)je^. On the other

hand, observe that any sequence (&)jeN of H"-valued predictable processes satis¬

fying (1.6) provides7 an L2(H)-\ahied predictable process $ by $(gj) := $J'.

Notice the particular case L^QH) = £2.

We summarize the construction of the stochastic integral, based on the theoryfrom [18, Sections 4.1-4.4]. Fix T £ R+.

Definition 1.5. We call /^(H) the Hilbert space of equivalence classes of

L^(H)-valued predictable processes $ with norm

\*\\c>Tm = E

T

\\$t\\lo(H)dt (1.7)

6Remember that I2 = Qz (i\) endowed with the implied scalar product, see (1.2) and com¬

pare with [18, (4.7)].7Notice that B(L%{H)) = <g)NB(.ff).


Then there exists an isometry $ H> $ W between £y(iî) and Tij, (FI), see

[18, Proposition 4.5], defined on the dense8 subset of elementary L(£2X; H)-valued9processes

k-l

$ = Yl ^rnX{tm,tm+1], *m is ^„-measurable, (1.8)m—0

by

fc-i

(* • W)t := Y, ^(Wi+1Ai - WimM), t £ [0,T].?n=0

Definition 1.6. The H-valued continuous martingale ($ W)t^[o,T] is called

the stochastic integral of $ with respect to W and is also denoted by

(* -w)t= IJo

t

*sdWK.

It is clear, how stochastic integration with respect to the d-dimensional Brow¬

nian motion Wd = (ß1,... ,ßd), d £ N, is included in this concept: any iï"d-valued

predictable process $ = ($1,..., $d) can be considered as L\ (H)-valued predictable

process $ by setting $J = $? if j < d and $J = 0 for j > d. This way one gets

easily

($ • W% = ($.W)t = Yf ^ d#- (L9).7=1

y°

Here are some elementary properties of the stochastic integral. Denote by E a

separable Hilbert space.

Proposition 1.7. Let $ £ £^(H).

i) For any stopping time r <T

($-W)tAT = (($l[0tT])-W)t. (1.10)

ii) The following series converges uniformly on [0,T] in probability

($-w)t = W*$ïd#. (1.11)

iii) let A G L(H; E). Then io$e £*,(£) and

A($ W) = (A o $) W. (1.12)

PROOF. Property i) is [18, Lemma 4.9]. Property ii) follows by considering

(1.9) and ($1,..., $d,0,... ) which converges in C\(H) towards $ for d -> 00.

Now apply Doob's inequality (1.4). Finally, iii) can be shown in the same way as

[18, Proposition 4.15].Notice that all equalities hold in %y

2

(H), hence up to indistinguishability. D

There is a larger class of integrable processes than £|,(H).

ssee [18, Proposition 4.7]9for an arbitrary choice of A = (Aj)jgK aa in Section 1.1

1.3. FUNDAMENTAL TOOLS 13

Definition 1.8. We call Ö£C(H) the space of all L%(H)-valued predictable pro¬

cesses $ such that

p jf ll**ll£oW<°° =i.

Observe that $ £ £yc (H) if and only if there exists an increasing sequence of

stopping times rn f T such that $lr0iTn] is in Cj.(H) for all nGl This allows us

to extend the notion of the stochastic integral, see [18, p. 94-96].

Proposition 1.9. Let $ £ £yc(ü). Then there exists a unique H-valued con¬

tinuous local martingale (Mt)t£[o,T] characterized by

MtAT = (($1[0>T]) W)t

whenever $1[o,t] £ £y(ü)- Again, M is called the stochastic integral of $ with

respect to W and it is written

Mt=:($-W)t= f $sdWs.Jo

Many properties of the stochastic integral for £y(iz")-integrands carry over for

£yc(ü)-integrands by localization. Here is one feature of this method.

LEMMA 1.10. Let (Mn)ngN be a sequence of H-valued continuous local martin¬

gales. Assume that for all S > 0 there exists a stopping time r with P[r < T] < 5

such that (MtnAr) -» 0 in U\(H). Then (Mn)*T -> 0 in probability.

Proof. Let 6, e > 0 and r be as in the lemma. Then

>[(Mn)*T>e]<ö + l sup \\MtnAT\\H>ete[o,T]

and the last term goes to zero as n —> oo by Doob's inequality (1.4). D

Using Lemma 1.10 the following can be shown.

Proposition 1.11. Equality (1.9) and Proposition 1.7 remain valid if £y(ü)is replaced by Cl^c(H).

Up to now, the stochastic integral was defined only on the finite time interval

[0, T]. But we can consider integrands in C2(H) := £2X5(Ü) (with T in (1.7) replacedby oo), respectively £loc(ü) := f]TW+ £^C(F).

Let $ £ £2(H). Write Mn := *l[o,n] W for the stochastic integral on [0,n].By (1.10), Mn+l coincides with M" on [0, n] for all n £ N. Therefore we can define

unambiguously a process $ • W - the stochastic integral of $ with respect to W - bystipulating that it is equal to Mn on [0, n]. This process is obviously an Ü-valued

continuous martingale.

By localization the same procedure applies for $ £ £loc(ü), and $ • W is an

H-valued continuous local martingale.

1.3. Fundamental Tools

Three fundamental theorems of stochastic analysis are Itô's formula, the sto¬

chastic Fubini theorem and Girsanov's theorem. The first is given by Lemma 7.11.

This section provides the other two.


1.3.1. The Stochastic Fubini Theorem. Let (E,£) be a measurable space

and let $ : (t, to, x) i->- $t(w, x) be a measurable mapping from (R+ x ft x E, V® £)into (L2(H), B(L2(H))). In addition let /i denote a finite positive measure on (E, £).

Fix T > 0. By localization we get the following version of [18, Theorem 4.18].

Theorem 1.12 (Stochastic Fubini theorem). Assume that

/ / \\§t(x)\\2Lo{H) ß(dx) dt < 00, P-a.s.

J 0 JE

Then there exists an Pt ® £-measurable version £(u>,x) of the stochastic integral

JQ $t(x)dWt which is ß-integrable P-a.s. such that

f Ç(x)iM(dx) = f ( f $t(x)fi(dx)\ dWt, P-a.s.

Je Jo \Je j

1.3.2. Girsanov's Theorem. Let 7 = (jj)jeN £loc(R)10. We define the

stochastic exponential £(7 W) of 7 W by

£ (7 W\ := exp ((7 -W)t-\Jo hs\\% ds^j . (1.13)

By [18, Lemma 10.15] there exists a real-valued standard (.T^-Brownian motion ß°

such that (7 W)t = J0 ||7s||^2 dß°. This way we are lead to the real-valued theory.

Lemma 1.13. The stochastic exponential £(7 • W) is a non-negative local mar¬

tingale. It is a martingale if and only i/E [£(7 • W)t] = 1 for all t £ R+.

Proof. By Itô's formula

£(7 . W)t = 1 + / (£(7 • W)s\\7s\\l2) dß°s.Jo

Whence the first assertion. For the second statement we refer the reader to [40,Remark 3, p. 141].

We recall [40, Proposition (1.15), Chapter VIII].

Lemma 1.14 (Novikov's criterion). If

E eXP\2.I ll7t|&* < 00, (1.14)

then £(7 W) is a uniformly integrable martingale. Accordingly, £(7 • W)t —>

£(7 • W)oo in ^(fi.^P) for t -» 00 and £(7 W)x > 0, P-a.s.

The last statement is a direct consequence of the fact that £(7 • W)oo > 0, P-a.s. on

the set {fR \\jt\\% dt < 00}, see [40, Proposition (1.26), Chapter IV].Let T G R+. The next theorem will assure the existence of an equivalent local

martingale measure.

Theorem 1.15 (Girsanov's Theorem). Let 7 = (jj)jen £ £yc(R) satisfy

E[£(7-W)r] = l-

10Recall that L°(

1.3. FUNDAMENTAL TOOLS 15

Then

et

ßi--=ßi- f lids, t£[0,T], J£NJo

are independent real-valued standard (Pt)te[o,T\-Brownian motions with respect to

the measure Py ~ P on Pt given by

——- =£(7

•W)t-

d¥u ;

Proof. See [18, Theorem 10.14]. D

We can now define, as above, the spaces HT(H), iiTfi(H), LT(H) and £yc(if) on

the filtered probability space (ft, Pt, (-?7t)t6[o,T], Pt)- Obviously we have £yc(ü) =

£yc (H). Notice that a sequence of .iy-measurable random variables converging in

probability P also convergences in probability Py and vice versa.

Proposition 1.16. The series

W = YßJ9i

converges in 'HT^(£2J and defines a Q-Wiener process in £\ with respect to Py.

Moreover, for any $ G £lfc(H),

($-W)t = (*-W)t- ! $s(ls)ds, VtG[0,T], P-a.s. (1.15)Jo

Proof. For n £ N we define Wn := Y!j=ißj9j- % Proposition 1.3, the

sequence (Wn)nen converges to W in HTfl(H), and W is a Q-Wiener process in £\for the measure Py. In view of (1.5) and Theorem 1.15

«

pt

W?=W?-Yi -ÛQjds, VnGN.

i=iJo

By Doob's inequality (1.4) and Proposition 1.3, the right hand side converges to

W* ~ SieN Jo ^s9j ds uniformly on [0,T] in probability, which is therefore indistin¬

guishable from W on [0,T].Equality (1.15) is certainly true for elementary £(^2; unvalued processes, re¬

call (1.8). Now let $ £ ClSc(H) = Ö°C(H). We define the stopping times

r„:=TAinf i6l+ / \\$ s\\2Lo{H) ds > n \,

n £ N.

Then rn f T and $n := $l[o,r„] G 4(^0 n ^y(Ü). By Lemma 1.17 below, (1.15)holds for each $n. By (1.10) therefore also for $. D

LEMMA 1.17. Let $ G CT(H)nCT(H). Then there exists a sequence ($n)neNof elementary L(£2; H)-valued processes converging to $ in CT(H) and in CT(H)simultaneously.

Proof. Standard measure theory.


1.4. Stochastic Equations

This section provides the basic concepts and results for stochastic equations in

infinite dimensions. Let W = (ß^)je^ be an infinite dimensional Brownian motion

as introduced in Section 1.1. Our standing set of ingredients is the following:

• A strongly continuous semigroup {S(t) | t £ R+} on H with infinitesimal

generator A.

• Two measurable mappings F(t,to,h) and a(t,to,h) = (a^(t,to,h))j^ from

(R+ x ft x H,V®B(H)) into (H,B(H)), resp. (L°2(H),B(L°(H))).• A non random initial value ho £ H.

1.4.1. Mild, Weak and Strong Solutions. We shall assign a meaning to

the stochastic equation in H

( dXt = (AXt + F(t, Xt)) dt + a(t, Xt) dWt

\ A0=/lo¬

in view of (1.11) this can be rewritten equivalently

dXt = (AXt + F(t, Xt)) dt + Y a' (*. Xt) dß

Xq = ho.

(1.16)

(1.16')

REMARK 1.18. Notice that F and a are random mappings and may depend

explicitly on to. In accordance with the notation of [18], however, we abbreviate

F(t,to,Xt(to)) and o-(t,LO,Xt(to)) to F(t,Xt) and a(t,Xt), respectively.

Here are three concepts of a solution.

Definition 1.19. Suppose X is an H-valued predictable process satisfying

jT

(\\XS\\H + \\F(s,Xs)\\H + \\a(s,Xs)\\lo{H))ds < oo = 1, Vt£ R+/o

for a stopping time t > 0, the lifetime of X. We call X

i) a local mild solution to (1.16), if the stopped variation of constants formulaholds11

rtf\T

Xt = S(t A r)/i0 + / S((t At)- s)F(s, Xs) dsJo

ftf\T+ Y S((tAT)-s)aj(s,Xs)dßi, P-a.s. Vi £ R+.

ii) a local weak solution to (1.16), if for arbitrary £ G D(A*)

((, Xt)H = (C, ho)h + JT

{(A*C, XS)H + (C, F(s, Xs))H^j ds

rtAT

+ /Z (C,o-j(s,Xs))Hdßl P-a.s. ViGR+.„•=wio

The stochastic convolution always has a predictable version, see [18, Proposition 6.2].

1.4 STOCHASTIC EQUATIONS 17

iii) a local strong solution to (1.16), if X £ D(A), dt ® dF-a.s.

ftAt

I AXs\\h ds < oo

L/o

and the integral version of (1.16) holds

rt/Vr,

.

ptAr

= 1, Vt G R+ (1.17)

/t/\r.

. pt/\r

(AXs+F(s,Xs))ds+ a(s,Xs)dWs, V-a.s. Vi G R+.

If t — oo we just refer to X as mild, weak, resp. strong solution.

This definition may be straightly extended to random .Fo-measurable initial values

ho. Notice that the lifetime r is by no means maximal.

REMARK 1.20. For stochastic equations in finite dimensions one usually dis¬

tinguishes between strong and weak solutions, see e.g. [32, Chapter 5]. There,

however, weak has a different meaning than in the present context. Here we followthe terminology of [18].

According to [18, Theorem 6.5] the concepts in Definition 1.19 are related by

iii) =£- ii) => i),

andi) îi)iicr(-,X) £ C2(H).Since the coefficients F and a are not homogeneous in time, we will frequently

consider the time shifted version of (1.16). The following result is classic.

Lemma 1.21. Let tq denote a bounded stopping time. Then

ß[Ta^:=ßi0+t-ßi0, iGR+, j£N

are independent standard Brownian motions for the filtration (Pt ) '= (PT0+t)-

Proof. See Lemma 7.3. D

We write W^ = (ß^'3)]^- It is obvious how W^ can be realized as Hilbert

space valued (.Tj')-Wiener process and how to define £loc(ü, (Pt )) etc. In fact,

($i) G £Ioc(ü) if and only if ($ro+t) G £}oc(H, (Pô))).The time To-shifted version of (1-16) now reads

f dXt = (AXt + F(t0 + t, Xt)) dt + o-(t0 +1, Xt) dWt{ro)\ X0 = h0

[ " '

and similarly the expanded form (1.16'). A local mild (weak, strong) solution to

(1.18) will be denoted by X^°'ha\ Suppose that X - X^ho~> is a local mild (weak,strong) solution to (1.16) with lifetime t > r0. Then (XT0+t) is local mild (weak,strong) solution to (1.18) with h0 replaced by XT0 and lifetime r — To. This can be

shown using the identity

f°

$s dWs = f $ro+s dW^, $ G £loc (H)Jt0 Jo

which is derived in the proof of Lemma 7.3.

In the sequel we shall mostly consider deterministic times tq= to £ R+.


1.4.2. Existence and Uniqueness. Lipschitz continuity plays an importantrole in the theory of stochastic equations. Assume that D and E are Banach spaces.

Definition 1.22. A mapping $ = $(£, u, h) : R+ x ft x D -> E is called locallyLipschitz continuous in h, if for all R £ R+ there exists a number C = C(R), the

Lipschitz constant for $, such that for all (t,ui) G R+ x ft

\m,LO,h1)-^(t,iv,h2)\\E<C\\h1-h2\\D, Wn,/i2 G Br(D). (1.19)

If C does not depend on R,12 we call $ simply Lipschitz continuous in h.

We can now formulate the main result on stochastic equations which is a com¬

bination of Theorems 6.5 and 7.4 in [18]. It is based on a classical fixed point

argument.

Theorem 1.23. Suppose F(t,u,h) and a(t,u,h) are Lipschitz continuous in

h and satisfy the linear growth condition

\\F(t,uj,h)\\2H + \\a(t,tü,h)\\lo{H) < C(l + \\h\\2H), M(t,to,h) £R+xftxH,

where C does not depend on (t,to,h).Then for any space time initial point (to, ho) G R+ x H there exists a unique

continuous weak solution X = X^to'h°^ to the time to-shifted equation (1.18). More¬

over, for any T £ R+ there exists a constant C = C(T) such that

E[\X^]<C(l + \\ho\\2H).

Here is the local analogue.

Corollary 1.24. Suppose F(t,co,h) and o-(t,oj,h) are locally Lipschitz con¬

tinuous in h.

Then for any space time initial point (to, ho) £ R+ x H there exists a uniquecontinuous local weak solution X = x^to'h°> to the time to-shifted equation (1.18).

Proof. See Lemma 7.5. D

REMARK 1.25. Theorem 1.23 and its corollary indicate that a (local) weak so¬

lution is the proper concept for equation (1.16). Indeed, (local) strong solutions

exist very rarely in general. Even if F,a3 £ D(A) and D(A) is invariant for S(t),this does not imply (1.17) yet. See e.g. [15, Proposition 3.26] for the case F = 0.

12Actually C may depend on T and then (1.19) holds for all (t, of) £ [0,T]xiî. The subsequentexistence results remain valid. For ease of notation we abandon this generalization.

CHAPTER 2

The HJM Methodology Revisited

2.1. The Bond Market

This chapter is devoted to a detailed discussion of the seminal paper by HJM [28].We consider a continuous trading economy with an infinite trading interval.

The case of a finite time interval for the term structure is not issue of this thesis

but will be exploited elsewhere.

The basic contract is a (default-free) zero coupon bond with maturity date T,

which pays the holder with certainty one unit of cash at time T. Let P(t, T) denote

its time t price for t <T. Clearly P(t,t) = 1, and we require that the following

term structure representation holds

P(i,T)=exp -/ f(t,s)ds\, VT>t, (2.1)

for some locally integrable function f(t, ) : [t, oo) —t R, the time t forward curve.

The number f(t,T) is the instantaneous forward rate prevailing at time t for a

risk-less loan that begins at date T > t and is returned an instant later. In the

particular case T = t we call f(t, t) the short rate.

HJM [28] represent each particular forward rate process (f(t, T))i6[0jr] as an

Itô process based on a fixed finite dimensional Brownian motion. We follow their

terminology at the beginning, but will then allow for an infinite dimensional un¬

derlying Brownian motion.

We adopt the space (ft, P, (Pt)tëR+^) fr°m Chapter 1 and consider first the

standard d-dimensional (JFt)-Brownian motion (ß1,...,ßd), d £ N. Define the

triangle subset of R2

A2 :={(t,T) GR2 |0<i<T}.

We recapture the HJM setup [28, Condition C.l] for the term structure move¬

ments. Let a(t,T,uj) and a^(t,T,to), 1 < j < d, be measurable mappings from

(A2 x ft,B(A2) <g> P) into (R,B(R)), such that for each T G R+ the processes

(a(t,T))t£[o,T] and (o~'J (t,T))teß,T] are progressively measurable. Moreover

|a(i,T)| + ]|cr(i,T)|||d) dt <oo, P-a.s. VT G

Then for fixed, but arbitrary T £ R+, the forward rate for date T evolves as the

Itô process

f(t,T) = f(0,T)+ I a(s,T)ds + y2 f ^(s,T)dßi, t£[0,T]. (2.2)Jo

i=1Jo

Here /(0, ) is a non-random initial forward curve which is locally integrable on R+.

19

20 2. THE HJM METHODOLOGY REVISITED

We emphasize that (2.2) represents a system of infinitely many Itô processes

indexed by T £ R+. In what follows we tend to transform this into one infinite

dimensional process.

Under some additional assumptions - which we do not further specify here,since we will take this point up in the next section - the savings account

B(t):=exp( j f(s,s)dsj , t£R+,

can be shown to follow an Itô process. So does (P(t,T))te[0,T]) see [28, Condi¬

tions C.2 and C.3].

2.2. The Musiela Parametrization

Musiela [37] proposed to change the parametrization of the term structure accord¬

ing to f(t,x + t), where x > 0 denotes time to maturity. There are indeed several

reasons for doing so.

Non-varying state space: The basic underlying state variable is the time t

forward curve {f(t,T) \ T > t}. In the HJM time framework, however, this

means that the state space is a space of functions on an interval varying with

t. The above modification eliminates this difficulty.Non-local state dependence: Models within the classical HJM framework

have state dependent volatility coefficients of the form a(t,T,u>, f(t,to,T)),see [28], [36] and [35]. For short rate models the dependence is on f(t,t),respectively. In any case, the dependence on the state variable is local. It

would be natural, however, if the volatility depended on the whole forward

curve. Hence if also non-local (like LIBOR rate) dependence were allowed,

incorporating all LIBOR or simple rate models in one framework. This aim

is provided by considering f(t,- + t) as state variable in a function space,

evolving according to functional sensitive dynamics.Consistency problems: One major issue of this thesis is to investigate when

the system of infinitely many processes (2.2) does allow for a finite dimen¬

sional realization. This question on one hand is of interest by its own. On

the other hand it occurs in connection with statistical purposes, as explainedin the introduction.

Suppose G = {G(-, z) | z £ Z} represents a forward curve fitting method.It is natural to ask whether there exists any HJM model (f(t,T))rt,T)eA2which is consistent with G- That is, which produces forward curves which lie

within the class G-

Let's consider for the moment the Nelson-Siegel family, see Chapter 5,

G(x,z) = Zl + (z2 + z3x)e~Z4X, 2cl4.

Formally speaking, we search a family of forward rate processes given by (2.2)and a Z-v&lued process Z such that

G(T-t,Zt) = f(t,T), \/(t,T)£A2. (2.3)

A naive approach is to fix a tenor 0 < T\ < < T$, define the mapping

G(t,z):=(G(T1-t,z),...,G(T5-t,z)):[0,T1]xZ^R5and try to (locally) invert G, which yields

(t,Zt) = G-1(f(t,T1),...,f(t,T5)), t£(0,Tx).

2.2. THE MUSIELA PARAMETRIZATION 21

The inverse mapping theorem requires non-singularity of the 5 x 5-matrix

-dxG(Tx-t,z) dXlG(Ti-t,z) ... dZ4G(Ti-t,z)]

-dxG(T5-t,z) dZlG(T5-t,z) ... dZ4G(T5-t,z)

But this is impossible since

dxG(x,z) = (-Z2Z4 + z3 - z3Z4x)e~ZéX

dZlG(x,z) = 1

dXaG(x,z) = e~'*x

dZsG(x,z)=xe~ZiX

dZiG(x,z) = (-z2x - z3x2)e~^

are linearly dependent functions in x\

The problem originates from the appearance of dxG(x,z) in the above

matrix, which is due to the explicit ^-dependence of G in (2.3). A way out is

given by the new parametrization f(t,x + t) since then (2.3) reads

G(x,Zt)=f(t,x + t), Vx,tGR+. (2-4)

This however causes a new difficulty: (f(t, x + t))t^m+ is not an Itô process in

general. So even if (2.4) can be inverted for all x £ WL+, what kind of process

is (Zt) = (G(x, -)~l(f(t, x + £)))? We will address these issues in Chapter 4.

We now shall see how the HJM dynamics can be expressed in the language of

stochastic equations in infinite dimensions (Section 1.4), allowing for the Musiela

parametrization.

Let {S(t) I t £ R+} denote the semigroup of right shifts which is defined byS(t)f(x) = f(x + t), for any function / : R+ ->• R.

Fix t, x £ R+. Then (2.2) can be simply rewritten, for T replaced by x +1,

f(t, x + t)= S(t)f(0, x) + j S(t - s)a(s, x + s)dsJo

drt (2-5)

+ }2 / S(t-a)cr>(s,x + s)dßij=i

Jo

where S(t) acts on f(0,x), a(s,x + s) and aj(s,x + s) as functions in x. We shall

work out in an axiomatic way the minimal requirements on a Hilbert space Ü such

that (2.5) can be given a meaning when

n :=f(t,- + t), t£R+

is considered as an ü-valued process. Also the HJM no arbitrage condition [28,Condition C.4] has to be recaptured. In particular, this means that (P(t,T))te[o,T]and (B(t))tsR+ have to follow Itô processes.

Clearly H C L110C(R+), by (2.1). Obseive that (2.5) is a pointwise equalityfor all x G R+. Hence pointwise evaluation has to be well specified for h £ H.

Moreover, we have to interchange integration and pointwise evaluation since x

appears under the integral sign. Accordingly, we assume

222.THEHJMMETHODOLOGYREVISITED

(HI):Thefunctionsh£Harecontinuous1andthepointwiseevaluation

Jx(h):=h(x)isacontinuouslinearfunctional2onH,forallx£M+.

Indeed,itisnotveryrestrictivetoassumethattheforwardcurvesf(t,T)givenby

(2.2)arecontinuousinT>t,see[28,Lemma2].Hereisanimmediateconsequenceofassumption(HI).

Lemma2.1.ForanyuGR+thereexistsanumberk(u)suchthat

\\Jx\\h<Ku),VxG[0,u].

Furthermore,(x,h)i->Jx(h):R+xH—»Risjointlycontinuous.

Proof.Letu£R+.By(HI),sup^^\Jx(h)\<ooforallheH.Nowthe

Banach-Steinhaustheorem[41,Theorem2.6]yieldstheexistenceofk(u).Clearly,x»->Jx:R+->Hisweaklycontinuous3.Thesecondassertionofthe

lemmafollowsbyconsidering

\JxM-Jx(h)\<WJxJMlhn-h\\H+\JXn(h)~Jx(h)\

andusingtheaboveestimate.D

InviewofSection1.4wehavetorequirefurthermore

(H2):Thesemigroup{S(t)\t£R+}isstronglycontinuousinHwithinfini¬

tesimalgeneratordenotedbyA.

Foranyh£D(A)wehaveftS(t)h=S(t)AhinHforallt£R+,seee.g.[41,

Theorem13.35].By(HI)thereforefth(t)=Ah(t)forallt£R+.HenceAh=

h!£H.Butthisimpliesh£C71(R+).Wehavethusshown

Lemma2.2.Under(H1)-(H2)wehave

D(A)c{h£Hr\C1(R+)\tiGÜ}.

Forthecoefficientsin(2.5)weformulatetheanalogonto[28,ConditionC.l].Atthispointwegeneralizethehypothesisonthenoiseandallowfromnowon

foraninfinitedimensionalunderlying(JT<)-BrownianmotionW=(ß3)j^,see

Section1.1.

Writeoit(co):=a(t,to,+t)anda=(cr^gjjwherecr|(w):=cr3(t,to,+t).

(CI):Theinitialforwardcurvero=/(0,•)liesinH.

(C2):TheprocessesaandaareH-,resp.L°(ü)-valuedpredictableand

/'Jo

(lKlliï+lkllio(ff))ds<oo

IntheclassicalHJMframeworka3=0forj>d.

=1,ViG

"^Thatis,foranyh£Hthereexistsacontinuousrepresentative.Weshallsystematically

replacetheequivalenceclasshbyitscontinuousrepresentativewhichwestilldenotebyh.

2HenceJ&iselementofHwhichweidentifywithitsdualhereandsubsequently.3Thatis,continuouswithrespecttotheweaktopologyonH.

2.2.THEMUSIELAPARAMETRIZATION23

Now(2.5)reads,foranyt,x£R+,

Jx(n)=Jx(S(t)r0)+fJx(S(t-s)as)ds+y2fMS(t-s)a{)tJo

,CNJo

ftrt

JxS(t)r0+fS(t-s)asds+YjIS(f-s)aidßlJ.

P"a-S-

(2.6)jeN'

see(1.12).Forifixed,theexceptionalP-nullsetinequality(2.6)dependsonx.

Butaccordingto(HI)bothsidesarecontinuousinx.Astandardargumentyields

equalityP-a.s.simoultaneouslyforallx£R+.Pointwiseequalityoftwofunctions

yieldsidentityofthefunctions,hence

rt=S(t)r0+IS(t-s)asds+£fS(t-s)a{dß{,P-a.s.(2.7)

By[18,Proposition6.2]therighthandsideof(2.7)hasapredictableversionwhich

westilldenotebyrt-Consequently,risamildsolutiontothestochasticequation

drt=(Art+at)dt+Y4dßt

iN(2.8)

,

ro=/(0,-).

Wemakeanadditionalassumption.

(C3):Thereexistsacontinuousmodificationofr,stilldenotedbyr.

Condition(C3)issatisfiedifS(t)formsacontractionsemigroupinH,see[18,

Theorem6.10],orifwerequirestrongerintegrabilityofa,see[18,Proposition7.3].

Notice,however,thatrisnotanPT-valuedsemimartingaleingeneral.

Remark2.3.Combining(C2),(C3)andLemma2.1weconcludethatthe

randomfieldsrt(to,x),at(to,x)andal(to,x)areV®B(R+)-measurableandrt(to,x)

isjointlycontinuousin(t,x)forf-a.e.to.

SofarwereformulatedtheclassicalHJMsetupinthespiritofMusiela[37]

byputtingadditionalassumptions(CI)and(C2)onthecoefficients(reflecting

essentially[28,ConditionsC.1-C.3]).Moreover,weallowforaninfinitedimensional

drivingBrownianmotion.Theprocessr,definedasthecontinuousmildsolution

to(2.8),isrelatedtof(t,T)from(2.2)by

rt(x)=f(t,x+1)\/xGR+,P-a.s.ViGR+.

Fromnowon,thebondpricesandthesavingsaccountaregivenby

P(t,T)=exp(-frt(x)dx),

(i,T)GA2

B(t)=exp(/rs(0)dsj,

t£M+.


2.3.ArbitrageFreeTermStructureMovements

Wederivethedynamicsofthediscountedbondprices.Sincerisnotanff-valued

semimartingale,wecannotuseItôformuladirectlybutweratherhavetodeal

withtherandomfield(rt(x))tiXeR+andapplythestochasticFubinitheorem.We

essentiallyadaptthetechniquefrom[28].Sufficientconditionsfortheexistence

ofanequivalentmartingalemeasure,whichensurestheabsenceofarbitrage,are

provided.

FirstweintroducethelinearfunctionalXuonHdefinedby

Pu(h):=/h(x)dx,u£R+.Jo

Thereisanimmediateconsequenceofassumption(HI).

LEMMA2.4.ThelinearfunctionalIuiscontinuousonH.Inparticular,

\\1U\\H<uk(u),VuGR+

wherek(u)istheconstantfromLemma2.1.

Moreover,themapping(u,h)i->-Iu(h):R+xH->Risjointlycontinuous.

PROOF.Theestimateistrivial.Thesecondassertionfollowsasintheproof

ofLemma2.1bytheweakcontinuityofui->-Tu:R+->•H.

Itfollowsfrom(C3)andLemma2.4thatP(t,T)=exp(-ZT-i(n))isPt-

measurableandjointlycontinuousin(t,T)£A2.

Fix(t,T)£A2.Thenwehaveby(2.7),Lemma2.4and(1.12)

i:=-logP(t,T)

l-trt

lT-t(S(t)r0)+flT.t(S(t-s)as)ds+YITT-t(S(t-s)ai)dß3sJoieNio

JT(r0)-lt(r0)+/(lT-s(as)-lt-s(as))ds

Jo

+E[\iT-s(o-i)-it-s(cfi))

dßi,„•CM

JO

r

{lT_s(o-3s)-lt-s(o-i))dßi,P-a.s.

wherewetookintoaccounttheobviousrelationIuoS(t)—Tt+u—TtTheprocesses

(Tt-s(o~{))se[o,T]arepredictableand

Y,\lT-s(o-i(u))\2<(Tk(T))2Y\Wl^)\\H<oo,V(s,w)G[0,T]xÎÎ.

Hence(It-s°fs)sG[o,T]is^2-valuedpredictable.Similarly

/Y\It-s(4)\2ds<

thus(Tt-s°o-s)££yc(R).ReplacingTbyi,thesameholdsfor(lt-s°Cs)se[o,t]-Bylinearitywecanthussplituptheintegralsandwrite

I=Ix-l2,P-a.s.

2.3.ARBITRAGEFREETERMSTRUCTUREMOVEMENTS25

where

h:=TT(ro)+IlT-s(as)ds+Y,flœ-.W,)dß>JojeNJo

I2:=lt(r0)+/lt-s(as)ds+Y/xt-s(o-.JoacwJo

i)dßl

Theintegralsini2needtobetransformed.Weintroducethemappings

_...

\a3(uj,u-s),iis<uo-3s(to,u):=<

sV'

I0,otherwise.

Substitutingu=x+s,wehave

ft—SftftXt-S(a3s)—

/a3s(x)dx=/a{(u—s)du=/â3s(u)du.JoJ

sJo

InviewofRemark2.3,ä=(äJ)3&îsmeasurablefrom(R+xflxl+,Ti®B(M.+))into(£2,B(£2)).Moreover,byLemma2.1and(C2)

//||crs(u)[Ifarfi*ds=//y"|â;j(u)|2dudsioJoJoJo

jen

=fE(Y'\°i(x)\2dx)ds

JOJGN

\J0J

ft<*(*(*))2//~2\\asÏÏHds<°°>P-a'S-

JO.-CM

Consequently,thestochasticFubinitheorem1.12appliesand,by(1.10)—(1.11),

£flt-M)dßl=Y,f{f~Û)Aà&=

f(Efti^dßi]du,cMiojGNôVo,•io\,cMio/

^\jenJo(u-s)dß3sdu,P-a.s

UsingtheordinaryFubinitheoremwederivesimilarly

rtrt/ru/Zts(as)ds=/I/as(u—s)ds)du,P-a.s.(2-10)

Combining(2.9)and(2.10)wecanwrite,by(1.12),

L2=

fJo{S(u)r0+[S(u-s)asds+Y"[S(u~SHdß33)duJoIJofcsiû/

=fru(0)rJo

j&

ldu,P-a.s.

262THEHJMMETHODOLOGYREVISITED

TakingintoaccountTr(ro)=—logP(0,T)wearriveatthefollowingrepresen¬tation

logP(i,T)=logP(0,T)+[(rs(0)-lT-s(as))dsJo

+W(-lT^s(al))dßi,P-a.s.V(t,T)GA2.

SinceP(t,T)isjointlycontinuousin(t,T)GA2,theP-nullsetcanbechosenfor

eachTGR+independentlyofiG[0,T].Whence(logP(i,TT))iS[0]T]isacontin¬

uousreal-valuedsemimartingale.Accordingly,sois(logP(t,T)—logß(i))tG[0)T].ApplyingItô'sformulatoexgivesforthebondpriceprocess

P(t,T)=P(0,T)+J*P(s,T)L(0)-Tr->s)+^(ïr-s(^))2)ds

+WP(s,T)(-TT-s(o-l))dßl,t£[0,T},^T,jJ0Jf

0

(2.11)

andforthediscountedbondpriceprocessZ(t,T):=^,'f,'

Z(t,T)=P(0,T)+[Z(s,T)-Jy_s(as)+^(JT_S(^))2)ds

/°V'eN/(2.12)

+YIZ(s,T)(-lT_s(a3s))dß3s,t£[0,T}.^Jo3e¥Jo

ThefollowingmappingwillplayanimportantroleForanycontinuousfunction

/onR+wedefinethecontinuousfunctionSf:R+—>Rby

(Sf)(x):=f(x)/f(n)dr,,x£R+.Jo

Hereisanelementaryresulton<S.

LEMMA2.5.Let$=($3)jeNL°2(H).Thenf(x):=|E^2*^))2lsa

continuouslydifferentiablefunctioninx£R+with

f(x)='£(S&)(x).

Bothseriesconvergeuniformlymxoncompacts.

Proof.Setfn(x)~\Y^î^x^3))2whichisobviouslycontinuouslydiffer¬

entiableinx£M_(.with

fn(x)=J2(SV)(x).

Moreover,form<n£N

n

i/nW-^wi+i/^-^wiâiî^+nîiifiîiH)E\\v\\h

j=m+l

23ARBITRAGEFREETERMSTRUCTUREMOVEMENTS27

whichtendstozeroforto,n—>oouniformlyinxoncompacts,byLemmas2.1and

2.4.Consequently,thefunctionf(x)=limn_>.0Ofn(x)existsandiscontinuouslydifferentiableina;GR+withf'(x)=limn-^f'n(x).D

BelowwewillconsideraparticularGirsanovtransformationfromPtosome

riskneutralmeasureQ~PonP^whichonlyaffectsachangeinthedriftafor

theQ-dynamicsofr.Solet'spretendforthemomentthatPitselfisariskneutral

measure.ThentheHJMdriftcondition[28,(18)]readsasfollows.

LEMMA2.6.Thediscountedbondpriceprocess(Z(t,T))t^[o,T\followsalocal

martingaleforallTGR+ifandonlyif

a=YSa°idt®d¥-a.s(2.13)

Proof.LetT£R+.By[40,Proposition(1.2)],(Z(i,T))te[0iT]isalocal

martingaleifandonlyiftheintegralwithrespecttodsin(2.12)isindistinguishable

fromzero.SinceZ(s,T)>0,thisisequivalentto

lT-t(at)=\J2(PT-t(o-Jt))2,dt®dP-a.s.(2.14)

JGN

seetheproofof(5.6).Thedt®dP-nullsetdependsonT.ButbyLemma2.5the

righthandsideof(2.14)iscontinuouslydifferentiableinT>tandsoisobviously

thelefthandside.Byastandardargumentwefindadt®dP-nullsetJ\f£Vwith

thepropertythat

M<*M)=ÊCWM))2'V(t,u>,x)ejVcxR+.

JGN

Differentiationyieldstheassertion.D

ObservethatLemma2.6imposesthrough(2.13)implicitlyameasurabilityand

integrabilityconditiononYljen^^•

Theirvalidityisguaranteedbythefollowing

pairofassumptions.

(C4):TheprocessesSa3areH-valuedpredictable.

(H3):ThereexistsaconstantKsuchthat

\\Sh\\H<K\\hfH

forallhGHwithShGH.

Ingeneral,Pisconsideredasphysicalmeasure.Accordingly,theconditionsof

Lemma2.6arenotsatisfiedunderP.

Definition2.7.WecallameasureQonJ-'ooanequivalentlocalmartingale

measure(ELMM)if

i)Q~PonPoo

ii)(Z(t,T))t£[o,T]isaQ-localmartingaleforallT£R_|_.

ThenextconditionassurestheexistenceofanELMM,see[28,ConditionC.4].

(C5):Thereexistsan£2-valuedpredictableprocess7=("f3)j^satisfying

Novikov'scondition(1.14)and

J213^=YSo-3-a,dt®dP-a.s.(2.15)

j6Nj6N

282THEHJMMETHODOLOGYREVISITED

ByHolder'sinequalityandLemma2.1,theseries

\iUM)

convergesuniformlyinxoncompacts.TakingintoaccountLemma2.5andinte¬

grating,weseethat(2.15)isequivalentto

^7t;Jy_t(^)=^(ly_i(^))2-IT„f(at),VT>t,di®dP-a.s.(2.16)

Thereisanintuitiveinterpretationof—7asmarketpriceofrisk,byconsidering

thebondpricedynamics(2.11):afterlocalization,therighthandsideof(2.16)is,

roughlyspeaking,thedifferencebetweenthe"averagereturnEpk'T)"ofthe

bondandthe"risk-lessrate"rt(0).Whencetheinterpretationof—7!asmarket

priceofriskinunitsofthevolatility,—Zy_t(cj(),forthe^-thnoisefactor.Whence

alsotheexpressionriskneutralmeasureforQ.

Condition(C5)allowsforLemma1.14.HencewecandefineQ~PonF^by

recallthedefinitionofthestochasticexponential(1.13).Girsanov'stheorem,see

Proposition1.16,yieldsthat

Wt=Wt-fjsds,t£R+

Jo

isaWienerprocessforthemeasureQ.Thedynamicsofrchangeaccordingly.

Proposition2.8.Undertheaboveassumptions,rsatisfies

rt=S(t)r0+f[S(t-s)YSa3s)ds+J2fS(t-s)a3sdß3s(2.17)Ja\lebJ3ëHJ°

withrespecttoQ.Consequently,QisanELMM.

Proof.Thestatementon(rt)followsbycombining(1.15),(2.7)and(2.15).

Lemma2.6yieldstheassertiononQ.

2.4.ContingentClaimValuation

Thissectionsketcheshowtovaluecontingentclaimsintheprecedingbondmar¬

ket.Yetwedonotaddressherethequestionofcompleteness,whichessentiallyis

equivalenttouniquenessoftheELMMQ.Sincetheunderlyingnoiseisaninfinite

dimensionalBrownianmotion,thenotionofinfinitedimensionaladmissibletrading

strategieswereconvenient.This,however,isbeyondthescopeofthisthesisand

willbeexploitedelsewhere.Foratreatmentofthefinitedimensionalcasewerefer

thereaderto[38,Section10.1].Measure-valuedtradingstrategiesareintroduced

in[9].WecontentourselfinpostulatingthatQistheriskneutralmeasure4and

allpricesarecomputedasexpectationsunderQ.Thisisjustifiedatleastifall

discountedbondpriceprocesses(Z(t,T))ie[0)x]followtrueQ-martingales.Since

Equivalently,—7isthemarketpriceofrisk.

2.4.CONTINGENTCLAIMVALUATION29

thenanyattainableclaim5willadmitauniquearbitrageprice.Itisdeterminedbythereplicatingstrategy.

Thesubsequentintroductionoftheforwardmeasure,theforwardLIBORrates

andcapletsispreliminaryforthediscussionofthepopularBGMmodelinSec¬

tion3.6.

2.4.1.WhenIstheDiscountedBondPriceProcessaQ-Martingale?

Wecangiveasufficientconditionforthediscountedbondpriceprocesstofollowa

trueQ-martingale.Thedynamics(2.12)readunderQ

Z(t,T)=P(0,T)+Y,!Z(s,T)(-lT^s(a3s))dßi,t£[0,T],

see(2.16).Itô'sformulayieldstherepresentationasthestochasticexponential

Z(t,T)=P(fi,T)£(£/o(-Iy_s(af)),

U'6p

(2.18)

ButnowwecanuseNovikov'scriterion,seeLemma1.14,toderive

Lemma2.9.If

Erexpf-/\\lT-t°o-t\/2dz<00

then(Z(t,T))te[o,T]*saQ-martingale.

Anothersufficientcriterionispositivityoftheforwardrates.Indeed,ifrt>0

foralliG[0,T]then0<Z(t,T)<1foralliG[0,T].Moreover,abounded

localmartingaleisauniformlyintegrablemartingale.Positivityofrisrelated

tostochasticinvarianceofthepositiveconeoffunctionsft>0inifunderthe

dynamics(2.8).Wedonotaddressthisproblemhereandreferthereaderto[34].

2.4.2.TheForwardMeasure.Forpricingpurposesitisconvenienttouse

atechniquecalled"changeofnuméraire",see[20]forathoroughtreatment.We

giveanapplicationforpricingbondoptions.

Assumethat(Z(t,T))te[o,T]isaQ-martingale.Wecandefinethemeasure

QT~Qon(ft,PT)by^=p(0T\B(T).By(2.18)wethenhave

PtP(0,T)

tâ))dftiG[0,T].(2.19)

Definition2.10.ThemeasureQTiscalledtheT-forwardmeasure.

AclaimXdueattimeTisinL1(ft,Pt,Qt)ifandonlyif-g^y£L1(ft,Pt,

Bayes'rule,seee.g.[32,Lemma5.3,Chapter3],yieldstheequality

XP(t,T)EQT[X\Pt]=B(t)]

B(T)Pt2-a.s.ViG[0,T].(2.20)

OntherighthandsidestandsthetimeipriceoftheclaimX.Thisisaveryuseful

relation,sincethelefthandsideallowsforclosedformexpressionsifXhasanice

distributionunderQT.Wewilllateruseitforpricingacap.

Bifeverdefinedinareasonableway


2.4.3.ForwardLIBORRates.Let'sintroduceanimportantQT-localmar¬

tingale.Wewillusethenotation1%~Iv—Xufor(u,v)£A.Fix6>0.

Definition2.11.Theforward5-period6LIBORrateforthefuturedateT

prevailingattimetis

L(t,T):=

i(eZtt-'M-l),

(t,T)GA2.

Wecanre-expressL(t,T)directlyintermofbondprices

WhencethemeaningofL(t,T)asforwardaveragesimplerateforaloanoverthe

futuretimeperiod[T,T+S\.LetTGR+andwrite£(t,T):=1?^(rt).CombiningLemma2.4and(2.17)

weconcludethat(£(t,T))te[0iT]lsareal-valuedcontinuoussemimartingaleand

£(t,T)=£(0,T)+fT^tl'1(S(t-s)J2So-3s1dsJo

V3^J

+YJfIT-St~t{S{t-s)a3s)d~ßij&Jo

=£(0,T)+lf^{(ZT+ö-s(o-i))2-(XT-s(o-i))2)ds

+zZfIT-l~S^s)dßi,t£[0,T].

WehaveusedtherelationIy^-*o5(i—s)=Iy^f_sandLemma2.5.Itô'sformula

yields

L(t,T)=L(0,T)

+èfei(S'T)E{Vr+*-M))2-&r-M)?+(P^Sr(4))2)dsZ°J°

iGN

=i(0,T)+lfé^Y.(XT-s~S^s)XT+S-s(^s))ds°J°

jGN

+Z2-Jte^Th^r(*3)dß3,t£[0,T}.

jGN°J°

Nowassumethat(Z(t,T+<5))t6[0,x+<?]isaQ-martingale.Using(2.19)and

Girsanov'stheorem1.15weconcludethat

ß(T+S)J=ft+jIt+s_sf(jj)^te[0,T+<S],jeNJo

6Typically,5standsfor3or6months.

2.4.CONTINGENTCLAIMVALUATION31

areindependent(Jrt)t6[0ir+(5]-BrownianmotionsunderQT+S.Consequently,the

process(L(i,T'))t6[0jy]isaQT+l5-localmartingalewithrepresentation

L(t,T)=L(Q,T)+YJ\[tet^^tss-'(ai)dßiT+^,t£[0,T].(2.21)iGN

°J°

2.4.4.Caps.Forathoroughintroductiontointerestratederivatives,suchas

swaps,caps,floorsandswaptions,wereferto[38,Chapter16].Werestrictour

considerationtothefollowingbasicinstrument.

AcapletwithresetdateTandsettlementdateT+5paystheholderthe

differencebetweentheLIBORL(T,T)andthestrikeratek.It'scash-flowattime

T+Öis

5(L(T,T)-k)+.

Acapisastripofcaplets.Ifwecanpricecaplets,wecanpricecaps.Thearbitragepriceofthecapletattimei<Tequals,see[38,p.391],

Cpl(t)=B(t)EQ-S(L(T,T)-K)+\PtB(T+5)

=P(t,T+ô)EQT+s[5(L(T,T)-k)+IPt],

wherewehaveused(2.20).Supposethereexistsadeterministic£2-valuedmeasurablefunctionir(-,T)=

rT(K3(-,T))jeNwithf0||7r(i,T)||22dt<ooandsuchthat

^ti~*(c4)=(!"e-£(*'T))7rJ(i,T),dt®dQ-a.s.on[0,T]xft.(2.22)

Then(2.21)reads

L(t,T)=L(0,T)+J2[L(s,T)n3(s,T)dßiT+^3,tG[0,T].iGN"70

Thisequationisuniquelysolvedbythestochasticexponential

L(t,T)=L(0,T)£(]£jTn3(s,T)dßiT+s^yjVJo

By[18,Lemma10.15]thereexistsareal-valuedstandard(J7i)4g[0,T]-Brownianmo¬

tionCöt°)iG[o,T]underQT+Ssuchthat

Ydf^{B,T)dß^+s^=fMs,T)\^dß°,te[o,r].jenJoJo

Accordingly,L(T,T)islognormallydistributedunderQT+<5,andthefollowingpric¬ingformulaholds,see[38,Lemma16.3.1].

Lemma2.12.Undertheaboveassumptions,thetimetpriceofthecapletequals

Cpl(t)=5P(t,T+S)(L(t,T)N(ex(t,T))-KN(e2(t,T)))(2.23)


where

logft^W^T)ei,2:=

v0(t,T)

v2(t,T);=J^|KS,T)||22ds.

andNstandsforthestandardGaussiancumulativedistributionfunction.

BGM[12]presentamethodformodellingtheforwardLIBORratessuchthat

(2.22)holdsforallTGR+.WewillrecapturetheirapproachinthepresentsettinginSection3.6.

ASummaryofConditions

Forthereader'sconveniencewesummarizetheconditionsmadesofar.Recallthe

definitionofS

(Sf)(x):=f(x)ff(n)dn,x£R+Jo

foranycontinuousfunction/:R+-4R.

(HI):Thefunctionsh£Harecontinuous,andthepointwiseevaluation

Jx(h):=h(x)isacontinuouslinearfunctionalonH,forallx£R+.(H2):Thesemigroup{S(t)|t£R+}isstronglycontinuousinHwithinfini¬

tesimalgeneratordenotedbyA.

(H3):ThereexistsaconstantKsuchthat

\\Sh\\H<K\\h\\2H

forallhGHwithShGH.

(Cl):Theinitialforwardcurvero=/(0,•)liesinH.

(C2):TheprocessesaandaareH-,resp..LP,(if)-valuedpredictableand

P/f||û!s||jï-+|ks|||o(ff)Jds<00=

(C3):Thereexistsacontinuousmodificationofr,stilldenotedbyr.

(C4):TheprocessesSa3areif-valuedpredictable.(C5):Thereexistsan£2-valuedpredictableprocess7=('y3)je?$satisfying

Novikov'scondition(1.14)and

J2j3a3=J2Scr3~a>dt®dP-a.s.(2.24)

jGNjGN

2.5.WhatIsaModel?

Uptonowweanalyzedthestochasticbehaviorofasinglegenericforwardcurve

processrwithintheHJMframework.Formodellingpurposesweratherwant

thestochasticevolutionbeingdescribedbyadynamicalsystemlike(2.8)with

statedependentcoefficientsat(to)=a(t,to,rt(cü))andat(oj)—cr(t,to,rt(uj)),which

allowsforarbitraryinitialcurvesro-

Fromthelastsectionitseemsplausibletospecifyasmodelingredientsthe

volatilitystructureaandamarketpriceofriskvector—7.Wethenletthedrift

abeingdefinedby(2.24).Proposition2.8,accordingly,assurestheexistenceofan

ELMMQ.

ViG

2.5.WHATISAMODEL?33

Asitcanbeseenfrom(2.17),7doesnotappearintheQ-dynamicsofr.Thus,insteadofsolving(2.8)directlyweratherstartwiththeWienerprocessWunder

Q,solve(2.17)andtransformW~-»W,byGirsanov'stheorem,tohave7back

inthedriftfortheP-dynamicsofr.Comparewith[28,Lemma1].Thatway,

however,themeasurePandtheWienerprocessWwilldependonrimplicitlyby7t(w)=j(t,to,rt(tü)).YetWwillbeadaptedto(Pt).Whichisconvenientsince

weneverneededthefiltration(Pt)beinggeneratedbyW.

Henceforthwearegivenacompletefilteredprobabilityspace

(ft,P,(Pt)teR+,Q)

satisfyingtheusualassumptions,themeasureQbeingconsideredastheriskneutral

measure.LetW=(ß3)je®beaninfinitedimensional(J7i)-Brownianmotionunder

Q.Therandommappingsa(t,to,h)(volatilitystructure)and—j(t,to,h)(marketpriceofrisk)arespecifiedinaccordancewiththefollowingassumptions.

(Dl):Themappingsaand7aremeasurablefrom(R+xftxH,V®B(H))into(L\(H),B(L\(H))),resp.(£2,B(£2)).(D2):ThemappingsSa3aremeasurablefrom(R+xftxH,V®B(H))into

(H,B(H)).(D3):ThereexistsafunctionPGL2(R+)suchthat

\\l(t,üü,h)\\p<r(i),V(i,w,/i)GR+xftxH.

Observethatassumptions(D1)-(D2)togetherwith(H3)implythat"Ylj^Sa3isameasurablemappingfrom(R+xftxH,V®B(H))into(H,B(H)).

Definition2.13.BytheHJMMequation(HJMequationintheMusielapara¬

metrization)wemeanthefollowingstochasticequation7inH

jdrt=(Art+FHJM(t,rt))dt+a(t,rt)dWt

1u

(2.25)[r0=h0

whereFHjM(t,to,h):=£;i6N<Scr-?(i,w,/j).

Definition2.14.Leta,7satisfy(D1)-(D3)andletIcHbeasetofinitial

forwardcurves.Wecall(a,y,I)a(local)HJMmodelinHifforeveryspacetime

initialpoint(to,ho)GR+xithereexistsauniquecontinuous(local)mildsolution

r=r(4o'/l°)tothetimet0-shiftedsHJMMequation(2.25).

Thisdefinitionisjustifiedbythefollowingresult.

Theorem2.15.Let(c,7,i)beanHJMmodel.Thenanyspacetimeinitial

point(t0,h0)£R+xIspecifiesameasureP~QonPândaWienerprocessW

withrespectto(ft,Poo,(Pt)t6R+,P)suchthat(C1)-(C5)holdfor

at(ui)=FHJM(to+t,co,rt(io))-^^(to+t,uj,rt(tj))a3(i0+t,u,rt(uj))

JGN

o-t(ui)=a(t0+t,w,rt(u))

7t(w)=7(i0+t,u,rt(uj))

wherer=r(-t°'h°S).Accordingly,QisanELMMforthissetup.

7SeeRemark1.18fornotation.

8recall(1.18)


Proof.Letr=r^to'h°'>bethecontinuousmildsolutiontothetimei0-shiftedversionof(2.25).From(D3)wededucethat

/||-7(t0+s,u,rs(to))\\%ds<||r||i2(K+)<oo,VwGft.(2.26)

HenceLemma1.14appliesandwecandefinethemeasureP~QonP^=P^0'by

^=£{_y.w(to))oo,(2.27)

AccordingtoGirsanov'stheorem,seeProposition1.16,

Wt=Wt{to)+f%dsJo

isaWienerprocessunderP.

By(H3)andHolder'sinequality

IKHh<K^\\aifH+H\\fi\Wt\\LoiH)<(K+\)\\at\\2Ll(H)+U^tWl-jew

"locf Now(2.26)yields(C2),since(ot)te)R+£(if)bytheverydefinitionofamild

solution.Using(1.15),theprocessrisseentobethecontinuousmildsolutionof

(2.8)with/(O,•)replacedbyh0,whence(C3).Conditions(C4)-(C5)areimpliedby(D1)-(D3)and,clearly,QisanELMMforthissetup.D

Remark2.16.Theorem2.15pointsoutthatamildsolutionrtotheHJMM

equation(2.25)istherightconceptfortheHJMframework.Thisisalsoinaccor¬

dancewithRemark1.25.Itisthereforeanessentialrequesttoderivetheresults

from[7]withoutassumingthatrisa(local)strongsolution,see[7,Assumption2.1]

Remark2.17.LocalHJMmodelsareuselessforpricingfinancialinstruments.

YetinsomecasesweonlycanassertlocalexistenceanduniquenessfortheHJMM

equation(2.25),whichisstilladaptedfortheconsistencyanalysisinChapter4-ThereforewealsoincludedthelocalcaseinDefinition2.14-

ThespecificationoftheinitialforwardcurvesetIisconvenientandallows

toincorporatealsothefollowingclassicalexamplesinourframework.Let—7be

anymarketpriceofrisksatisfying(Dl)and(D3).SeeSections4.4.1-4.4.2for

terminologyandresults.

Example2.18(CIRModel).HereI={g0+yg\\y>0}whereg0,giare

givenfunctions,a1=<r1(ft)=\/oiJo(h)g-\.forallh£Ianda3=0forj>2.

Then(a,7,1)isalocalHJMmodel.

Example2.19(VasicekModel).Thevolatilityisconstant,cr1=\fa~g\,where

gi(x)=e@xanda3=0forj>2.Moreover,I={g0+ygi|y£R}andgoisa

givenfunction.Then(a,7,1)isanHJMmodel.

Itisdesirabletofindeasyverifiablecriterionsforatoprovidea(local)HJM

model.WewillapproachthisbytheclassicalLipschitzcondition,seeSection1.4.2.

Butifa(t,w,h)is(locally)Lipschitzcontinuousinh,whataboutFffjM(t,to,h)7InthenextchapterwepresentaHilbertspaceofforwardcurveswhichisbotheco¬

nomicallyreasonableandconvenientforderiving(local)existenceanduniquenessresultsfortheHJMMequation(2.25).

CHAPTER3

TheForwardCurveSpacesHu

3.1.DefinitionofHw

WeintroduceaclassofHilbertspacessatisfying(H1)-(H3)1andwhichareco¬

herentwitheconomicalreasoningabouttheforwardcurvex>->•rt(x).Sinceinpracticetheforwardcurveisobtainedbysmoothingdatapointsusing

smoothfittingmethods,seeChapters5and6,itisreasonabletoassume

/K(*)|2d,<oc.J$U

Moreover,thecurveflattensforlargetimetomaturityx.Thereisnoreasonto

believethattheforwardrateforaninstantaneousloanthatbeginsin10years

differsmuchfromonewhichbeginsonedaylater.Wetakethisintoaccountby

penalizingirregularitiesofrt(a;)forlargexbysomeincreasingweightingfunction

w(x)>1,thatis,

|r[(x)|2w(x)dx<oo.

However,thisdoesnotdefineanormyetsinceconstantfunctionsarenotdistin¬

guished.Soweaddthesquareoftheshortrate|rt(0)|2.Weshallgiveamathematicallycorrectapproach.RecallthefactthatifftG

Lloc(M.+)possessesaweakderivative2ft'GLl0C(R+),thenthereexistsanabsolutely

continuousrepresentativeofh,stilldenotedbyh,suchthat

h(x)-h(y)=h'(u)du,Vx,y£R+.(3.1)

Thiscanbefounde.g.in[14,SectionVIII.2].Accordingly,thefollowingdefinition

makessense.

Definition3.1.Letw:R+->[l,oo)beanon-decreasingC1-functionsuch

that

w"3GL1(R+).(3.2)

Wewrite

\\h\\l:=\h(0)\2+[\h'(x)\2w(x)dxJr+

anddefine

Hw:={ftGL}0C(R+)|3ft'GL1loc{R+)and\\h\\w<oo}.

ThechoiceofHwisestablishedbythenexttheorem.

"seepage32

2uniquelyspecifiedby/Kh(x)ip'(x)dx=—JRh'(x)<p(x)dxforallif6C£((0,oo))

35

363THEFORWARDCURVESPACESHw

THEOREM3.2.ThesetHwequippedwith\\\\wformsaseparableHilbertspace

meeting(H1)-(H3).

Beforeprovingthetheoremwegivetwoguidingexamplesofadmissibleweight¬ingfunctionswwhichsatisfycondition(3.2).

Example3.3.w(x)=eax,fora>0.

Example3.4.w(x)=(1+x)a,fora>3.

PROOF.Itisclearthat||•\\wisanorm.ConsidertheseparableHilbertspace

RxL2(R+)equippedwiththenorm(||2+[|•H2^\)•

Thenthelinearoperator

T:Hw-»RxL2(R+)givenby

Th=(h(0),h'wi],h£Hw

isanisometrywithinverse

(T-1(u,f))(x)=u+ff(r,)W--(r,)dn,(u,f)eRxL2(R+).Jo

HenceHwisaseparableHilbertspace.

Weclaimthat

P0:={/eC72(R+)[/'GC7c1(R+)}

isdenseinHw.Indeed,Cg(K+)isdenseinL2(R+),see[14,CorollaireIV.23].

Fixft,£Hwandlet(fn)beaapproximatingsequenceofh'wïinL2(R_|_)with

/„GC£(R+)foralln£N.Writehn:=T-1(ft,(0),/„).Obviously,hn£V0and

hn—>hinHw.Whencetheclaim.

Sincew-1isbounded,(3.2)impliesinparticularw"1Gi/1(R+).Thesubse¬

quentSobolev-typeinequalities(3.4)-(3.8)arecrucial.Throughout,hdenotesa

functioninHw.TheconstantsC1-C4areuniversalanddependonlyonw.

Firstweprove

IIÎU1(R+)<Ci|HL.(3.3)

ThisisestablishedbyHolder'sinequality

/|ft'(a;)|da;=/\h'(x)\w^(x)w~^(x)dxJm+Jr+

<n\hl(x)\2w(x)dx)(j^w-1(x)dx\^WhlUw-^l^y

Animportantconsequenceof(3.3)isthath(x)convergestoalimitft(oo)GR

forx—y00,see(3.1).Thisisofcourseadesirablepropertyforforwardcurves.

ForC2=1+C\itfollowseasilyfrom(3.1)and(3.3)that

IWIlo=(r+)<C2\\h\\w.(3.4)

Writen(:r):=f^°w~1(if)dn.Thenwehave,bymonotonicityofw,

/•OO

U(x)<w~i(x)w~3(n)dr]<w~i(x)\\w~^\\Li(M+),x£R+JX

31DEFINITIONOFff„37

andhence,by(3.2),

Moreover,

in2tulL~(m+)<\\w3|U1(k+)<°°-

-iii2|n2||ii(K+)<\\w3llli(R+)<0O.

Bythesamemethodasabovewederive

\h(x)—h(oo)\dx=

<

<IHUin*||Li

h'(r))w2(f])w2(rj)dr]dx

0\2

\ti(n)\2w(n)dn)U^(x)dx

Hence,by(3.6),

||ft-ft(co)|U1(E+)<C3||ft|U.

Finallywecompute,usingHolder'sinequalityagain,

/poo4/h1(rj)wï(rj)w~ï(n)dnw(x)d.

(3.5)

(3.6)

(3.7)

<//\h'(rj)\2w(n)dr]\IL2(x)w(x)dx

<l|/i||illn*u;||i~(R+)II11JIlL1nil

Therefore,by(3.5)-(3.6),

||(ft-ft(co))4w;||L1(K+)<a(3.,

Nowwecanprove(H1)-(H3).Estimate(3.4)yieldscontinuityofthelinear

functionalJxforallx£M.+.

Sincebytheprecedingremarks,see(3.1),Hwconsists

ofcontinuousfunctions,(HI)isestablished.

Leth£Hwand<p£C\((0,oo)).ThentheshiftsemigroupS(t)satisfies

/S(t)h(x)<p'(x)dx—

/h(x)<p'(x—t)dx=—

/h1(x)ip(x—t)dxJm+Jr+Jr+

h'(x+t)ip(x)dx

(3.9)

sinceip(-—t)£C\((t,oo)).Consequently,(S(t)h)'existsandequalsS(t)h'.In

viewof(3.4)itfollows

\\S(t)h\\2w=\h(t)\2+[\h'(x+t)\2w(x)dx<(C2+l)\\h\\2w.(3.10)

Wehaveusedthemonotonicityofw.HenceS(t)h£HwandS(t)isboundedfor

alliGR+.

383.THEFORWARDCURVESPACESHw

WehavetoshowstrongcontinuityofS(t).Westartwiththefollowingobser¬

vation.By(3.1)

h(x+t)-h(x)=th'(x+st)ds,h£Hw.(3.11)Jo

LetgGV0.Sinceg'GHw,(3.10)and(3.11)yield

\\S(t)g-g\\l=\g(t)-g(0)\2+f\g'(x+t)-g'(x)\2w(x)dx

<\g(t)~g(0)\2+t2ff\g"(x+st)\2w(x)dxdsJoJm+

<lsW-5(0)|2+i2/\\S(st)g'\\2wds^0fori->0.io

Hence5(i)isstronglycontinuousonVq.ButforanyftGHwande>0there

existsg£T>owith||ft—g^^,<4(Ce+1)Combiningthiswith(3.10)yields

\\S(t)h-h\\w<\\S(t)(h-g)\\w+\\S(t)g-g\\w+\\g-h\\w

^C2^Äd^)+m)9-^+wtv)êforismallenough.WeconcludethatS(t)isstronglycontinuousonHw,whence

thevalidityof(H2).Itremainstoprovetheestimatein(H3).LetftGHw.Similarlyto(3.9)one

showsthat(Sh)'existswith

(Sh)'(x)=h'(x)/h(n)dn+h2(x).Jo

Wedefine

H°w:={h£Hw\ft(oo)=0}.(3.12)

By(3.4),H%isaclosedsubspaceofHw.Weclaimthat

Sh£Hw^^h£H°w.(3.13)

Assumethath(oo)>0.ThenthereexistsxoGR+suchthath(x)>-^-forall

x>xo-Butthen

h(oo)\..

1(x—Xo),vx>a;o

rx0

Sh(x)>h(x)/h(n)dn+JoV2

whichexplodesforx—>oo.Thesamereasoningappliesifft(oo)<0.Whencethe

necessityin(3.13).

Conversely,assumethatft(oo)=0.Using(3.7)and(3.8)weestimate

\\Sh\\2w=[h'(x)fh(n)dn+h2(x)w(x)dxJs.+Jo

<2fJ\h'(x)\2(f\h(Tj)\di]\+hA(x)Iw(x)dx

<2(||^|||1(K+)\\h\\l+C4\\h\\i)<2(C2+C4)

HenceSh£Hwand(3.13)isproved.Moreover,weconcludethat(H3)holdswith

K=^2(C2+Ci).U

3.1.DEFINITIONOFHw39

Remark3.5.Itisnotdifficulttoseethat(HI)and(H2)holdwithoutas¬

sumption(3.2).Yetitmakestheproofmorehandy.

TheprecedingproofallowsforidentifyingthefullgeneratorAofS(t)inHw.

Corollary3.6.WehaveD(A)={hgHw|ft'gHw}andAh-

h!.

Proof.WriteV:={ft£Hw\h!£Hw}.ByLemma2.2,D(A)CV.

Conversely,letftGV.Applying(3.11)weget

h'(x+t)-h'(x)

Consequently,

i(t):=

-h"(x)=

f(ft"(a;+st)-h"(x))ds.Jo

h'(x+t)-h'(x)

<

h"(x)w(x)dx

\h"(x+st)-h"(x)\2w(x)dxds

<f\}S(st)h'-h'\\2wds^0fori->0Jo

bydominatedconvergence,see(3.10).Sinceft'iscontinuousonR+,see(3.1),we

conclude

S(t)h-ft

t

ft'h(t)-ft(0)

ft'(0)+n(i)->0foriHÔ,wI

whenceh£D(A).D

MostimportantfortheHJMMequationisthefollowingresult.

Corollary3.7.ThemappingS:H®->HîslocallyLipschitzcontinuous,

thatis

\\Sg-Sh\\w<C5(\\g\\w+\\h\\w)\\g-h\\w,Vg,h£H°w

wheretheconstantC5onlydependsonw.

Proof.Inviewof(3.13)and(3.7)it'simmediatethatSh£H°forallftGH^.Letg,h£H^.UsingHolder'sinequalityand(3.7)-(3.8)wecalculate

\\Sg-Sh\\2w=[g'(x)fXg(V)dV-h'(X)fft(n)dV+g2(x)-h2(x)*

w(x)dxJe.+JoJo

Jm+

<[["KV)Jm+Jo

dn\g'(x)-h'(x)\w(x)dx

<3/\g'(x)\2Iig(v)-h(r]))dvw(x)dx

+3

+3/(g(x)+h(x))2w*(x)(g(x)-h(x))2w^(x)dxJr+

<3(\\g\\2w\}g-ft|li1(R+)+||ft|l!1(R+)\\g-h\\l)

+3||(5+ft)4w|||1(R+)\\(g-h)4w\\lHR+)

<3(C32+2C74)(1M]<\\g-h\\l


andwehaveusedtheuniversalinequality

\xi+---+xk\2<k(\xi\2+---+\xk\2),

feGN.(3.14)

SettingC75=y^Cf+2C4)yieldstheresult.

3.2.VolatilitySpecification

Workinginthespaceifu,wecangivesimplecriteriafora(t,u>,h)toprovidea

(local)HJMmodel.

Condition(D2)and(3.13)clearlyforcea3£Hi,recall(3.12).Thisisof

courseadesirablepropertysincewedon'twanttheforwardratesrt(x)beingtoo

volatileforlargex,see(2.6).Infact,Corollary3.7yields

LEMMA3.8.Letameet(Dl)withH=HwandL°,(H)replacedbyL^(Hl).Then(D2)issatisfiedaswell.

FromnowonwerequirethatasatisfiestheassumptionsofLemma3.8.In

viewofCorollary1.24it'senoughtorequirelocalLipschitzcontinuityofa(t,to,h)andFhjm^jU,ft)inftforassertinglocalexistenceanduniquenessfortheHJMM

equation(2.25).ThechoiceofHwgainsininterestifweconsiderthefollowing

property.

Letcr(t,oo,ft)belocallyLipschitzcontinuousinft,withLipschitzconstantC=

C(R)andsuppose

lk(*.w,0)||Lo(irM)<C,V(t,to)£R+xft(3.15)

foraconstantC.Thisclearlygives

\\a(t,to,h)\\Lo{Hio)<C+C(R)R,Vh£BR(Hw),ViîGR+.(3.16)

Whenceaislocallybounded.

Lemma3.9.i)Letasatisfytheaboveassumptions.ThenFHjM(t,to,h)is

locallyLipschitzcontinuousinft,,

ii)Supposeinadditionthata(t,ui,h)isLipschitzcontinuousinftanduniformlybounded:

\\o-(t,to,h)\\Lo{Htij)<C,V(t,u,h)£R+xftxHw,(3.17)

forsomeconstantC.ThenthesameholdstrueforFHjM(t,oo,h).

Proof.Forsimplicityofnotationweabandontemporarilythe(i,(^-depen¬denceofaandFhjm-

Letfti,ft,2GBR(HW)andwriteA:=\\FHjM(hi)-FHjM(h2)\\wByCorol¬

lary3.7andHolder'sinequality

A<J2\\^j(hi)-Sa3(h2)\\wjGN

<CbJ2(Hî)IL+\\o-j(h2)\\w)IK(fta)-a3(h2)\\w

<C5(H^MHlo^)+h(h2)\\L°(Hm))h(hi)-o-(/i2)||Lo(Jïu:)

<G5C(R)(iKftOHiO^)+\W(h2)\\LUHw))lifta-h2\\w.

Combiningthiswith(3.16)provesi)andthefirstpartofii).Theboundedness

assertiononFhjmisaconsequenceof(H3).

3.3.THEYIELDCURVE41

CombiningLemmas3.8-3.9withTheorem1.23andCorollary1.24wegetour

mainexistenceanduniquenessresultfortheHJMMequation.Denoteby—7a

marketpriceofriskvectormeeting(Dl)and(D3).

THEOREM3.10.i)LetabeasinLemma3.9.i).Then(a,y,Hw)isalocal

HJMmodelinHw.

ii)LetabeasinLemma3.9.U).Then(ct,j,Hw)isanHJMmodelinHwand

alldiscountedbondpriceprocesses(Z(t,T))te[0,T]aretrueQ-martingales.

Moreover,anymildsolutionr^to,h°ôfthetimeto-shiftedHJMMequation

(2.25)isalsoaweaksolution,(to,ho)£R+xHw.

Proof.Onlythestatementon(Z(t,T))tç[o,T\remainstobeproved.Butthis

isastraightforwardconsequenceofLemma2.9andtheestimate

\\lT-toa(t,rt)\\%<(Tk(T))2\\a(t,rt)\\2Lo{H^<(Tk(T))2G2,

seeLemma2.4.D

Weshallexaminethreetypesofpossiblevolatilityspecifications.Thefirstone

correspondstotheclassicalHJMframeworkwherethevolatilitydependslocallyon

thestatevariable.ItturnsoutthatinthiscaseonemerelygetslocalHJMmodels.

Forexampletheonewhichispresentedin[28,Section7].Thesecondtypeismuchmoreconvenient.Herethevolatilitydependsona

functionalvalueofthestatevariable.Itiseasytofindsufficientconditionsfor

providinganHJMmodelinthiscase.

Lastly,werecapturetheBGM[12]modelwhichwealreadymentionedinSec¬

tion2.4.4.However,sincealsothistypeofvolatilityisoflocalnature,wemerely

getalocalHJMmodel.

Subsequently,therefollowsanintermediarysectionwhereweintroduceanim¬

portantfinancialvariable.

3.3.TheYieldCurve

WealreadyknowbyRieszrepresentationtheoremthatthefunctionalsJxandTx

canbeidentifiedwithelementsinHw,seeTheorem3.2andLemma2.4.Let'sjust

mentionhowtheylooklikeasfunctions.

Lemma3.11.Wehave

Jx(rj)=1+/—rrdu,n£

Jow(u)

-uA

w(u)_

.

j"nx-uAxTx(v)=x+r^—du,j]£

aselementsinH,,

PROOF.WeonlyproofthestatementonTx.LetftGif«,.Integrationbyparts

yields

(Ix,h)w=xh(0)+j(x—nAx)ft'(77)drjJm+

=xh(0)+f(x-n)h'(n)dr]Jo

fXfX

=xh(0)-xh(0)+/h(rj)dn=/h(n)dr).JoJo

423THEFORWARDCURVESPACESH,

D

Thenextresultwillproveveryuseful.RecallthatA*denotestheadjointofA.

Lemma3.12.Wehavelx£D(A*)andA*XX=Jx-J0forallx£R+.

Proof.FromCorollary3.6weknowthL.t

(lx,Ah)w=fti(n)dvi=h(x)-h(0)=(Jx-Jo,h)w,Vh£D(A).Jo

D

Besidestheforwardcurveandthetermstructureofbondpricesthereisanother

measurementofthebondmarketinbetween.

Definition3.13.Giventhecontinuoustimetforwardcurvex1-4rt(x),the

timetyieldcurvext->-yt(x)isdefinedby

1fxyt(x):=-/rt(r})dn,x£

xJo

Noticethatyt(0)=r*(0)iswelldefined.

Incontrasttoforwardrates,yieldsofzerocouponbondscanbedirectlyob¬

servedonthemarket.WewillexploitthisfactinExample3.18below.

Forcompletenessweshalldescribetheimplieddynamicsfory.Itturnsout

thattheyieldcurveprocessenjoysniceregularityproperties.

LetasatisfytheassumptionsofLemma3.9.Ü)andletrbeaweaksolutionof

theHJMMequation(2.25)inHw.Forafixedx>0wehavebyLemma3.12that

(xyt(x))=(Ix(rt))isareal-valuedsemimartingale.TakingintoaccountLemma2.5

wederivethedecomposition

xyt(x)=lx(r0)+f[(Jx-Jo)(rs)+\^(J,(^(s,rs)))2)dsJo

jGN

+Wlx(a3(s,rs))dß3s.„r-T^JO

H

Hencetherandomfield(Y(t,x)):=(xyt(x))isaclassicalsolution(thatis,(t,x)-pointwise)ofthestochasticpartialdifferentialequation(SPDE)

dY(t,x)=(^-Y(t,x)-rt(0)+^CWfrrt)))2)dt+£lx(a3(t,rt))d~ß\

\^;GN

JjGN

k

Y(0,x)=Ix(r0).

3.4.LocalStateDependentVolatility

HJMpresentin[28,Section7]aclassofmodelswithlocalstatedependentvolatility.

Inthepresentsettingthisreads

a3(t,u,h)(x)=&(t,uo,x,h(x)),jGN(3.18)

forsomerealvaluedfunctions$J(i,w,x,s).Weshallformulatesufficientcondi¬

tionson($-7)i76Nsuchthata—(cr3)jçngivenby(3.18)satisfiesthehypothesesof

34.LOCALSTATEDEPENDENTVOLATILITY43

Lemma3.9.i).Writingf(t,T)=rt(T-t)foramildsolutionr=r(0'/l°)tothe

HJMMequation(2.25)(ifitexists)weget

f(t,T)=h0(T)+fW<F(s,T-s,/(s.T))/$3(s,T-s,f(s,u))du)dsJo

jgnVJ*J

+W&(s,T-s,f(s,T))dß3s^mJ0

(3.19)

whichis[28,Equation(42)].

Proposition3.14.Let$3(t,to,x,s):R+xftxR+xR=:D->R,jGN,

satisfythefollowingassumptions

i)$3(t,to,x,s)isV®B(R+)®B(R)-measurableii)$3(t,to,x,s)iscontinuouslydifferentiablein(x,s)forall(t,co,x,s)£D

iii)limôo$J(i,to,x,s)=0forall(t,oo,s)£R+xftxR.

Suppose,moreover,thereexistacontinuousmappingc=(c3)je^:R—>£2with

c3convexandc3(—s)=cJ(s),andameasurablemapping\?=(93):iefq:R+—>£2

with

/|]*(a;)||22w(x)dx<oo

meeting

iv)\<t>3(t,oo,0,s)\<c3(s)v)|ôB*J(t,w,a;)s)|<^(a;)c»(s)vi)\ds$3(t,to,x,s)\<c3(s)vii)I^^Ô.sO-^^w.Oâ)!<(c,(*i)+c'(*2))|si-S2|viii)\dx$3(t,oo,x,Sl)-dx&(t,to,x,s2)\<iH3(x)(c3(s1)+c3(s2))\s1-s2\ix)|ôs$J(i,w,a;,si)-ds$3(t,to,x,s2)\<(c3(si)+c3(s2))\si-s2|

forall(t,oo,x,s)£D,s\,s2GMandj£N.Thena=(a3)je^definedby(3.18)

fulfillsthehypothesesofLemma3.9.i).

PROOF.Wheneverthereisnoambiguityweshallskipthenotionaldependence

on(t,oo).Theproofisdividedintofoursteps.

Step1:Let'srecallthechainrulefortheweakderivative.Forh£Hwwehave

4-&(x,h(x))=dx$3(x,h(x))+ds$3(x,h(x))h'(x).

dx

Thiscanbeshownusingii)asintheproofof[14,CorollaireVIII.10].

Step2:Nowweshowthata(h)£L^(Hl)forallh£Hw.

LetftGHw.Clearlylima:^.00cr;'(ft)(x)—0,byii)andiii).Assump¬

tionsiv)-vi)yield

\\a3(h)\\2w<\<S>3(0,h(0))\2

+2/(\dx$3(x,h(x))\2+\ds&(x,h(x))\2\ti(x)\2)w(x)dxJm+

<\<?(\\h\\w)\2(l+2^{\V3(x)\2+\ti(x)\2)w(x)dx\

whichimpliestheassertion.

443.THEFORWARDCURVESPACESH,

Step3:Nextwecheckmeasurabilityof(t,to,h)>->a3(t,oo,h).It'senoughto

showmeasurabilityof(t,to,h)(->•(a3(t,to,h),g)wforallgGHw,see[18,Proposition1.3].

Fixg£Hw.Byi),ii)andLemma2.1themapping

(i,to,x,h)h-»dx§3(t,oo,x,h(x))

isV®B(R+)®B(ifTO)-measurable.Thesameappliesfords&(t,oo,x,h(x)).Henceforany/GHwthefunction

(t,u,h)Î(t,oo,h)(f):=/ds&(t,oj,x,h(x))f'(x)g'(x)w(x)dxJr+

isV®ß(ifu,)-measurable.Hereweusedvi)andamonotoneclassargument,

see[4,Lemma23.2],Since/GHwwasarbitrarythisimpliesthat(i,to,h)>->

L(t,oo,ft)isaV®B(H^-measurableapplicationintothedualspaceofHw

whichweidentifywithHwasusual(NoticethatI(t,to,ft)isaboundedlinear

functionalonHw,byvi)).Similarreasoning,usingv),showsthat

(t,to,h)(->J(t,to,h):=/dx$3(t,to,x,h(x))g'(x)w(x)dx

isV®ß(if„,)-measurableinto'.

Butnowwecanwrite

(a3'(t,oo,h),g)w=&(t,oj,0,J0(h))g(0)+J(t,u,h)+I(t,oo,h)(h)

anddeducethedesiredmeasurabilityassertion.

Step4:ItremainstochecktheboundednessandLipschitzproperties.Observe

that(3.15)isadirectconsequenceofiv)-v).Letfti,ft2GHw.Inview

of(3.14)wehave

\\<T*{h1)-cT>(h2)\\l<\&{0M0))-&(P,h2(0))\2

+3f\dx&(x,h(x))-dx&(x,h2(x))\2w(x)dxJm+

+3/\ds¥(x,h1(x))(h'l(x)-h'2(x))\2w(x)dxJm+

+3/\h'2(x)(ds^>j(x,h1(x))-ds&(x,h2(x)))|2w(x)dxJm+

<C(h2)(c»(||hi|U)+^(||ft2|U)2||ft!-

ft2"2

where

C(h2)=1+3f||*(a;)||22w(x)dx+3+3||ftJvu

'2

22\\w

andwehaveusedvi)-ix).Thereforea(t,oo,ft,)islocallyLipschitzcontinuous

inft,andthepropositionfollows.

D

Thefollowingtwoexamplesarepresentedin[28,Section7].

Example3.15.Letcj>£Hi.Define^1(t,oo,x,s)=$x(x,s):=4>(x)sand

&=0forj>1.TheassumptionsofProposition3.14areeasilyseentobefulfilled

withc1(s)=(1+IMUXl+|s|)and^(x)=\4>'(x)\,andc3'=W=0forj>1.

AccordingtoTheorem3.10,thesospecifiedvolatilitymappingaprovidesalocal

3.5.FUNCTIONALDEPENDENTVOLATILITY45

HJMmodel.However,itisshowninMorton'sthesis[36]thateverymildsolution

rtotheHJMMequation(2.25)explodeswithprobabilityoneinfinitetime.

Wedefine

x(s):=Tjr(|s|AÀ),sel(3.20)

whereÀstandsforsomepositivenumber.

Example3.16.LikeExample3.15,butthistime$1(x,s):=<j)(x)x(s).A

slightmodificationoftheprecedingproofshowsthatalsohereafulfillsthehypothe¬

sesofLemma3.9.i).Itisprovedin[28,Proposition4]that(3.19)hasajointlycontinuoussolution(f(t,T))(t,T)gA2-Yetwestillcannotassertglobalexistencefor

theHJMMequation(2.25)sincethenorm\\rt\\wisriotaprioricontrollable.

InviewofExample3.16onemaywishtoconsidertheHJMMequation(2.25)inalargerspacelikei2(R+)equippedwithnorm

ll/ll!2(K+):=/\f(x)\2e-axdx,a>0.JllL+

ThisapproachisusedbyGoldysandMusiela[27].Indeed,Example3.16providesa

globalmildsolutiontoequation(2.25)ifinterpretedinL2(R+).However,L2(R+)doesnotmeet(HI)and(H3)anditisnotclearhowtheanalysisofChapter2

couldbeperformedinthisspace.

3.5.FunctionalDependentVolatility

Veryconvenientforthepresentframeworkisafunctionaldependentvolatility

cr3(t,to,h)(x)=&(t,to,x,(3\h)),jGN(3.21)

where(3arecontinuouslinearfunctionalsonHwand$J(i,to,x,s)arereal-valued

functions.Thefollowingpropositioncontainsappropriateconditionson&tomake

surethata—(a3)jenin(3.21)providesanHJMmodel.

PROPOSITION3.17.Let&(t,oo,x,s):1+xftxR+xR=:D-»R,j£N,

satisfythefollowingassumptions

i)&(t,u,x,s)isV®B(R+)®B(R)-measurableii)<&3(t,oo,x,s)iscontinuouslydifferentiableinx

iii)lim^ôo<F(i,w,x,s)=0forall(t,oo,s)£R+xftxR

iv)\dx&(t,w,x,8)\<&{x)v)\dx&(t,u,x,sl)-dx&(t,oj,x,s2)\<W(x)\sl-s2\

forall(t,oo,x,s)£D,si,s2GR,and$=(ty3)jS®a*inProposition3.14-Then

a—(a3)jefidefinedby(3.21)fulfillsthehypothesesofLemma3.9.H).

PROOF.WeproceedasintheproofofProposition3.14.Againweskipthe

notionaldependenceon(i,oo)wheneverthereisnoambiguity.


Assumptionsii),iii)andiv)yield$3'(-,s)GHland,usingHolder'sinequality,

\\*i(;s)\\l<\&(0,s)\2+f\V(x)\2w(x)dxJm+

<[[\dx$j(x,s)\dx)+[\¥(x)\2w(x)dx\Jm+JJr+

<U\&(x)\2w(x)dx)(Ww-'h^+l)

forallsGR.Fromthiswededucea(h)£L°,(Hl)forallftGHw.Moreover,(3.17)holds.

ByasimilarargumentasinStep3intheproofofProposition3.14,usingi)andii),wederivethemeasurabilityof

(t,oo,h)^{a3(t,oo,h),g)w=&(t,oo,0,Cj(h))g(0)

dx<è3(t,oo,x,Ç3(h))g'(x)w(x)dx

forarbitraryg£Hw.

Itremainstoshowthata(t,to,h)isLipschitzcontinuousinft.Butthisisa

directconsequenceofv)andiii)whichimpliestheequality

&(t,oo,0,s)=-

/dx$3(t,to,x,s)dx.Jr+

Thefollowingexamplelookspromising.

Example3.18.Fixm£Nandmbenchmarkyields,seeDefinition3.13,

—l5l(rt),...,T-Pôm(n),0<<5i<••<öm,oiàm

whicharedirectlyobservableonthemarket.Let(j)1,...,cf>m£Hiandset

^(t,oo,H)ô-Kh):=^Xi^{h))^j^m

[0,j>m

see(3.20).ThenProposition3.17applieswith

fttäxis),j<m

id®3=\

$3(t,oj,x,s)=$3(x,s)[0,j>m

(cf>3)forj<m,resp.$3=0forj>m.

Al Thefunctions<j>,...,

(fimcanbeinferredbystatisticalmethods.

3.6.TheBGMModel

AprominentrepresentativewithintheHJMframeworkistheBGM[12]model

whichwasalreadymotivatedinSection2.4.4.Itisdesirabletohaveitincluded

inthepresentsetup.Inthissectionweconstructthereforeavolatilitystructurea

satisfyingtheassumptionsofLemma3.9.i)andwhichmeets(2.22).

3.6.THEBGMMODEL4T

RecallthenotationfromSection2.4.AssumethatwewanttopriceanycapletwithsettlementdateT+6<T*accordingtoformula(2.23),forsomeT*£

R+.WithoutlossofgeneralityT*=K*8forK*£N.Motivatedby(2.22)we

arelookingforafamilyofdeterministicfunctions(63(t,x))je^suchthatforany

(i,ft)GR+xHw

l*+s(a3'(t,h))=(l-e-z'S'W'}93(t,x),Va;G[0,(K*-1)6}.(3.22)

Wesuppresstemporarilythenotionaldependenceoniandj.Supposeforthe

momentthat6(x)isarbitrarilysmoothinx.Takingderivativein(3.22)weget

o-(h)(x+5)-<j(h)(x)=4>(h,x),Vx£[0,(K*-l)S),(3.23)

where

<t>(h,x):=(h(x+6)-h(x))e-^+Sê(x)+(l-e^'W)8x0(x).

Recurrencerelationship(3.23)definesavolatilitystructurea(h)(x)forx£[6,K*5)provideda(h)(x)isdefinedon[0,6).Noticethat(3.23)impliesequality(3.22)only

uptoaintegrationconstantCwhichisspecifiedbysettingx=0

Is(o-(h))=(l-e-Is^)e(0)+C.

Wewilllaterset0(0)=0,see(3.24).ThusC=0ifandonlyifls(a(h))=0for

allh£Hw.FollowingBGM[12]weseta(h)(x)=0on[0,6).Solving(3.23)we

derive

k

a(h)(x)=J2^(/l>x~iô)forxG\.kS>(k+1)J)and1<fc<IT-1.

i=l

Thisimplieslim^rf.+^ga(h)(x)—Yî=i<f>(h,i6)andweseethata(h)(x)iscontin¬

uousina;G[0,K*6)forallftGHwifandonlyif(p(h,0)=0forallftGHw.That

is,ifandonlyif

0(0)=030(0)=0.(3.24)

Wehavetoextendthedefinitionofa(h)(x)tothewholeR+.Set

K*-l

o(h)(K*5):=lima(h)(x)=]T</>{h,iô)

i=l

andextrapolatelinearlyto0

a(h)(x):=a(h)(K*6)(l-ix~^*S)\y

x>K*S_

Thisconstructionimpliesthata(h)'existsini,110C(R_)_)and

o-(h)'(x)=<

fO,x£[0,6)

Y!l=idx<t>(h,x-i6),x£(k6,(k+l)6),1<k<K*-1

-\o-(h)(K*5),x£(K*5,(K*+1)6)

[0,x>(K*+l)ô.

(3.25)

483.THEFORWARDCURVESPACESJgw

Wehavetocheckwhethera(h)£HiandlocalLipschitzcontinuity.Inviewof

(3.25)wecompute

dx<j>(h,x)=(h'(x+6)-h'(x))e-^+Sê(x)

-(h(x+5)-h(x))2e-^+S^9(x)

+2(h(x+6)-h(x))e-T*+SWdx6(x)

+(i-e-^s^)d2xe(x).

Thefollowingestimatesarebasic.From(3.14)weobtain

|a^(ft,x)|2(l+e2*IIMU^[2(\h'(x+6)\2+\h'(x)\2)\9(x)\2 <

\4

*+S\h'(V)\dT,\\9(x)\2

+4U*+S\ti(r1)\dV]l\dx9(x)\2

+2\d2x9(x)\2

andthusby(3.3)

\dx<t>(h,x)\2<(l+e2*PII„)U(\h'(x+S)\2+\ti(x)\2)\9(x)\24

+C714||ft[|t|0(a;)|2

+4C12||ft||2u|ôcc0(a;)|2(3"26)

+l\d2J(x)\

Letfti,ft,26Hw.WedecomposeA:—dx<f>(h\,x)—dxcj)(h2,x)accordingto

A=(h[(x+6)-h[(x))(V1**4^)-e"1^'^2))6(x)

+e-IS+ä(^)(ft'1(a;+S)-h'2(x+6)+h[(x)-h'2(x))6(x)

~

(f+SK(v)dri)(e-^+ä(^)-e-^+ä(^))9(x)

_

e-i:+ä(^2)(fx+S(h[(n)+h'M)dJ([+\K(v)-h'2(v))dVj9(x)

+2(j***h[(rj)dn)(e~^+S^-e-^+5(^)dx9(x)

+2e-I*+iîjX+5(h'M-

h'2(n))dr\dx9(x)

3.6.THEBGMMODEL49

Using

g-ZÎ+'CAl)_

e~i:+S(h2)<SWh-h2\\we6^h^+Wh^\

(3.3)and(3.14)weget

JAl!<^(IIMU+liM»)(2(\h[(x+6)\2+|Jii(aO|2)<S2||/ii-h2\\l\9(x)\2

+2(\h[(x+6)-h'2(x+5)\2+\h[(x)-h'2(x)\2)\9(x)\2

+Ct\\h1\\i82\\h1-h2\\2w\9(x)\2

+2C71(||ft1||2„+||ft2[|2ü)||ft1-ft2||2„|0(x)|2

+4C72[M|2u52||ft1-ft.2||2J&0(a;)|2

+4C712||/i1-ft2||2„|Ôœ0(a;)|2

+ö2\\h1-h2\\2w\d2x9(x)\2).(3.27)

Inthesequel,Cdenotesarealconstantwhichonlydependsonw,6andK*

butrepresentsdifferentvaluesfromlinetoline.

Inviewof(3.25)

ffK*5/|a(ft)'(x)\2w(x)dx=/\a

Jr+Jo

ButHolder'sinequalityyields

rK*5

'*£\|2

(h)'(x)\2w(x)dx

"Jk*6

K*5

\a(h)(K*6)\2<(/\a(h)'(x)\dx\<C\\a(h)'(x)\2w(x)dx,

andso,by(3.14)and(3.25)

/\a(h)'(x)\2w(x)dx<Cf\a(h)'Jr+Jo

[x)\2w(x)dx

K*-lf(k+l)ök

=C2_j/|y'idx(j)(h,x—i6)\2w(x)dx

fc=i'kö

i=i

K*-lkf(k+l)5-CEE/\dxcp(h,x-i6)\2w(x)dx

T._1;_1

JkS

(3.28)

fc=li=l

<Cw(K*5)j\dx<p(h,x)\2dx.

whereI:=[0,(K*—1)5].Inthesamemannerwecanseethat

f\<r(hi)'(x)-a(h2)'(x)\2w(x)dx<Cw(K*5)f\dx<t>(hux)-dx</)(h2,x)\2dx.J«+Ji

(3.29)

503.THEFORWARDCURVESPACESHx

From(3.26)and(3.28)itfollowsnoweasily

\\a(h)\\2w<C(l+e2^»»)(||0||^(7)+||0||22(/)+R0||22(/)+||<920||22(/))<c{i+e2^)\\dle\\2LHI)

(3.30)

andwehaveused(3.24).Combining(3.27)and(3.29)givessimilarly

\\a(h)-a(h2)\\2w<Ce^llM-.+IIM«.)^~h2\\2w||<920[||2(7).(3.31)

Wearethusledtothefollowingresult.

Theorem3.19.Let(93(t,x))jeNsatisfy

i)(t,x)i->93(t,x):R+xI—¥Risjointlymeasurable

ii)93(t,-)£C2(I),ViGR+iii)93(t,0)=dx63(t,0)=0,ViGR+

iv)supiGffi+VjigN\\d2x63(t,•)lll2(/)<oo.

Thentheaboveconstructionyieldsameasurablemapping

a=(a3)jm:(R+xHW,B(R+)®B(HW))-+(L°2(Hl),B(L°2(Hl)))

whichmeets(3.22)andtheassumptionsofLemma3.9.i).

PROOF.Thestructureandargumentsoftheproofaresimilartothoseofthe

proofofProposition3.14.Weshallgiveasketch.

Conditioniii)is(3.24).Estimate(3.30)andiv)clearlyyielda(t,h)£L^(Hl)and(3.15).Letg£Hw.Measurabilityof(t,ft)i->(a3(t,h),g)wfollows,usingi)

andii),bysimilarargumentsasinStep3intheproofofProposition3.14.

Finally,iv)and(3.31)implylocalLipschitzcontinuityofa(t,ft)inft.D

ThesofoundlocalHJMmodelisaslightmodificationoftheBGM[12]model.

Thedifferenceisthat(2.22)holdsforT<T*-5withn3(t,T)=93(t,T-t).

Consequently,onlycapletswithsettlementdateT+6<T*canbepricedaccording

toformula(2.23).Fromapracticalpointofviewthisisnorestrictionatall,since

T*canbechosenarbitrarilylarge.Fromthemathematicalpointofviewthis

restrictionisessential:T*=K*5hastobefiniteinestimates(3.28)and(3.29).Unfortunately,wecannotproveexistenceofanHJMmodelwhichhasprop¬

erty(3.22).Incontrasttotheprecedingresult,BGM[12]canassertglobalexis¬

tenceforequation(2.25)ifconsideredasanSPDE,see[12,Corollary2.1].Yettheir

SPDEmethodsgivenoinformationabout||rt[|w.Moreover,theydonotprovidea

generalframeworkfor(local)HJMmodelsasinthepresentthesis.

Insummary,wehaveshownthattheBGM[12]modelis(essentially)recap¬

turedbyoursetting.Inparticular,itisavailablefortheconsistencyresultsin

Chapter4.

CHAPTER 4

Consistent HJM Models

4.1. Consistency Problems

In practice, the forward curve x i->- rt(x) is often estimated by a parametrized

family G — {G(-,z) \ z £ Z] of functions in Hw with parameter set Z C Rm,

m G N. By slight abuse of notation, we use the same letter G for the mapping

z i-> G(-, z) : Z -» Hw. The parameter z £ Z is daily chosen in such a way that the

forward curve x >-> G(x,z) optimally fits the data available on the bond and swap

market. This means that one observes the time t evolution (rt)teR+ of the forward

curve by its Z-valued trace process (Zt)teR+ given by

G(Zt)=rt. (4.1)

Relation (4.1) implies rt G G- If we assume that r is given by an HJM model

(a, 7, i) in Hw then this requires a certain consistency between (a, 7,1) and the

family G- This problem was first formulated by Björk and Christensen [7]. Two

questions naturally arise:

i) What are the conditions on (a, 7.1) such that for any (io, fto) G R+ x (I fl Q)there exists a non-trivial Z-valued process Z and (4.1) is (locally) satisfied

for r = r-(to'h°)?

ii) If so, what kind of process is Z1

Both questions are related to invertibility of (4.1) and have been studied in the

subsequent articles for exponential-polynomial families (Chapters 5-6) and for the

regular but general case where G is a submanifold in Hw (Chapter 7). Accordingly,we considered different concepts of consistency.

Definition 7.8 straightly extends to the set M = G-

Definition 4.1. A local HJM model («7,7,1) in Hw is consistent with G if

i) G CI

ii) G is locally invariant for the HJMM equation (2.25) equipped with a.

This definition replies to question i).

REMARK 4.2. It is evident that consistency is a property of a and I only. This

fact is also reflected by the invariance of the Nagumo type consistency conditions

(7.2)-(7.3) with respect to the Girsanov transformation (2.27).

Here is a stronger version of the previous definition, which corresponds to ques¬

tion ii).

Definition 4.3. A local HJM model (c,j,I) in Hw is r-consistent1 with G if

i) G CI

1This terminology is in accordance with Definition 4.1 in [7].

51

52 4. CONSISTENT HJM MODELS

ii) For any (to, ho) G R+ x G there exists a Z-valued Itô process2

rt rt

z(tQ,h0) = Zq+ f bgds+ f PsdW^Jo Jo

such that G(Z) is a local mild solution to the time to-shifted HJMM equa¬

tion (2.25) equipped with a and initial condition G(Zq) = ho-

Any such state space process Z in turn is called consistent with G if G(Z) is a

global mild solution.3

By uniqueness of r = r(to./lo)) r-consistency of (a, 7,i) with G implies (4.1) for

r and Z = Z(to'h°^ and therefore implies consistency. The converse is not true in

general. In Chapter 7 we give criterions for the latter to hold. A restatement of

Corollary 3.7 and Lemma 3.9 yields their applicability. Conditions (A1)-(A5) are

formulated at the end of Section 7.2.

Lemma 4.4. Suppose that a satisfies the assumptions of Lemma 3.9Ä). Then

(A4) is satisfied. If in addition o~(t,oo,h) is right continuous in t £ R_)_ for all

h £ Hw and oo G ft, then (A5) holds too.

We can now rephrase Theorem 7.13 as follows.

Theorem 4.5. Let a satisfy (A2) and the hypotheses of Lemma 4-4- Supposethat G is an m-dimensional C2 submanifold of Hw. Then r-consistency of (a, 7,1)with G is equivalent to consistency.

If G is linear, then the same conclusion can be drawn without assumption (A2).

There is a subtle but important difference between G being a regular or merely

an immersed submanifold.

Assume that G is a C2 immersion at some zo £ Z. From Proposition A.l we

deduce that there exists an open neighborhood V of zq in Z such that G(V) is an

m-dimensional C2 submanifold in Hw and G : V — G(V) is a parametrization.But G can have self-intersections or can accumulate on G(V) as shown in Figure 1.

Thus, in principle, G(V) may very well fail to be consistent with (a, 7,1) but

still (a, j, I) is consistent with G- The same reasoning applies for seeing that con¬

sistency does not imply r-consistency: even if G is an immersed submanifold, the

trace process Z may have a jump at z0 if G(z0) is an accumulation point of G-

2for some Rm-, resp. L§(Km)-valued (^t ^-predictable processes b and p = (/><)., eN with

/ (IIMr +||ps[£o(Km)) ds<oo, Q-a.s. VteR+

3Compare with Definition 5.1, resp. Definition 6.1.

4 2 A SIMPLE CRITERION FOR REGULARITY OF G 53

On the other hand it is true that r-consistency with G implies r-consistency with

G(V). If furthermore a satisfies the hypotheses of Theorem 4.5, then Theorem 7.13

applies. We shall express the consistency conditions (7.2)-(7.3) in local coordinates,see (4.4) below. But first we have to check differentiability of G.

4.2. A Simple Criterion for Regularity of G

Consider G £ C2(R+ x Z) with G(z) = lG(-, z) G Hw for all z £ Z. We can give a

simple criterion for differentiability of G in Hw.Let £i, • ,£m denote the standard basis in Rm and let zo £ Z. The partial

derivative D^G(zo) is the derivative of the mapping s h» G(zq + s^) : R —>• Hw in

s = 0. For the differential calculus in Banach spaces we refer to [1, Chapter 2]. By

(HI) we have

Jx [ ^G(zo + s&)|.=od

dsJx (G(z0 + s£A) \s=o = dzß(x, zo), x £ R+.

Therefore DtG(zo) = dZtG(-,z0) if it exists.

Lemma 4.6. Let w : R+ —> [1, oo) be a non-decreasing function such that

ww~l £ L1 and let V be an open subset of Z. Suppose

SUP \dz,dxG(x,z)\2w(x) < oo.

(x,z)em.+ xv

Then the partial derivatives DlG(z) exists for all z £ V and z \-¥ D%G(z) is con¬

tinuous from V into Hw, for all 1 < % <m. Consequently, G £ C1(U;ifw) and

DG(z) = (dZlG(-,z),...,dZmG(-,z)) £ L(Rm;Hw). (4.2)

Proof. Let z £V. Write ez = eÇj, where e > 0. We have to show that

T(e) :=

»ose T(t

Tx(e) :=

G(* + *)-G{*)_8ztGMWe decompose T(e) = Tx(e) + T2(e), where

G(0,z + ez)-G(0,z)dz,G(0,z)

0 for e -^ 0.

0 for e -> 0

and

T (0 == /Jsu

dxG(x,z + el)-dxG(x,z)2

'~ ~ azßxG(x,z) w(x)dx

< ifJR+ JO

dz,dxG(x,z + sez) — dzßxG(x,z) w(x) dsdx.

For e small enough we have z + sez G V for all s G [0,1] and by assumption

dz,dxG(x,z + sez) -dzßxG(x,z) w(x) < 2Gw(x)w-1(x) £ LX([0,1] x R+)

where G is independent of x and e. Hence T2(e) -> 0 for e -> 0, by dominated

convergence. We conclude that DtG(z) exists. Continuity of z h> DlG(z) : V ->

Hw follows similarly. This proves the first part of the lemma.

For the second part we refer to [1, Proposition 2.4.12]. D

4Terminology introduced at the beginning of Section 4.1.


By [1, Proposition 2.4.12], G G C2(Z;HW) if and only if Dß £ C1(Z;HW) for

1 < i < m, which again can be checked with Lemma 4.6. Moreover

D2G(z)=(dZidZjG(-,z))1SiJSm (4.3)

which is a bilinear mapping from Rm x Rm into Hw.

Assume that G £ C2(Z;HW) and that dZlG(-,z0),... ,dZmG(-,z0) are linearly

independent functions, for some Zo £ Z. Then G is a C2 immersion at z0 and we

can go on with the reasoning at the end of the preceding section. Thus let a meet

the hypotheses of Theorem 4.5 and let (a,j,I) be r-consistent with G-

By Theorem 7.9 necessarily G(V) G D(A). Combining (4.2)-(4.3) with (7.41)-(7.42) and Theorem 7.13 yields the consistency condition in local coordinates

m * m,

dxG(x,z) = YJbi(J,u,z)dZiG(x,z) +gE ak>l(t,uo,z)dZkdZlG(x,z)

(4.4)mfX v '

)dn

"'fX

- V ak>l(t,oo,z)dZhG(x,z) / dZlG(n,z)<

for all (t,to,z) £ R+ x ft xV (after removing a nullset from ft). Here ak'1 :=

YljenP^'kP^'1 denotes the diffusion matrix and b the drift vector of the coordinate

process.

Compare condition (4.4) with Propositions 5.2 and 6.2. It is just the stronger

(since pointwise) version of (5.4) and (6.3), respectively, which is due to the different

meanings of "consistency" in Chapters 5-6 and Chapter 7, see Definition 4.3.

If G — G(Z) is a regular submanifold, then condition (4.4) can be used to

check whether any local HJM model is consistent with G- In some cases this leads

to the complete characterization of all consistent local HJM models. So for regular

exponential-polynomial families.

4.3. Regular Exponential-Polynomial Families

We restate the analysis of Chapters 5-6 from the preceding point of view. As

mentioned at the end of the previous section, this needs a slight modification.

Throughout this section we choose as weighting function w(x) — (1 + x)a for

some a > 3, see Example 3.4.

We adapt the notion of Section 6.3 and write

K

G(x,z) = Y^pi(x,z)e-*^* (4.5)i=l

where Pi(x,z) = Z)io zi,ix*> f°r s°me vector n — (m,... ,uk) £ N^ and K £ N.

Since Hw contains only bounded functions, we restrict to the bounded exponential-

polynomial families BEP(K,n), see Definition 6.3. But BEP(K,n) is not a sub¬

manifold in Hw. We have to extract the singularities. An easy computation yields

fx3e-**.»i+i*, 0<j<ntoZi ,G(x,z) = <

/'•'

\-xVi(x,z)e'zî+^, j = m + l.

4.3. REGULAR EXPONENTIAL-POLYNOMIAL FAMILIES 55

Ze:=Rx < (z2fl,..., zK,nK+i)

Hence {<9Zi G(-, z)\0<j<ni + l,l<i< K} are linearly independent functions

if and only if

Zi,m+i f zj,n,+i and zi<ni ^ 0, VI 0 or both Zi,ni+i = 0

and ni = 0, compare with (6.8). But the parameter set Z has to be open. To

include the latter case, we will therefore assume that

m = 0 (4.7)

£1,1= 0 and zitTli+i > 0 for 2 < i < K.

Consequently G(oo,z) = Zii0. For regularity reasons, see Remark 4.9 below, we

have to sharpen condition (4.6) and let the distance between different exponents be

bounded away from zero by some fixed e > 0. This leads to the following parameter

set

Zi,rii ~T~ U>

z^m+i > 0, V2 e>

in Rm where m = 1 + (n2 + 2) -) 1- (nK +2). To avoid self-intersections of G we

further require that

ni + nj, 2<i^j<K. (4.8)

Now the representation (4.5) of G(-, z) with z £ Zt is unique: G(-, z) = G(-, z) with

z,z £ Zt implies z = z.

Definition 4.7. For K G N, n G Nf satisfying (4.7)~(4.8), e > 0 and Ze as

above we define

REP(K,n,e) := G(Ze)

the regular exponential-polynomial family.

Clearly, we have REP(K,n,e) C BEP(K,n).Setting w(x) = (1 + x)a+2, we conclude by Lemma 4.6 that G £ C2(Ze,Hw).

Up to now, G : Ze -3 Hw is an injective immersion and REP(K, n, e) is an immersed

submanifold in Hw.

Proposition 4.8. The mapping G : Ze -3 Hw is an embedding. Its image

REP(K,n,e) is consequently an m-dimensional C2 submanifold in Hw, with m

given above.

Proof. We have to show that G : Zt -3 REP(K, n, e) is a homeomorphism.Let (zk) be a sequence in Zt and z £ Zt such that G(zk) -3 G(z) in Hw.

Inequality (3.4) yields G(zk) -3 G(z) in L°°(R+). The proof is completed by

showing that zk -3 z.

In view of (3.3) we conclude zkQ = G(oo,zk) -3 G(oo,z) = Zi>0. Thus

(G(oo,zk) — zkfi) —> (G(co,z) — z\fi) in Hw and we may assume in the sequel

that zk0 = Zifi = 0.

Write 9i :— limsupj,^^ zk^n.+1 < oo, i = 2,..., K. Then there exists a subse¬

quence (zfcf) of (zk) such that limôo zi^.+1 = 9~i for all i.

Suppose first 0j < oo for all i. By definition of Zt it follows that

\9i -9j\>e, V2<iJ:j<K. (4.9)

56 4 CONSISTENT HJM MODELS

We write N := m — K and define the R^-vectors

vß(x) := (e~z* "2+1*, xe~z^+lX,..., xn2e~Z2+lX,..., xnK e~z*-nK+lX\

v(x) - (e-~e2X,xe-~e*x,...,xn*e-~e2X,. .,xnKe~°KX)

IJj / l"U, K U, f^LC "<L£ \

V '— \z2fl>z2,l>---iz2,n2>--->ZK,nKJ-

Then

(v»(x),y»)m = G(x,zk») -> G(x,*), Vx G R+. (4.10)

Fix a sequence Xi < < xjv in R+. The vectors wM(xi ),..., v,M(xpf) are linearly

independent for any \i and so are v(x\),... ,v(xn), by (4.9). Hence the N x N-

matrices

'v'i(x1)\ fv(xi)\M = M(xi,. ..,xN) :=

v(xN)J

M» = Mlx(x1,...,xN) :=

\v»(xN)Jare regular and Af -» M for ^t -> oo. Consequently, (MM)_1 -3 M_1 and

/G(xi,2

2/" -3 M-1 :

\G(xjv,2)

Hence y%tJ := lim^-xx, 2S* exists in R for 0 < j < n%, 2 Vx G R+. (4.11)

Combining (4.8) with (4.11) we get 0» = ^j,n,+i and J/«,j = 2»,j- That way one

concludes that any subsequence of (zk) contains a convergent subsequence with

limit z. Hence limôo zk = z, and the proposition is proved in case 9l < oo for all

i = 2,...,K.Now assume that I := {2 x*.

j=0

We write N := N — J2tei(ni + -0 an(^ define the R^-vectors vh(x), v(x) and yM by

skipping all components of v^(x), v(x), resp. yß which include z^ or 0» for i £ I.

Set x* := maxjp/ x*. Instead of (4.10) we now have

{v^(x),r)MN = G(x,zk») - ^Pl(x,zk-)e-z-^+ix -3 G(x,z) Vx > x*.

tel

Choose a sequence x* < xi < < x^. Inequality (4.9) still holds for i,j £I. Accordingly, the vectors v(xi),... ,v(x^) are linearly independent and so are

v'x(xi),..., vß(xff) for all ß. A passage to the limit similar to the above implies

limaG(x,zk^) =Y^(jLy^x°)e~®lX =G(x,z), Vx > x*,

4.3. REGULAR EXPONENTIAL-POLYNOMIAL FAMILIES 57

which is impossible since [x*, oo) is a set of uniqueness for analytic functions. Hence

I is empty, and the proof is complete. D

Remark 4.9. Estimate (4-9) is essential for the preceding proof. Although

Proposition 4-8 seems to be true also for e = 0, we do not see how to prove it

for that case.

We can now summarize the main results from Chapter 6 as follows.

Theorem 4.10. Let a satisfy the assumptions5 of Lemma 4-4'> and suppose the

local HJM model (o~,y,I) is consistent with REP(K,n,e). Then

a(t,oo,-) = 0 on REP(K,n,e).

Proof. The proof is divided into two steps.

Step 1: Write

B := {ß = (0,&, ...,ßK)£RK \ßi>0, \ßi - ßA >e,\/2<i^j<K}.

For ß G B we define

Eß — REP(K,n, e) n span {x^'e"^ | 0 < j < m, l<i<K).

We claim that (er,7, i) is consistent with Eß. Indeed, let (to, ho) G R+ x Eß.That is, ft0 = G(z*) with z* £ Zt and 4iTl<+1 = ßi for 1 0 such that r^ = G(ZtAr) for all i G R+.

Accordingly, Ato'h°^ is a local strong solution to the time i0-shifted HJMM

equation (2.25). This in turn implies relation (6.3) for G(x,Z.At). By The¬

orem 6.4 we conclude that there exists a stopping time 0 < r' < r such

that

&t,n,+i = \£ (p».n.+i;fc)2 =o, dt® dQ-a.s. on [0,r'].ken

Therefore Z\^, = ßi and the claim follows.

Step 2: Write

B := {ß £ B I 2ßi = ßj for some i^j}.

The set B is contained in a finite union of hyperplanes in RK. Therefore

(B \ B) n B - B.

Since Eß is a linear submanifold, Theorem 4.5 yields that (a,j,I) is r-

consistent with Eß. From Theorem 7.13 we derive the consistency condition

in local coordinates (4.4) which in turn implies (6.3). Hence Theorem 6.15

applies and cr(t,to, ft) = 0 for ft, G Eß, for all ß £ B \ B. By continuity of

h 1-4 a(t,oo,h) the assertion follows.

D

5We don't need (A2) here.

6In fact, Z = (G_1 o </>)(Y) where <j> is as in the proof of Theorem 7.6 and Y is given by

(7.27).

58 4 CONSISTENT HJM MODELS

Theorem 4.10 contains a stricter statement than Theorems 6.14-6.15. It is be¬

cause "consistency" in Chapter 6 is only related to one particular initial point ho

in REP(K,n,e), see Definition 6.1 and Proposition 6.2 and compare with Defini¬

tions 4.1-4.3.

Let's recall the two leading examples of Chapters 5-6.

4.3.1. The Nelson-Siegel Family.

GNS(x,z) = zifi + (z2}o + z2Ax)e~Z2,2X

In the preceding context we have Gns(^c) = REP(2, (0, l),e), where actually e is

redundant.

4.3.2. The Regular Svensson Family.

Gs(x, z) = z1>0 + (z2fi + z2Ax)e-Z2-2X + z3Axe-z^x

Obviously, condition (4.8) is not satisfied. So the preceding discussion docs not

apply immediately. Yet we can "regularize" also the Svensson family. Set e > 0

and

Zs,e :=Rx {(22,0,22,1,22,2,23,1, z3>2) I |z2,o| > e, 22)2,23,2 > 0, \z2,2 -^3,21 > el-

Similar as in the proof of Proposition 4.8 one shows that Gs ' 2s,e -> Hw is

an embedding. As before we conclude a(t,oo,h) = 0 for ft G Gs(2s,e)> f°r every

consistent local HJM model (a,7,1).In Section 6.9.2 we identify the coordinate process Z for the non-trivial HJM

model7 which is r-consistent with the full (bounded) Svensson family. It is amazing

that the support of the non-deterministic component Zt' contains 0 for arbitrary

small t (it is Gaussian distributed). Thus R[Zt ^ Zs,t] > 0 for every i > 0. However,as far as I can see, there is no deeper connection between failure of (4.8) and the

existence of a non-trivial consistent state space process for the Svensson family.

Remark 4.11. Consistency of HJM models with exponential-polynomial fami¬lies has been checked first by Björk and Christensen [7] for deterministic a. Their

result was extended by the two articles in Chapter 5 and 6. There, however, we

merely checked r-consistency for EP(K,n), resp. BEP(K,n). Question ii) in Sec¬

tion 4-1, asserting the generality of that approach, was left open and it is now

clarified by Theorem 4-10.

4.4. Affine Term Structure

In this section we apply Theorem 7.13 to identify the particular class of affine local

HJM models. That is, those local HJM models which are consistent with finite

dimensional linear submanifolds.

It is required throughout that a meets the assumptions of Lemma 4.4. Supposethat Af is an m-dimensional linear submanifold of Hw which is locally invariant

for the HJMM equation (2.25). Without loss of generality, M is connected. Ac¬

cording to Theorem 7.10 there exists V C Rm open and connected and a global

parametrization (f> : V -3 M with <f>(y) = gQ + Y^T=i 2/«#» where go,... ,gm £ D(A)are linearly independent. Clearly D(f)(y) = (gi,... ,gm).

It's a generalized Vasicek model.

4 4 AFFINE TERM STRUCTURE 59

Applying Theorem 7.13 yields, see (4.4),

mm ,/ m \

so+Evd = E&^>w' y)s« -2

Ea*J (*>w' ^G^ (4-12)1=1 1=1 \î,J=l /

for all (t,oo,y) G R+ x 0 x U (after removing a P-nullset from ft), where Gt(x) :=

j gl(rj) dr]. To shorten notation, we let'stand for dx.

Write 7j = g%(0). Integrating (4.12) yields

m m *m

go-

to + E »»G** - 7.) = E &^> w> y)G»_2EßtJ (*»w' y)G*Gi (4-13)

x=i i=i »j=i

Now if Gi and G%G3, 1 < z, j < m, are linearly independent functions, we can

invert the linear equation (4.13) for bl(t,oo,y) and ah3(t,oo,y). Since the left hand

side of (4.13) is affine in y, we obtain that also they are affine functions in y:

bl(t,oo,y) = b\t,oo) + J2ßl'3(t,u)y„ l<i< m

3=i

a1'3 (t, oo,y)= aM (t, oo) + ]T a«'fc (t, w)^, 1 < i, j < m

fc=i

for some predictable functions bl(t,oo), ß^3(t,oo), ah3(t,oo) and ah3'k(t, to) being

right continuous in i.

Since affine functions are analytic and V is open, equation (4.13) splits into the

following system of Riccati equations for the G% 's

m ~ m

G'0=7o + ^Tbk(t,to)Gk-- £ ak'l(t,oo)GkGL

(4.14)

G'i = 7* + Y^ßk'\t,oo)Gk - - ]T aît^GkGt, l<i<m

fe=i fe,i=i

with initial conditions Go(0) = • • = Gm(0) = 0.

Hence we derived some of the results in [21] as a special case of our general

point of view. Clearly, Theorem 7.13 imposes constraints on the choice of V in order

that b and p are locally Lipschitz, see (7.44), and that (7.43) has a local solution in

V. Moreover, there is a tradeoff between the functions g% and the coefficients a1'3,

ah3'k, b\ ß%,d and 7l by the Riccati equations (4.14). We shall discuss this problem

for the case m = 1. For a fuller treatment we refer to [21].For simplicity we assume that there is only a one dimensional driving Brownian

motion W and that all coefficients Eire autonomous, that is

b(t,u,y)=b1(y)=b + ßy

a(t,oo,y) =a1'1(y) =a + ay.

Accordingly

p(t,oo,y) = pl'x(y) = ^+ay.


We rewrite the Riccati equations (4.14) for Go and Gi:

G'o = 7o + 6Gi - \aG{, G0(0) = 0

Gi1=7l+ßGl-±aGl Gi(0) = 0. (4.15)

Equation (4.15) admits anon-exploding solution Gi : R+ —> R if and only if aji > 0

or both crfi < 0 and ß < — y/2|a7i|.Suppose a > 0. Theorem 7.13 tells us that y >-3 ^/a + ay has to be locally

Lipschitz in V, see (7.44). Consequently, a+ ay has to be bounded away from zero.

Thus V C [1:zS,oo) for some e > 0. The dynamics of the coordinate process, see

(7.43), is given by the stochastic differential equation

dYt = (b + ßYt) dt + Va + aYtdWt, Y0 £ V. (4.16)

It can be shown8, see [29, p. 221], that (4.16) admits a (global) unique continuous

strong solution on [=-}, oo) if b > ^ß.If a < 0, then V C (-oo, ^-] for some e > 0 and (4.16) remains valid.

4.4.1. The Cox-Ingersoll-Ross (CIR) Model. For a > 0, 71 = 1 and

a = 70 = 0 (and hence b > 0) we get the CIR [17] short rate model. There exists

a closed form solution to (4.15), see [17, Formula (23)],

Gx(x)

(yW+2a- - ß) U^^)x -l\+ 2VF+2^

and Go(x) = b Jq Gi(rj) dm It holds by inspection9 that go = G'0 = bG\ £ Hw and

gi = G[ £ Hi for any polynomial weighting function w as in Example 3.4. Hence

the CIR model is contained in our setting.From (7.41) and the identity y = go(0) +ygi(0) = Jo(4>(y)) we obtain

a(t,oo,h) = o-l(h) = y/aj0(h)gu ft, G cf>((0, 00)).

Denote by â : Hw -3 Hi a measurable continuation of a. Let Y be the continuous

solution of (4.16) with Yq > e and write re = inf{i | Yt < e}. It is straightforwardthat (rtArJ = (<f>(YtATc)) with ro = 4>(Y0) is the unique continuous local strongsolution to the HJMM equation (2.25) equipped with â, see Theorem 7.13

However, er(ft) - and hence 5(h) - cannot be extended Lipschitz continuously to

ft, = (f)(0) (naturally a((p(0)) = 0). Thus, although <j>(Y) is a strong solution to the

HJMM equation (2.25) with ä($(0)) = 0, we cannot assert its global uniqueness.

Uniqueness, however, holds for Y in R+, by the preceding discussion.

Substitute X := Y + a

a

9More general it can be shown that the critical point j/o = £ R+ for (4.15)is globally exponentially stable, see [2, Chapter IV]. A fact which also applies for the multi¬

dimensional system (4.14). We will investigate this further in connection with general affine

models.

4 4. AFFINE TERM STRUCTURE 61

4.4.2. The Vasicek Model. The case a = 0 leads to a Gaussian coordinate

process Y. There are no constraints on V, but a has to be non-negative.For 7o = 0, 7i = 1, b > 0 and ß < 0 this is the Vasicek [48] short rate model.

The solution for (4.15) is given by

G1(x) = -ß(e^-l)and G0(x) = b fx Gi(rf) dn - âJxG2(n)dn. Obviously, g0 — ftGi - \aG2 £ Hw

and gi(x) = e/3x £ Hi. Hence also this classical model is contained in our setting.From (7.41) we deduce that a(t,oo, ft) = al(h) = \fagx for ft G <f>(R).

Part 2

Publications

CHAPTER 5

A Note on the Nelson—Siegel Family

ABSTRACT. We study a problem posed in Björk and Christensen [7]: does

there exist any nontrivial interest rate model which is consistent with the

Nelson-Siegel family? They show that within the HJM framework with

deterministic volatility structure the answer is no.

In this paper we give a generalized version of this result including sto¬

chastic volatility structure. For that purpose we introduce the class of consis¬

tent state space processes, which have the property to provide an arbitrage-free interest rate model when representing the parameters of the Nelson-

Siegel family. We characterize the consistent state space Itô processes in

terms of their drift and diffusion coefficients. By solving an inverse prob¬

lem we find their explicit form. It turns out that there exists no nontrivial

interest rate model driven by a consistent state space Itô process.

5.1. Introduction

Björk and Christensen [7] introduce the following concept: let M be an interest rate

model and G a parameterized family of forward curves. M and G are called consis¬

tent, if all forward rate curves which may be produced by M. are contained within

G, provided that the initial forward rate curve lies in G- Under the assumption of

a deterministic volatility structure and working under a martingale measure, theyshow that within the Heath-Jarrow-Morton (henceforth HJM) framework there

exists no nontrivial forward rate model, consistent with the Nelson-Siegel family

{G(., z)}. The curve shape of G(., z) is given by the well known expression

G(x, z)=Z!+ z2e~z*x + z3xe-z*x, (5.1)

introduced by Nelson and Siegel [39].For an optimal todays choice of the parameter z £ R4, expression (5.1) rep¬

resents the current term structure of interest rates, i.e. x > 0 denotes time to

maturity. This method of fitting the forward curve is widely used among central

banks, see the BIS [6] documentation.

From an economic point of view it seems reasonable to restrict z to the state

space Z := {z = (zi,..., Zi) £ RA \ zi > 0}.The corresponding term structure of the bond prices is given by

n(x, z) := exp ( — / G(rj, z) dn I.

ThennGC°°([0,oo) x Z).In order to imply a stochastic evolution of the forward rates, we introduce in

Section 5.2 some state space process Z — (Zt)o<t<oo with values in Z and ask

whether G(. ,Z) provides an arbitrage-free interest rate model. We call Z consis¬

tent, if the corresponding discounted bond prices are martingales, see Section 5.3.

65

66 5 A NOTE ON THE NELSON-SIEGEL FAMILY

Solving an inverse problem we characterize in Section 5.4 the class of consistent

state space Itô processes. Since a diffusion is a special Itô process, the very im¬

portant class of consistent state space diffusion processes is characterized as well.

Still we are able to derive a more general result. It turns out that all consistent Itô

processes have essentially deterministic dynamics. The corresponding interest rate

models are in turn trivial.

Consistent state space Itô processes are, by definition, specified under a mar¬

tingale measure. This seems to be a restriction at first and one may ask whether

there exists any Itô process Z under some objective probability measure inducing

a nontrivial arbitrage free interest rate model G(., Z). However if the underlyingfiltration is not too large we show in Section 5.5 that our (negative) result holds

for Itô processes modeled under any probability measure, provided that there ex¬

ists an equivalent martingale measure. Hence under the requirement of absence of

arbitrage there exists no nontrivial interest rate model driven by Itô processes and

consistent with the Nelson-Siegel family.

Using the same ideas, still larger classes of consistent processes like Itô processes

with jumps could be characterized.

5.2. The Interest Rate Model

For the stochastic background and notations we refer to [40] and [31].Let (ft,T, (Pt)o<t<oa,R) be a filtered complete probability space, satisfying

the usual conditions, and let W = (W£,... ,Wt)o<t<oo denote a standard d-

dimensional (,Ft)-Brownian motion, d > 1.

We assume as given a Z-valued Itô process Z = (Z1,..., ZA) of the form

Z\ = Zl+ f blds + jzf <JdW3, i = l,...,4, 0<i<oo, (5.2)Jo J=1Jo

where Zq is .Fo-measurable, and b, a are progressively measurable processes with

values in R4, resp. R4xd, such that

/ {\bs\ + \o-s\2) ds < oo, P-a.s., for all finite i. (5.3)Jo

Z could be for instance the (weak) solution of a stochastic differential equation,but in general Z is not Markov.

Write r(t,x) := G(x, Zt) for the instantaneous forward rate prevailing at time

i for date t + x.

It is shown in [19, Section 7] that traded assets have to follow semimartingales.Hence it is of importance for us to observe that the price at time i for a zero coupon

bond with maturity T

P(t,T):=Tl(T-t,Zt), 0<t<T<oo,

and the short rates

r(t,0) = lim r(t,x) = G(0, Zt) = -7^(0, Zt), 0 < i < 00,x—yO ox

form continuous semimartingales, by the smoothness of G and n. Therefore the

same holds for the process of the savings account

B(t) := exp ( / r(s, 0) ds j ,0 < t < 00.

5.3. CONSISTENT STATE SPACE PROCESSES 67

5.3. Consistent State Space Processes

We are going to define consistency in our context, which slightly differs from that

in [7]. We focus on the state space process Z, which follows an Itô process or may

follow some more general process.

Definition 5.1. Z is called consistent with the Nelson-Siegel family, if

(EMI)

is a R-martingale, for all T < oo.

The next proposition is folklore in case that Z follows a diffusion process,

i.e. if bt(oo) = b(t, Zt(oo)) and at(oo) = a(t, Zt(oo)) for Borel mappings b and a from

[0, oo) x Z into the corresponding spaces. That case usually leads to a PDE includingthe generator of Z. The standard procedure is then to find a solution u (the term

structure of bond prices) to this PDE on (0, oo) x Z with initial condition u(0, . ) =

1. It is well known in the financial literature that Z is necessarily consistent with

the corresponding forward rate curve family. In contrast we ask the other way

round and are more general what concerns Z. Our aim is, given G, to derive

conditions on b and a being necessary for consistency of Z with {G(. ,z)}z^z- But

the coefficients b and a are progressively measurable processes. Hence Z given

by (5.2) is not Markov, i.e. there is no infinitesimal generator. By the nature

of b and a such conditions can therefore only be stated dt ® dP-a.s. (note that

equation (5.4) below is not a PDE). On the other hand the argument mentioned

in the diffusion case works just in one direction: consistency of Z with a forward

curve family Q = {v(- ,z)}zez does not imply validity of the PDE condition for

u(x, z) = exp(— fx v(rj, z) dn) in general.

Actually one could re-parameterize f(t,T) := r(t,T -1), for 0 < t < T < oo,

and work within the HJM framework. Equation (5.4) below corresponds to the

well known HJM drift condition for (f(t,T))o<t<T- But as soon as Z is not an Itô

process anymore, this connection fails and one has to proceed like in the following

proof (see Remark 5.3), which therefore is given in its full form.

Set a := oo*, where a* denotes the transpose of o, i.e. at3 = ^2k=1 o-lka{k, for

1 < i,j < 4 and 0 < i < oo. Then a is a progressively measurable process with

values in the symmetric nonnegative definite 4 x 4-matrices.

Proposition 5.2. Z is consistent with the Nelson-Siegel family only if

lxG(x,Z)=J:^lG(x,Z)

+\ E °« (ä$^(*. *) -1^>z) [ 4G(??'z) ^

(5,4)

for all x > 0, dt ® dP-a.s.

68 5. A NOTE ON THE NELSON-SIEGEL FAMILY

Proof. For / G G2 (Z) we set

At(to)f(z):=J2bi^)^(z) + li^^(oo)^--(z), 0<i<co, z £ Z.

i=ll

i,j=ll 3

Using Itô's formula we get for T < oo

P(t, T) = P(0, T)+ f (ASU(T - s, Zs) - ^-U(T - s, Zs)) ds

+ f ayzU(T-s,Zs)dWs, 0<t<T, P-a.s.Jo

where Vz denotes the gradient with respect to (zi,z2,Zs, z<ù, and

157^= 1+ / 7JTT7rn(0,Zs)dS, 0<i<co, P-a.s.

B(t) Jo B(s) dx

For 0 Zt

and the local martingale

M(t,T):= f -±-<r*sV,IL(T-8,Za)dWt.Jo ^{s)

Integration by parts then yields

1 /"*-—P(t,T) = P(0,T)+H(s,T)ds + M(t,T), 0<t<T, P-a.s.B(t) Jo

Let's suppose now that Z is consistent. Then necessarily for T < 00

rt

H(s,T)ds = 0, ViG[0,T], P-a.s. (5.5)/o

Since b and u are progressive and n is smooth, H(. ,T) is progressively measurableon [0,T] x ft. We claim that (5.5) yields

H(.,T) = 0, on[0,T]xft, dt®dP-a.s. (5.6)

Proof of (5.6). Define N := {(i,w) G [0,T] x ÎÎ | H(t,T)(oo) > 0}. Then A is a

B®P- measurable set. Since H(.,T) is positive on N we can use Tonelli's theorem

f H(t,T)(oo)dt®dP= f ( f H(t,T)(to)dt) dR(to) = 0,Jn Jq \Jnw j

where Nu := {i | (t,oo) £ N} £ B and we have used (5.5) and the inner regularityof the measure dt. We therefore conclude that Af has dt ® dP-measure zero. Byusing a similar argument for —H(. ,T) we have proved (5.6).

Note that (5.6) holds for all T < 00, where the dt <g> dP-nullset depends on T.

But since H(t,T) is continuous in T, a standard argument yields

H(t,t + x)(to) =0, Vx>0, for di<g>dP-a.e. (t,u).

Multiplying this equation with B(t) and using again the full form this reads

ATi(x,Z)--—Il(x,Z) + 1--TI(0,Z)Il(x,Z)=0, Vx > 0, di®dP-a.s. (5.7)

fJo

5.4. THE CLASS OF CONSISTENT ITO PROCESSES 69

Differentiation gives

[ A0^ z>d" - \ t •* {[i^2) *i) {£ £«<.. z>d"

- G(x, Z) + G(0, Z) = 0, Vx > 0, dt <g> dP-a.s.

where we have divided by n(x, Z), since n > 0 on [0, oo) x Z. Differentiating with

respect to x finally yields (5.4). D

REMARK 5.3. Definition 5.1 can be extended in a natural way to a wider class

of state space processes Z. We mention here just two possible directions:

a) Z a time homogeneous Markov process with infinitesimal generator C. The

corresponding version of Proposition 5.2 can be formulated in terms of equa¬

tion (5.7), where A has to be replaced by C. The difficulty here consists of

checking whether n goes well together with Z, i.e. x h» n(x, .) has to be a

nice mapping from [0,oo) into the domain of C

b) Z an Itô process with jumps. Again one could reformulate Proposition 5.2

on the basis of equation (5.7) by adding the corresponding stochastic integralwith respect to the compensator of the jump measure, which we assume to be

absolutely continuous with respect to dt. The jump measure could be implied

for example by a homogeneous Poisson process.

5.4. The Class of Consistent Itô Processes

Equation (5.4) characterizes b and a, resp. a, just up to a dt <g> dP-nullset. But the

stochastic integral in (5.2) is (up to indistinguishability) defined on the equivalenceclasses with respect to the dt ® dP-nullsets. Hence this is enough to determine the

process Z, given Z0, uniquely (up to indistinguishability). On the other hand Z

cannot be represented in the form (5.2) with integrands that differ from b and a on

a set with dt CS> dP-measure strictly greater than zero (the characteristics of Z are

unique up to indistinguishability). Therefore it makes sense to pose the followinginverse problem on the basis of equation (5.4): For which choices of coefficients b

and a do we get a consistent state space Itô process Z starting in Zo?The answer is rather remarkable:

Theorem 5.4. Let Z be a consistent Itô process. Then Z is of the form

7i_

7i

^t — Lo

-Znt , rr'i. —Znt

Z? = Z*e-z°t

Zt = Z2e-^ + Z*te-

zf^z^+i i &>+>;/ v?dwiiino

for all0<t<oo, where ft0 := {Z% = Z$=0}.

Remark 5.5. On fto the processes Z2 and Z3 are zero. Hence G(x,Zt) = Zqon fto, i.e. the process Z4 has no influence on G(., Zt) on fto. So it holds that the


corresponding interest rate model is of the form

r(t, x) = G(x, Zt)

= Z\ + {Zle-Z*1 + Zlte-2*1) e~z°x + Z^^xeT^

= Z\ + Z2e~z^+X) + Zl(t + x)e~z^t+xl= r(0,t + x), Vi,x>0,

and is therefore quasi deterministic, i.e. all randomness remains Pq-measurable.

REMARK 5.6. One can show a similar (negative) result even for the wider class

of state space Itô processes with jumps on a finite or countable mark space, see

Remark 5.3. However allowing for more general exponential-polynomial families{G(. ,z)} there exist (although very restricted) consistent ltd processes providing a

nontrivial interest rate model, see [23].

Proof. Let Z be a consistent Itô process given by equation (5.2). The proofof the theorem relies on expanding equation (5.4).

First of all we subtract -§^G(x, Z) from both sides of (5.4) to obtain a null

equation. Fix then a point (i, oo) in [0, oo) x ft. For simplicity we write (z\, z2, z3,z\)for Zt(oo), aij for aJJ (oo) and hi for b\(oo). Notice that Zt(oo) £ Z, i.e. z± > 0. Observe

then that our null equation is in fact of the form

Pl(x) + p2(x)e-z*x + P3(x)e-2z*x = 0, (5.8)

which has to hold simultaneously for all x > 0. The expressions Pi,p2 and p3 denote

some polynomials in x, which depend on the z;'s, 6;'s and a^'s. Since the functions

{l,e~ZiX,e~2ZiX} are independent over the ring of polynomials, (5.8) can only be

satisfied if each of the Pi's is 0. This again yields that all coefficients of the p^'shave to be zero. To proceed in our analysis we list all terms appearing in (5.4):

—G(x,z) - (-z2Zi +z3- z3Zix)e~Z4X,

VzG(x, z) = (1, e~ZiX, xe~ZiX, (-z2x - z3x2)e~ZiX) ,

d2

dzidzjG(x,z) = 0, for 1 <i,j <3,

—V,G(x, z) = (0, -xe~Z4*, -x2e~ZiX, (z2x2 + z3x3)e-ZiX) .

Finally we need the relation

777 '

rTe-'« dr, = -qm(x)e-**x + —^ m = 0,1, 2,...,o z4

where qm(x) = Yfk=o (m'fcv xt-n IS a polynomial in x of order m.

First we shall analyze P\. The terms that contribute to pi are those containingö|^G(x,z) and -^G(x,z) Jx ^rG(ri,z)dn, for 1 < j < 4. Actually P\ is of the

form

pi(x) =anxH h&i,

where... stands for terms of zero order in x containing the factors a\j = aji, for

1 < J < 4. It follows that an = 0. But the matrix (a^) has to be nonnegative

5 4 THE CLASS OF CONSISTENT ITO PROCESSES 71

definite, so necessarily

aij = flji = 0, for all 1 < j < 4,

and therefore also &i = 0. Thus P\ is done.

The contributing terms to p3 are those containing -J=^-G(x,z) fx -^-G(n,z) dn,

for 2 < i,j < 4. But observe that the degree of P3 and P2 depends on whether z2

or z3 are equal to zero or not. Hence we have to distinguish between the four cases

i) z2 # 0, z3 £ 0 iii) z2 = 0, z3 £ 0

ii) ^2 ^ 0, 23 = 0 iv) z2 = z3 - 0.

case i): The degree of P3 is 4. The fourth order coefficient contains 0,44, i.e.

2

p3(x) = 0,44—x4 +...,

Z4

where...

stands for terms of lower order in x. Hence

a4j = aj4 = 0, for 1 < j < 4.

The degree of p3 reduces to 2. The second order coefficient is sf2-. Hence

a3o = Oj3 = 0, for 1 < j < 4. It remains P3(x) = ^-. Thus the diffusion

matrix (atJ ) is zero. This implies that (o~l3 ) is zero, independent of the choice

of d (the number of Brownian motions in (5.2)). Now we can write down p2:

P2(x) = -b4z3x2 + (b3 - biZ2 + z3z4)x + b2 + z2z4 - z3.

It follows that 64 = 0 and

b2 = z3 - z2z4,

h = -z3z4-

For the other three cases we will need the following lemma, which is a direct

consequence of the occupation times formula, see [40, Corollary (1.6), Chapter VI].

Lemma 5.7. Using the same notation as m (5.2), it holds for 1 < 1 < 4 that

aMl{z»=o} = Wl{z<=o} = 0, dt® dP-a.s.

As a consequence, since we are characterizing a and 6 up to a dt ® dP-nullset,

we may and will assume that zt = 0 implies al3 = aJX = bt = 0, for 1 < j < 4 and

i = 2,3.

case ii): We have VzG(x,z) ~ (l,e~Z4X,xe~ZiX, —z2xe~Z4X). Hence the degreeof p3 is 2. Since a3j = aj3 = b3 = 0, for 1 < j < 4, the second order coefficient

comes from

d fx d z2-a44-—G(x,z) / —G(n,z)dn = a4i^-x2e'2z4X + ...,

dz4 J0 dz4 z4

where...

denotes terms of lower order in x. Hence a4j = aj4 = 0, for

1 < J < 4. The polynomial p3 reduces to p3(x) = £^3-. It follows that also in

this case the diffusion matrix (ai3) is zero. From case i) we derive immediatelythat now

P2(x) = -b4Z2X + b2 + £2£4,

hence b2 — —z2z4 and 64 = 0.


case iii): Since a2] = aj2 = b2 = 0, for 1 < j < 4, the zero order coefficient

of p2 reduces to —z3. We conclude that z2 = 0 implies z3 = 0, so this case

doesn't enter di ® dP-a.s.

case iv): In this case al3 = bk = 0, for all (i,j) ^ (4,4) and k ^ 4. Also

-^-G(x,z) = -g^G(x,z) = 0. Hence ^2(x) =p3(x) = 0, independently of the

choice of 64 and 044.

Summarizing the four cases we conclude that equation (5.4) implies

61 = 0 b3 = -z3z4

b2=z3-z2z4 a1}=Q, for (i,j) ^ (4,4).

Whereas b4 and 044 are arbitrary real, resp. nonnegative real, numbers whenever

z2 = z3 = 0. Otherwise 64 = 044 = 0.

This has to hold for di®dP-a.e. (i, to). So Z is uniquely (up to indistinguisha¬bility) determined and satisfies

zt - zo

72 — 72^t

—

^0f (Zl - Z2Zf) dsJo

Z\ = Zl-\ Z\Z\dsJo

Z4=Z4 + fbîds + Ysf °\2dWiJo

J=1Jo

for some progressively measurable processes b4 and a^3, j = 1,... ,d, being com¬

patible with (5.3) and vanishing outside the set {(i, 00) \ Z2(oo) — Zf(oo) — 0}.Note that Z2 and Z3 satisfy (path-wise) a system of linear ODE's with con¬

tinuous coefficients. Hence they are indistinguishable from zero on ft0. So the

statement of the theorem is proved on fto- It remains to prove it on fti := ft \ re¬introduce the stopping time r := inf{s > 0 | Z2 = Z3 = 0}. We have just

argued that fto C {t = 0}. By continuity of Z also the reverse inclusion holds,hence ft0 = {r = 0}. The stopped process ZT =: Y satisfies (path-wise) the

following system of linear stochastic integral equations

ds

(5.9)

We have used the fact that fr4 = frH^co] and a4 — <74l[ri00]. Then the last equationfollows from an elementary property of the stopped stochastic integral:

Yt1 -zlft/\T

Yt2 = z2 +Jo

rt/\T

Y2s zt

Yt" = zl + I Y3Z4 ds

YtA — ^0' for 0 dWi = J2 / (<#l[r,oo]) l[0,r] dW3 = Y, / ^l[r] dW> = 0,,

= 1J°

7=1J°

7=1J°

d

£3

=

by continuity of W.

I

5.5. E-CONSISTENT ITÔ PROCESSES 73

The system (5.9) has the unique solution for 0 + Z3(t A T)e~zo(tA^

Y3 = Z3e-z^tArî

Yt4 = Z4.

Since Z = Y on the stochastic interval [0,r] and since Yt ^ 0, Vi < oo, P-a.s. on

fti, it follows by the continuity of Z, that ft\ — {t > 0} = {r = oo}. Inserting this

in the above solution, the theorem is proved also on ft\.

5.5. E-Consistent Itô Processes

Note that by definition Z is consistent if and only if P is a martingale measure for

the discounted bond price processes. We could generalize this definition and call a

state space process Z e-consistent if there exists an equivalent martingale measure

Q. Then obviously consistency implies e-consistency, and e-consistency implies the

absence of arbitrage opportunities, as it is well known.

In case where the filtration is generated by the Brownian motion W, i.e. (Pt) =

(P^), we can give the following stronger result:

PROPOSITION 5.8. If (Pt) = (P^), then any e-consistent Itô process Z is ofthe form as stated in Theorem 5.4- In particular the corresponding interest rate

model is purely deterministic.

Proof. Let Z be an e-consistent Itô process under P, and let Q be an equiv¬alent martingale measure. Since (Pt) — (J-^), we know that all P-martingaleshave the representation property relative to W. By Girsanov's theorem it follows

therefore that Z remains an Itô process under Q. In particular Z is a consistent

state space Itô process with respect to the stochastic basis (ft, P, (PtV)o<t<oo,Q)-Hence Z is of the form as stated in Theorem 5.4.

Since P^ consists of sets of measure 0 or 1, the processes Z1 ,Z2, Z3 are (afterchanging the ZqS on a set of measure 0) purely deterministic and therefore also

(r(t, .))o<t<oo, see Remark 5.5. D

CHAPTER 6

Exponential-Polynomial Families and the Term

Structure of Interest Rates

ABSTRACT. Exponential-polynomial families like the Nelson-Siegel or Svens¬

son family are widely used to estimate the current forward rate curve. We

investigate whether these methods go well with inter-temporal modelling.

We characterize the consistent Itô processes which have the property to pro¬

vide an arbitrage free interest rate model when representing the parameters

of some bounded exponential-polynomial type function. This includes in

particular diffusion processes. We show that there is a strong limitation on

their choice. Bounded exponential-polynomial families should rather not be

used for modelling the term structure of interest rates.

6.1. Introduction

The current term structure of interest rates contains all the necessary information

for pricing bonds, swaps and forward rate agreements of all maturities. It is used

furthermore by the central banks as indicator for their monetary policy.There are several algorithms for constructing the current forward rate curve

from the (finitely many) prices of bonds and swaps observed in the market. Widelyused are splines and parameterized families of smooth curves {G(., z)}z^Z} where

Z C RN, N > 1, denotes some finite dimensional parameter set. By an optimalchoice of the parameter z in Z an optimal fit of the forward curve x i-> G(x, z) to

the observed data is attained. Here x > 0 denotes time to maturity. In that sense

z represents the current state of the economy taking values in the state space Z.

Examples are the Nelson-Siegel [39] family with curve shape

GNS(x, z)=Z! + (z2 + z3x)e~Z4X

and the Svensson [44] family, an extension of Nelson-Siegel,

Gs(x, z) = zx + (z2 + z3x)e~Z5X + z4xe"Z6X.

Table 1 gives an overview of the fitting procedures used by some selected central

banks. It is taken from the documentation of the Bank for International Settle¬

ments [6].Despite the flexibility and low number of parameters of Gns and Gs, their

choice is somewhat arbitrary. We shall discuss them from an inter-temporal point of

view: A lot of cross-sectional data, i.e. daily estimations of z, is available. Therefore

it would be natural to ask for the stochastic evolution of the parameter z over time.

But then there exist economic constraints based on no arbitrage considerations.

75

76 6. EXPONENTIAL-POLYNOMIAL FAMILIES

TABLE 1. Forward rate curve fitting procedures

central bank curve fitting procedure

Belgium Nelson-Siegel, Svensson

Canada Svensson

Finland Nelson-SiegelFrance Nelson-Siegel, Svensson

Germany Svensson

Italy Nelson-Siegel

Japan smoothing splines

Norway Svensson

Spain Nelson-Siegel (before 1995), Svensson

Sweden Svensson

UK Svensson

USA smoothing splines

Following [7], instead of Gns and Gs we consider general exponential-poly¬nomial families containing curves of the form

K 71,

G(x,z) = j2(T,z^xß)e~z'n-+lx-t=l A'=0

Hence linear combinations of exponential functions exp(—zlin,+i%) over some poly¬nomials of degree n% £ No- Obviously Gns and Gs are of this type. We replacethen z by an Itô process Z = (Zt)t>o taking values in Z. The following questionsarise:

• Does G(. ,Z) provide an arbitrage free interest rate model?

• And what are the conditions on Z for it?

Working in the Heath-Jarrow-Morton [28] - henceforth HJM - framework

with deterministic volatility structure, Björk and Christensen [7] showed that the

exponential-polynomial families are in a certain sense to large to carry an interest

rate model. This result has been generalized for the Nelson-Siegel family in [25],including stochastic volatility structure. Expanding the methods used in there, we

give in this paper the general result for bounded exponential-polynomial families.

The paper is organized as follows. In Section 6.2 we introduce the class of Itô

processes consistent with a given parameterized family of forward rate curves. Con¬

sistent Itô processes provide an arbitrage free interest rate model when driving the

parameterized family. They are characterized in terms of their drift and diffusion

coefficients by the HJM drift condition.

By solving an inverse problem we get the main result for consistent Itô pro¬

cesses, stated in Section 6.3. It is shown that they are remarkably limited. The

proof is divided into several steps, given in Sections 6.4, 6.5 and 6.6.

In Section 6.7 we extend the notion of consistency to e-consistency when P is

not a martingale measure.

The main result reads much clearer when restricting to diffusion processes, as

shown in Section 6.8. It turns out that e-consistent diffusion processes drivingbounded exponential-polynomial families like Nelson-Siegel or Svensson are very

limited: Most of the factors are either constant or deterministic. It is shown in

6.2. CONSISTENT ITO PROCESSES 77

Zl=Zi+ f bids + j^fa^dW3,Jo tîJQ

Section 6.9, that there is no non-trivial diffusion process which is e-consistent with

the Nelson-Siegel family. Furthermore we identify the diffusion process which is

e-consistent with the Svensson family. It contains just one non deterministic com¬

ponent. The corresponding short rate model is shown to be the generalized Vasicek

model.

We conclude that bounded exponential-polynomial families, in particular Gns

and Gs, should rather not be used for modelling the term structure of interest

rates.

6.2. Consistent Itô Processes

For the stochastic background and notations we refer the reader to [40] and [31].Let (ft,P, (Pt)o<t<oo,R) be a filtered complete probability space, satisfying the

usual conditions, and let W — (W/,..., Wf)o<t<oo denote a standard d-dimensional

(Pt)-Brownian motion, d > 1.

Let Z = (Zl,..., ZN) denote an R^-valued Itô process, N > 1, of the form

d

1,...,N, 0 <i< oo,

where Zq is JF0-measurable, and b, o are progressively measurable processes with

values in RN, resp. RNxd, such that

/ (\bs\ + \o~s\2) ds < oo, P-a.s., for all finite i.

Jo

Let G(x, z) be a function in G1,2(R+ x RN), i.e. G and the partial derivatives

(dG/dx), (dG/dzi), (d2G/dzidzj), which exist for 1 < i,j < N, are continuous

functions on R+ x R-^. Interpreting Zt as the state of the economy at time i, we let

x i-3 G(x, Zt) stand for the corresponding term structure of interest rates. Meaningthat G(x, Zt) denotes the instantaneous forward rate at time i for date t + x.

Notice that

n(x, z) :— exp ( - / G(rj, z) dn

is in G1,2(R_|_ x R^) too. Therefore the price processes for zero coupon T-bonds

P(t,T):=TL(T-t,Zt), 0 < i < T < oo, (6.1)

and the process of the savings account

B(t) := exp(- f g-n(0, Zt) ds) ,

0 < i < oo,

form continuous semimartingales.Let Z denote an arbitrary subset of RN. The function G generates in a canon¬

ical way a parameterized set of forward curves {G(., z)}zez- We shall refer to Z

as the state space of the economy.

DEFINITION 6.1. Z is called consistent with {G(. ,z)}zez, if the support of Z

is contained in Z and

'P(t,TYm) l (6"2)-°W / 0<t<T

is a R-martingale, for all T < oo.


Set a := aa*, where a* denotes the transpose of a, i.e. a])3 = ^2k=1 o\' af' ,

for 1 < i, j < N and 0 < i < oo. Then a is a progressively measurable process with

values in the symmetric nonnegative definite N x TV-matrices.

Using Itô's formula, the dynamics of (6.2) can be decomposed into finite vari¬

ation and local martingale part. Requiring consistency the former has to vanish.

This is the well known HJM drift condition and is stated explicitly in the followingproposition.

Proposition 6.2. If Z is consistent with {G(. ,z)}zez then

+ \ t «'> (g^GC 2> - £°(*. *> [ £«<«. *> *

(6.3)

for all x>0,dt® dR-a.s.

Proof. Analogous to the proof of Proposition 5.2. D

6.3. Exponential-Polynomial Families

In this section we introduce a particular class of functions G. As the main result

we characterize the corresponding consistent Itô processes.

Let K denote a positive integer and let n = (ni,..., n#) be a vector with

components n^ £ No, for 1 < i < K. Write \n\ := n\ + V uk- For a point

Z= (Ziß,...,Z1,ni+i,Z2,o,---,Z2,n2+l,---,ZKß,---,ZK,nK +l) R^+2K (6.4)

define the polynomials Pi (z) as

71;

Pi(z) =Pi(x,z) :=J2zitßx^, l<i<K.

fj.=0

The function G is now defined as

K

G(x,z):=J2Pi(x,z)e-z^x. (6.5)i=l

Obviously G G G1,2(R+ x R.\n\+2K). Hence the preceding section applies with

A=|n|+2A:.From an economic point of view it seems reasonable to restrict to bounded

forward rate curves. Let therefore Z denote the set of all z £ RN such that

suPx£K+ \G(X,Z)\ <00.

Definition 6.3. The exponential-polynomial family EP(K, n) is defined as

the set of forward curves {G(., z)}zeRN.The bounded exponential-polynomial family BEP(K, n) C EP(K, n) is de¬

fined as the set of forward curves {G(. ,z)}z<-z-

6 3 exponential-polynomial families 79

Clearly GNS(x, z) £ BEP(2, (0,1)) a,nd Gs(x,z) G BEP(2,, (0,1,1)), if at each

case the parameter z is chosen such that the curve is bounded. From now on, the

Nelson-Siegel and Svensson families are considered as subsets of BEP(2, (0,1)) and

BEP(3, (0,1,1)) respectively.If two exponents ^,«,+i and ^,nj+i coincide, the sum (6.5) defining G reduces

to a linear combination of K — 1 exponential functions. Thus for z £ RN we

introduce the equivalence relation

i ~z i :^=> 2»,n,+l = ZhnJ + l (6.6)

on the set {1,..., K} and denote by [i] = [i]z the equivalence class of i. We will

use the notation

nw = n[t](z) := maxjn,, | j £ [t]z}

Z[i],ß = Z[z],v.(z) := {j £ [i]z \n3 > ß}, 0 [*](*) := Yl P3^-

In particular P[z](z) = ^„=0 z[i],ßxß and (6.5) reads now

G(x,z)= Y ow^)e-z'-+ia:-We{i, ,K}/„

Observe that for z £ Z we have

Zi,nt+i=0, only if p^(z) = z[l]fi

^,n,+l<0, only if p[z](z) = 0.

We shall write the R-^-valued Itô process Z with the same indices which we

use for a point z £ RN, see (6.4),

pt d „t

Zzt'ß = Z^ + / by ds + Y aiyx dWA, 0 < j± < n% + 1, 1 'v<x, 0<p<ra, + l, 0<v<n3 + l, l<i,j<K.A=l

Notice that for 1 ß

\ [j{z\ zhni+1 = ^,„,+l}J

80 6 EXPONENTIAL-POLYNOMIAL FAMILIES

is not closed in general but nevertheless a Borel set in R^. We introduce the

following, thus optional, random sets of singular points (i, oo)

A := {Px{Z) = 0 or P[l](Z) = 0} ,1 +1)»ij=i

and the optional random sets of regular points (i, oo)

K

V:= (R+ xfi)\(|jAUBUC)i=i

V :=(R+ xft)\(BUC).

Let us recall that for S and T stopping times, a stochastic interval like [S, T] is

a subset of R+ x ft. Hence [S] — [S, S] is the restriction of the graph of the mappingS : ft -3 [0, oo] to the set R+ x ft.

For any stopping time t with [t] G (R+ x fl) \ i, we define

t'(oo) := inf{i > t(oj) \ (t,oo) £ Al\,

the debut of the optional set [r, oofnÛ- Observe that in general it is not true that

t' > t on {t < oo}. This can be seen from the following example: For

G(x, z) = zlfie'Zl lX+ z2ße~Z2 lX

+ z3fie~z^x G BEP(S, (0,0,0))

let Zl'° = Z\fi = 1, Z2'0 = -1, Zl'1 = 1 + i and Z\'x = Zl'1 = 1 for t £ [0,1]. Then

pi(Zo) = P[i](Z0) = 1 and P[i](Zt) = 0 for all i G (0,1]. Hence [0] G (R+ xft)\Aubut t' —0. However, by continuity of Z we always have

t <t' P-a.s. on {oo \ (t(oo),oo) £ V1}. (6.11)

Recall the fact that there is a one to one correspondence between the Itô pro¬

cesses Z starting in Zq (up to indistinguishability) and the equivalence classes of

b and a with respect to the dt ® dP-nullsets in P+ ® P. Hence we may state the

following inverse problem to equation (6.3): Given a family of forward curves. For

which choices of coefficients b and a do we get a consistent Itô process Z starting

inZ0?The main result is the following characterization of all consistent Itô processes,

which is remarkably restrictive. The proof of the theorem will be given in Sections

6.5 and 6.6.

Theorem 6.4. Let K G N, n = (n1:... ,nK) £ Njf and Z as above. If Z is

consistent with BEP(K,n), then necessarily for 1 < i < K

a,,„,+i,t,n,+i = 0j on /R(Z) ^ q^ df 0 rfp.o^ (612)

b.,n,+i = 0j on fyt(Z} ^ 0| n {p[i](z) Ï 0}, dt ® dF-a.s. (6.13)

Consequently, Zl'n'+1 is constant on intervals where px(Z) ^ 0 and P[t](Z) 7^ 0.

That is, forR-a.e. oo

Zr-+1(to)=Z^+l(oo) fort£[u,v],

6.4. AUXILIARY RESULTS 81

ifPi(Zt(oo)) ^ 0 andP[i](Zt(oo)) £ 0 for t £ (u,v).For a stopping time r with [r] C V let t'(oo) := inf{i > r(oo) \ (t,oo) £

V} denote the debut of the optional random set (B U C) H [r, oo[. üften ii /Wds

furthermore that t < r' on {t < oo} and

Z££ = z;-"e-z',n,+1* + Z*-"+1te-^'"i+1*

171,11, r7i,m —ZlO' t

on [0, t' — r[, /or 0 < n < ni — 1 and 1 and r' is the debut of (U^L-Âi U B U C) PI [r, oo[,i/ien r' = oo and m addition

/or 1 < i < K, R-a.s. on {r < oo}.

Remark 6.5. It will be made clear in the proof of the theorem that it is actually

sufficient to assume Z to be consistent with EP(K,n) for (6.12) to hold.

As an immediate consequence we may state the following corollaries. The

notation is the same as in the theorem.

Corollary 6.6. If Z is consistent withBEP(K,n) and if the optional random

sets {pi(Z) = 0} and {p[q(Z) = 0} have dt ® dR-measure zero, then the exponent

2i,m+i jS indistinguishable from Zo'n'+ ,1 < i < K.

Proof. If {pi(Z) = 0} and {p^(Z) = 0} have dt ® dP-measure zero, then

{pi(Z) ± 0} n {p[i\(Z) ^ 0} = R+ x ft up to a dt ® dP-nullset. The claim follows

using (6.12) and (6.13). D

COROLLARY 6.7. If Z is consistent with BEP(K,n) and if the following three

points are P-a.s. satisfied

i) pi(Zo) ï 0, for alll<i<K,

ii) there exists no pair of indices i ^ j with Zq71 = Zq3 ,

iii) there exists no pair of indices i ^ j with 2Zl0'niJr = Zq3 ,

then Z and hence the interest rate model G(x,Z) is quasi deterministic, i.e. all

randomness remains Pq-measurable. In particular the exponents Zt,ni+1 are indis¬

tinguishable from Zo'n , for 1 < i < K.

Proof. If i), ii) and iii) hold P-a.s. then [0] C V. The claim follows from the

second part of the theorem setting t = 0. D

6.4. Auxiliary Results

For the proof of the main result we need three auxiliary lemmas, presented in

this section. First there is a result on the identification of the coefficients of Itô

processes.

6.8. Let

xt--= X0 + f ß? ds + £ fV,j dW3Jo j=1Jo

Yf-= Y0+ [ ßJds + zZf rtJdW3Jo i=1Jo


be two Itô processes. Then dt ® dR-a.s.

3= 1 3=1 3=1

1{X=Y}ßX = l{A-=Y}/3y-

Proof. We write (., . ) for the scalar product in Rd. Then

\(lX,lX) ~ (lX,lY)\ = \(lX,7X - 7Y)I < \/<7x,7X>V/(TX-7Y,7x-7y)By the occupation times formula, see [40, Corollary (1.6), Chapter VI],

l{x.=Y.}tä ~ ^> 7S* " 1Ï) ds = °> for a11 * < oo. P"a-S-/Jo

10

Hence by Holder inequality

[ hx,=Y,}\(lX,lX)-(7X,^)\dsJo

 = l{x=Y}(7*,7y> = 1{X=Y}(7F,7K), di® dP-a.s.

By continuity of the processes X and Y there are sequences of stopping times

(Sn) and (Tn), Sn < Tn, with [Sm,Tm] n [5„,T„] = 0 for all m ^ n and

{X = Y} = M [5n, r„], up to evanescence.

ngN

To see this, let n £ N and let S(n, 1) := inf{i > 0 | \Xt - Yt\ = 0}. Define

T(n,p) := inf{i > S(n,p) \ \Xt - Yt\ > 0} and inductively

S(n,p+ 1) := inf{i > S(n,p) | |Xt - Yt| = 0 and sups(n>p)<s 2~n}.

Then by continuity we have limôo S(n,p) = oo for all n £ N and it follows that

{X — Y} = \Jnj>e¥![S(n,p),T(n,p)]. Now proceed as in [31, Lemma 1.1.31] to find

the sequences (Sn) and (Tn) with the desired properties.From above we have 1{x=y}(ix — 1Y)2 = 0, dt ® dP-a.s. For any 0 for 0 < i < oo, P-a.s.

Jo „ei^.Af

Using the same arguments as in the proof of [25, Proposition 3.2], we derive the

desired result.

6 4 AUXILIARY RESULTS 83

Secondly there are listed two results in matrix algebra.

Lemma 6.9. Let 7 = (7^) be a N x d-matnx and define the symmetric non-

negative definite N x N-matnx a := 77*, i.e. al>3 = a3tt = Xa=i 7j,a7?,a- Let I

and J denote two arbitrary subsets of {1,... ,N}. Define

oti,j = ajj = 535Zaw3EJ iei

Then it holds that aij > 0 and \oti,j\ < ^/ctijyjajj.

Proof. For 1 < A < d define 7/^ := ^lieIî,\- Then by definition

d d d

3Ç.J «e/ a=i A=l l£l j£J A=l

Hence

aij = ^2 (7/,a)2 > 0

A=l

and by Schwarz inequality

\al,j\ < £(*a 2^ m,Aj . 2^\j \=1 \j X=l

X! (7J,a)2 = ^/äÏJ^/äTJ.

Lemma 6.10. Let a = (alt}) be a n x n-matnx, n £ N, which is diagonally

dominant from the right, 1. e

n

I a»,« I > X^la«l3=13fr

n n

\a*A> zLZ la*'-?l' \set z~2 •:=°)>3=1+1 3=n+l

for all 1 Y^=2\ai^\ —

0> particular

aiti ^ 0. If n = 1 we are done. If n > 1, the elimination step

aW

=

1,3'

"«,i——"i,j, 2<i,j<n,"1,1

leads to the (n — 1) x (n — l)-matrix a(i) (°S)

1,1 )2<i,3<n- We show that a^ is

diagonal dominant from the right. If alti — 0, there is nothing to prove for the z-th

row. Let at>i ^ 0, for some 2 K,l- «1,1

0:1,1Fi 3\> 2<j<n.


Therefore

n n

£k(ïi<Ei°ïïi3=1+1 3=2

1&

n

= E3=2

n

- |CK»,13-1

"î,7 "1,7

ai,i<£kj + ^ £Kj

.7=23&

ai,i.7=2

.7=2

< K,i| - |a,,i| +

aj,i

ai,i

Ctll(|«i,i| — |ai,»|)

Proceed inductively to aS>,...,

a

"1,1

(2)

ai,»| < |a[?|

v(n-l)D

6.5. The Case BEP(l,n)

We will treat the case K — 1 separately, since it represents a key step in the proofof the general BEP(K,n) case. For simplicity we shall skip the index i = 1 and

write n — n± £ No, p = pi, b3 — b1'3, ah3 — a1,h1'3, etc. In particular we use the

notation of Section 6.2 with N = n + 2.

Lemma 6.11. Letn £ No and Z be as above. If Z is consistent with BEP(l,n),then necessarily

Zl = Z\fi»„-•z? H

+ Ztxte~z*+H

Zt Z£e-Z°

zr^zr'+u'b^ds+Y^jy?+1>3 dWi I lO0

for 0 < i < n — 1 and 0 < i < oo, P-a.s., where fto := {p(Zq) = 0}.

Consequently, if Z is consistent with BEP(l,n), then {p(Z) = 0} = R+ x ft0.Hence {Z+1 ^ ZJ+1} c Mz) = °}- Therefore we may state

COROLLARY 6.12. If Z is consistent with BEP(1, n), then Z is as in the lemma

and

G(x,Z)=p(x,Z)e~z° x.

Hence the corresponding interest rate model is quasi deterministic, i. e. all random¬

ness remains Po-measurable

Proof of Lemma 6.11 Let n G N0 and let Z be an Itô process, consistent

with BEP(l,n). Fix a point (t,oo) in R+ x ft. For simplicity we write zt for Z\(oo),al3 for a\'3 (oo) and bx for b\(oo). The proof relies on expanding equation (6.3) in the

:.5. THE CASE BEP(l,n) 85

point z = (zo, , zn+i)- The involved terms are

d2G(x,z)_

d2G(x,z)=

dztdzj dzjdzi

^G(x, z) = (J^p(x, z) - zn+1p(x, z)\ e~z^x (6.14)

lG(x,z)={^ZrX; °^"

(6.15)dzi

v '

\-xp(x,z)e-z»+lX, i = n + l

'0, 0<i,j<n

_xi+ie-zn+1x^ 0<i<n,j = n + l (6.16)

x2p(x,z)e~~Zn+lx, i = j = n + 1.

Finally it's useful to know the following relation for m G No

m -z +1V, \-qm(x)e-^x + t^çt, zn+l £ 0

0 Ife. ^n+l = 0,

where gm(x) = SaLo rro-*0! ^+r" is a polynomial in x of order m.^ '

n + 1

Let's suppose first that zn+i ^ 0. Thus, subtracting J^G(x, Z) from both sides

of (6.3) we get a null equation of the form

qi(x)e~Zn+lX + q2(x)e~2Zn+lX = 0, (6.18)

which has to hold simultaneously for all x > 0. The polynomials q\ and q2 dependon the Zj's, &I's and a^s. Equality (6.18) implies q\ = q2 = 0. This again yieldsthat all coefficients of the o/,'s have to be zero.

To proceed we have to distinguish the two cases p(z) ^ 0 and P(z) = 0. Let's

suppose first the former is true. Then there exists an index i £ {0,..., n} such that

Zi y^ 0. Set m := max{i <n\zt^0}. With regard to (6.15)—(6.17) it follows that

degçfë = 2m + 2. In particular

72

q2(x) — an+i,n+i x +...,

Zn+l

where...

denotes terms of lower order in x. Hence an+i,n+i = 0. But the matrix

a has to be nonnegative definite, so necessarily

an+itj = a,j>n+i = 0, for all 1 < j < n + 1.

In view of Lemma 6.8 (setting Y = 0), since we are characterizing a and 6 up to

dt ® dP-nullsets, we may assume a^3 = a,j^ = 0, for 0 < j < n +1, for alH > m +1.

Thus the degree of q2 reduces to 2m. Explicitly

/ \ am,m 2m ,

Q2(x) = ——xzm + ... .

Zn+l

Hence am>m = 0 and so amt0 — aj,m = 0, for 0 < j < n + 1. Proceeding inductivelyfor i = m — 1, m — 2,..., 0 we finally get that the diffusion matrix a is equal to zero

and hence q2 = 0 is fulfilled.

Now we determine the drift 6. By Lemma 6.8, we may assume bl = 0 for

m + 1 < i < n. With regard to (6.14) and (6 15), q\ reduces therefore to

qi(x) = -bn+1zmxm+1 + ... .


It follows bn+i = 0 and it remains

m --1

qi(x) = (bm + zn+1zm)xm + ^2(bi- zl+1 + zn+1Zi)xli=0

n-l

= (bn + Zn+iZn)xn + £(0« -

Zi+1 + Zn+1Zi)x\*=0

We now turn to the singular cases. If p(z) — 0, that is Zq = = zn = 0, we

may assume atj = a3tt = hi = 0, 0 < j < n + 1, for all i <n. But this means that

q± = q2 — 0, independently of the choice of bn+± and a„4-i,n+i-

For the case where zn+i = 0 we need the boundedness assumption z £ Z. By

(6.8) it follows that z\ = • = zn = 0. So by Lemma 6.8 again a^j = a^ = o; = 0,

0 < j < n + 1, for alH > 1. Thus in this case equation (6.3) reduces to

0 = &o - öo,oa;

and therefore bo = ao,o = 0.

Summarizing all cases we conclude that necessarily

h = -Zn+iZt + Zi+i, 0 n+i = 0.

The rest of the proof is analogous to the proof of [25, Proposition 4.1]. D

6.6. The General Case BEP(K,n)

Using again the notation of Section 6.3 we give the proof of the main result for

the case K > 2. The exposure is somewhat messy, which is due to the multi-

dimensionality of the problem. The idea however is simple: For a fixed point

(t, to) £ R+ xîlwe expand equation (6.3), which turns out to be a linear combination

of linearly independent exponential functions over the ring of polynomials, equalingzero. Consequently many of the coefficients have to vanish, which leads to our

assertion.

The difficulty is that some exponents may coincide. This causes a considerable

number of singular cases which require a separate discussion.

Proof of Theorem 6.4. Let K > 2, n = (n-i,... ,nx) G N^, and let Z be

consistent with BEP(K, n). As in the proof of Lemma 6.11 we fix a point (i, oo) in

R+ x ft and use the shorthand notation zhß for Zl'ß(to), aitß]jiU for al'll'J'l'(oo) and

bittl for blt'ß(oo), etc. Since we are characterizing a and 6 up to a dt ® dP-nullset, we

assume that (i, oo) is chosen outside of an exceptional dt ® dP-nullset. In particularthe lemmas from Section 6.4 shall apply each time we use them.

The strategy is the same as for the case K = 1. Thus we expand equation (6.3)in the point z = (ziß,... ,ZK,nK+i) to get a linear combination of (ideally) linearly

6 6 THE GENERAL CASE BBP(K, n) 87

independent exponential functions over the ring of polynomials

K

J2^(x)e-z>-'+lX + Yl qr,3(x)e~{z^+1+Z3 n>+l)x = 0. (6.19)?=1 1<1<3<K

Consequently, all polynomials qz and q%t3 have to be zero. The main difference to

the case K = 1 is that representation (6.19) may not be unique due to the possibly

multiple occurrence of the following singular cases

i) 2i,n,+l = Z]>ni+1, for l je j,

Ü) 2zt^+\ — z3,n,+l + Zk,nL+l,

iii) 2^î;ra>+i = z3>rij+i,

iv) ztjm+i = Zj>nj+i + Zk,nk+i>

for some indices 1 < i,j,k < K. However, the lemmas in Section 6.4 and the

boundedness assumption z £ Z are good enough to settle these four cases.

Let's suppose first that pt(z) ^ 0, for all i £ {1,...,K}. To settle Case i),let ~ denote the equivalence relation defined in (6.6). After re-parametrization if

necessary we may assume that

{1,...,K}/^ = {[1],...,[K}}

and £i,ni+i < • < Zk n _ +1for some integer K < K. Write I ~ {1,..., K}. In

view of Lemma 6.8 we may assume

aj,n]+l,3,n}+l= Gj.n.+M.n.+l and °3,n3+l = \n,+i for all j £ [i], i £ I. (6.20)

The proof of (6.12) and (6.13) is divided into four claims.

Claim 1. altn,+i,t,n,+i = 0, for all % G i.

Expression (6.19) takes the form

Y/qt(x)e~z—+^ + J2 g,J(a;)e-^-+1+z'-».+1^ = 0, (6.21)i£l 2,ji"

for some polynomials q% and qt!3. Taking into account Cases ii)—iv), this repre¬

sentation may still not be unique. However if for an index i £ I there exist no

j,k £ I such that 2zhnl+i = z3tnJ+1 + zkjnk+1 or 2zltnt+1 = zhnJ+i (in particular

Zi>n,+i 7^ 0) then we have

ql>t(x) = aj,n,+i,t,n,+i^

xMm+^

-I-...,Zi,nt+1

where fim := max{f \ v < n3 and z3i„ ^ 0 for some j £ [i]} £ No- Hence

ai,n,+i,t,n,+i — 0 and Claim 1 is proved for the regular case.

For the singular cases observe first that ^,n,+i = 0 implies ahnt+iihrii+i = 0,which follows from Lemma 6.8. Now we split i into two disjoint subsets I\ and i2,where

h := {i £ I \ Zitni+i ^ 0 and there exist j, k £ I, such that

2Zj,7i,+l = ZJ]nj_|_i + Zk,nk+1 or ^zi,nt+l — Z3^n}+{\

I2:=I\h.

Observe that Zf, n.+1 > 0 implies K £ I2 and zi,ni+i < 0 implies 1 G i2- Since at

least one of these events has to happen, the set I2 is not empty. We have shown


above that aj,n,+i,j,n,-i-i = 0, for i £ I2. If ii is not empty, we will show that for

each i £ p, the parameter ^,n,+i can be written as a linear combination of zhn +i 's

with j G i2- From this it follows by Lemmas 6 8 and 6 9 that Oj^+i^^+i = 0

for all i G ii and Claim 1 is completely proved. We proceed as follows. Write

ii = {h, ...,ir} with z%lln +i < • • • < Zi^n^+i. For each ik £ I\ there exists one

linear equation of the form

fzH,ntl+l\

*,...,*, 2, *,...,*) zih,ntk+l dk,

\zir,nVr +lJ

where * stands for 0 or —1, but at most one —1 on each side of 2. The ak on the

right hand side is either 0 or Zj,n,+i or ^,n,+i + z3,n3+i f°r some indices i,j £ I2.

Hence we get the system of linear equations

/2 *••• *\ (ziun%1+\\ /a{\

V * 2/ Xz^n +1J \arJBy Lemma 6.10, the matrix on the left hand side is invertible, from which follows

our assertion.

Claim 2. a3>n]+hk,p = afc,„J?nj+i = 0, for 0 < v < nk, for all l<j,k<K.

In view of (6.20), Claim 2 follows immediately from Claim 1 and Lemma 6.9.

Analogous to the notation introduced in (6.7) we set

hbU '•= E ^^

°~[i],»,\ = E ai^x

a{i\,IJ,,k,v :=/ j

a3,ß,k,v,

for 0 < ß < ri[j] ,0<i/<nk, l<k<K,l<X<d, i£l, and

a[i],v,[k],v~ E E a3^1^'

for 0 < ß < 7i[,], 0 < v < n^j, i, k £ I.

Claim 3. If z\^tll = 0, for i £ I and ß £ {0,... ,Ti[j]}, then

\%],H = aW,7i,W,M = a[t],ii,k,v = ak,u,[i],fj, = 0,

for all 0 < v < nk, 1 < k < K.

Notice that 0[i]iM)[i],M = Sa=i "m a \-Hence Claim 3 follows by Lemma 6.8 and

Lemma 6.9.

Claim 4. bîin,+i = 0, for all % g I such thatp^(z) ^ 0.

6 6 THE GENERAL CASE BEP(K, n) 89

Suppose first that z%^n%+\ ^ 0, for all % £ I. Let i £ I such that P[x\(z) ^ 0, and

let's assume there exist no j, k G I with zltrit+i — z3,nj+i + zk^nk+\. How does the

polynomial qt in (6.21) look like? With regard to (6.20), Claim 2, Lemma 6.8 and

equalities (6.14)-(6.17) the contributing terms are

fimAn, ßmAn3

£-p3(x,z)e-*>-^X=(( E ^^-1)-^,n,+l( E z3,^)y-Z'^+lX,

rij~\-l ~ ßmAn3 MmAï^j

E^^GM=(( E b3,ßx»)-bhnt+i( y, v+1)>ß=0

- J>M M=o M=o

and

,+ii

(6.22)

ßmAn,

— —( V"*n ,

nfe" TcV-z..n.+i» //polynomial^ ,n-(g..nt-4-i+Zfc.n,+i)a~

^ 2-, ahß,k,v nh+lx je { mx )e '

11=0 zk,nk+l

(6.23)

for 0 < v < nk, for all 1 < k < K and j £ [i]. We have used the integer

ßm := max{À | A < ni and ^ ^ 0 for some Z G [i]}.

Define ßm : = max{A | A < n^ and Z[^tx ^ 0} G No- Obviously /îm < /xm. ByClaim 3 we have a^^k.v = 0, for all ßm < ß < n^y Thus summing up the above

expressions over j £ [i] we get

Ux) = -bl^+lz[lUmx^+l + .... (6.24)

Consequently &î,ni+i = 0 in the regular case.

For the singular cases the boundedness assumption z G Z is essential. We spliti into two disjoint subsets Ji and J2, where

Ji := {i £ I \ there exist j,k £ I, such that zl>n%+i = ^,n3+i + zk,nk+i

and z3,n}+i > 0 and zk,nk+i > 0}

J2:=J\Jl.

Notice that in any case 1 e J2. We have shown above that for each i £ J2 such

that 21,71,4-1 is not the sum of two other z3^n +is it follows that 6t)Tl,.fi = 0. We

will now show that ol,nl+i = 0 for all i £ J2. Let i £ J2 and assume there exist

j,k £ I with zt,m+i — z3,n,+i + Zk,nk+\. Then necessarily one of the summands is

strictly less than zero. Without loss of generality z3iUl+i < 0. Since z G Z, we have

P[3](z) = 0, see (6.8). Thus a[j]lAtl[j],^ = 0 by Claim 3 and therefore a^^^ = 0, for

all 0 < ß < Ti[j], 0 < v < nk, 1 < fc < K. The contributing terms to the polynomialin front of e~z'-n*+lX, i.e. q% + q3^k + ...,

are those in (6 22) and (6.23) and also

1 (dG(x,z) fxdG(r,,z) dG(x,z) fx dG(V,z) n

= -ai,li,m,Jxße-z'^+lx f rive-Zk^+^dri+xve-ZK~h+ix f ê'^^^dn),(6.25)


for 0 < ß < ni, 0 < v < nm, l G [j], m £ [k]. However, summing up- for fixed ß,

m and v - the right hand side of (6.25) over I £ P\3\p, gives zero. Hence the terms

in (6.25) actually don't contribute to the meant polynomial. The same conclusion

can be drawn for all j,k £ I with the property that Zi,n;+i = zj,nj+i + zktnk+i- It

finally follows as in the regular case that 0j>n;+i = 0 for all i £ J2.

If Jx is not empty, we show that for each i £ J\, the parameter zitTli+i can

be written as a linear combination of Zj>nj+i's with j £ J2. From this it follows

by Lemma 6.8 that bi<ni+i = 0 for all i £ J\. We proceed as follows. Write

di = {ii, ,ir'} with Ziltnn+i < • - < Zi^,%ni ;_|_i. For each ik £ J\ there exists

one linear equation of the form

(*,...,*,1,0,...,0)

/ zii,ntl+l \

zik,n,,+l ~ akl

\zir',nir,+lJwhere * stands for 0 or —1, but at most two of them are —1. The a'k on the righthand side is either 0 or £i,n,+i or ^,rai+i + Zjtn,+i for some indices i, j £ J2 with

Zi,m+i > 0 and Zj%n.+\ > 0. Obviously a[ is of the latter form. Hence we get the

system of linear equations

(1 0

V*

°\'

zi1:ntl+i \/ Zi,m +1 + Zj,nj+1 \

* 1/ \zir,,nt ,4-1 y V y

for some i,j £ J2. On the left hand side stands a lower-triangular matrix, which is

therefore invertible. Hence Claim 4 is proved in the case where Zi,ni+\ ^ 0 for all

i£l.

Assume now that there exists i £ I with z^n.+i = 0. Then i £ J2. We have to

make sure that also in this case 0^+1 = 0, for all j £ J2. Clearly bilTll+i is zero

by Lemma 6.8. The problem is that z3,nj+i — Zi,n;+i + zj,n+i f°r ah j £ J2- But

following the lines above it is enough to show a[i])/i;[j])M = 0, for all 0 < ß < n^yFrom the boundedness assumption z £ Z we know that P[i\(z) = z^ß, see (6.8).Hence a[i]jAi;[j]jA = 0, for 1 < ß < n^. Suppose there is no pair of indices j, k £ I\{i}with Zj,nj+i+zk,nk+i = 0. Summing up the contributing terms in (6.22) and (6.23)over j £ [i] we get the polynomial in front of e°, i.e.

Qi(x) + qiA(x) = -aHi0;[j])0a; + • •, (6.26)

hence aô^j.o = 0- If there exist a pair of indices j,k G I \ {i} with z3>n.+\ +

zk,nk+i — 0, then one of these summands is strictly less than zero. Arguing as

before, the polynomial in front of e° remains of the form (6.26) and again a[i]ß-[i]ß =

0. Thus Claim 4 is completely proved.

Up to now we have established (6.12) and (6.13) under the hypothesis that

Pi(z) ^ 0, for all i £ {1,..., K}. Suppose now, there is an index i £ {1,..., K}withpi(z) = 0. By Lemma 6.8, we may assume a^yj^^ = biitl = 0,forall0 < ß<ni.But then Lemma 6.9 tells us that none of the terms including the index i appears

6.6. THE GENERAL CASE BEP(K, n) 91

in (6.19). In particular a^m+i-^ni+i and b^ni+\ can be chosen arbitrarily without

affecting equation (6.19). This means that we may skip i and proceed, after a

re-parametrization if necessary, with the remaining index set {!,...,K — 1} to

establish Claims 1-4 as above.

This all has to hold for dt <g> dP-a.e. (t,oo). Hence (6.12) and (6.13) are fully

proved. A closer look to the proof of (6.12), i.e. Claim 1, shows that the boundedness

assumption z £ Z was not explicitly used there. Whence Remark 6.5.

Next we prove that the exponents Zl'ni+1 are locally constant on intervals

where pt(Z) and p\q(Z) do not vanish. Let v > 0 be a rational number and let

Tv := inf{i > v | Pi(Zt) = 0 or p^(Zt) = 0} denote the debut of the optional set

[v,oo[nAi. By (6.12) and (6.13) and the continuity of Z we have that Zl<ni+1 is

P-a.s. constant on [^,2^,], hence P-a.s. constant on every such interval [u, T„]. Since

every open interval where Pi(Zt) ^ 0 or p^ (Zt) ^ 0 is covered by a countable union

of intervals [w,T„] and by continuity of Z the assertion follows and the first part of

the theorem is proved.

For establishing the second part of the theorem let t be a stopping time with

[r] G V and R(t < oo) > 0. Define the stopping time t'(oj) :— inf{i > t(oj) \(t,oo) $ £>'}. By continuity of Z, we conclude that r < r' on {r < oo}. Choose a

point (i, oo) in [r, r'[. We use shorthand notation as above.

By definition of V we can exclude the singular cases Zi,ni+i — Zj,n2+\ or

2z{,tm+i = Zjtn,+i, for i ^ j. In particular K — K, hence i = {1,...,K}. First we

show that the diffusion matrix for the coefficients of the polynomials Pi(z) vanishes.

Claim 5. cn^-j.v — a3.v-i.ß = 0, for 0 < ß < rij, for 0 < v < n3, for all i,j £ I.

By Lemma 6.8 it's enough to prove that the diagonal ai^ß-itß vanishes for 0 < ß < ni

and i G i. If there is an index i £ I with Pi(z) = 0 then argued as above ai,ß;i,ß =

bi,ß = 0, for all 0 < ß < ni, and we may skip the index i. Hence we assume now

that there is a K' < K such that Pi(z) ^ 0 (and thus 2i,n,+i > 0, since z £ Z) for

all 1 0 and there exist j, k £ I',

such that 2ziirii+1 = z3m+i + zk,nk+1}

I'2:=I'\I[.

Hence Zi,n;4-i = 0 for i G i' implies i £ I'2. We have already shown in the proofof Claim 4 that in this case at^-i^ = 0, for all 0 < ß < ni. The same follows for

i £ I2 with Zi,ni+i > 0, as it was demonstrated for the case K = 1.

Now let i £ I[ and let l,m £ I1, such that I < m and 2zi>ni+\ = ^,n,4-i +

zm,nm+i- Thus the polynomial in front of e~2zi'n*+lX is q^ + q^m + ...,and among

the contributing terms are also those in (6.25). If I or m is in I'2, those are all

zero. Write I[ = {i\,..., V'} with ziltnzi+i < • • < z^,,^ //+i-Then necessarily

/ G i2 in the above representation for ZiltTli 4-1. Thus the polynomial in front

of e" z'i-nii+lX is qiltil. It follows ailjß;iltß = 0, for all 0 < ß < n^, as it was

demonstrated for the case K = 1. Proceeding inductively for i2,... ,irn, we derive

eventually that ai!ß-itß— 0, for all 0 < ß < n^ and i £ P. This establishes Claim 5.


We are left with the task of determining the drift of the coefficients in p%(z).By (6.13), we have 0j,nj4-i = 0 for all i £ I'. Straightforward calculations show that

(6.19) reduces to

K'

J2qi(x)e-Z>^x = 0,i=i

with

m-l

Qi\X) — \yi,ni ' Zi,m+lzi,rii)X T / \"i,ß~

Zi,ß+l T Zi^ni-\.\Zi^ß)X .

ß=0

We conclude that for all 1 < i < K (in particular if pi (z) = 0)

bi,ß = ziyß+i-

Zitni+iZitß, 0<ß<ni-l-, (6.27)Oi,m — zi,ni+lZi:rii-

By continuity of Z, Claim 5 and (6.27) hold pathwise on the semi open interval

[t(oj) ,t' (oo) [ for almost every to. Therefore ZT+. is of the claimed form on [0, r' — r[.

Now replace V by T> and proceed as above. By (6.11) we have r < r' on

{r < oo}, and since V C V, all the above results remain valid. In addition

Pi(z) = P[i\(z) 7^ 0 and thus aiini+1^ni+i = Oj^+i = 0, for all 1 < i < K, by

(6.12) and (6.13). Hence ZzTf+1 = Zpni+1 on [0,r' - t[, for all 1 < i < K, up to

evanescence. But this again implies r' = oo by the continuity of Z. D

6.7. E-Consistent Itô Processes

An Itô process Z is by definition consistent with a family {G(., z)}zçz if and onlyif P is a martingale measure for the discounted bond price processes. We could

generalize this definition and call a process Z e-consistent with {G(. ,z)}zez if there

exists an equivalent martingale measure Q. Then obviously consistency implies e-

consistency, and e-consistency implies the absence of arbitrage opportunities, as it

is well known.

In case where the filtration is generated by the Brownian motion W, i.e. (Pt) =(P), we can give the following stronger result:

PROPOSITION 6.13. Let K £ N and n = (nx, ...,nK) £ Nf. If (Pt) = (F^),then any Itô process Z which is e-consistent with BEP(K,n), is of the form as

stated in Theorem 6.4-

PROOF. Let Z be an e-consistent Itô process under P, and let Q be an equiv¬alent martingale measure. Since (Pt) = (P^), we know that all P-martingaleshave the representation property relative to W. By Girsanov's theorem it fol¬

lows therefore that Z remains an Itô process under Q, which is consistent with

BEP(K, n). The drift coefficients b^ change under <Q> into V^. Whereas o*'^ = bôn -[a*'^'1'^ = 0}, dt ® dP-a.s. The diffusion matrix a remains the same. Therefore

and since the measures dt ig> dQ and dt ® dP are equivalent on R+ x ft, the Itô

process Z is of the form as stated in Theorem 6.4. D

Notice that in this case the expression quasi deterministic, i.e. J^-measurable, in

Corollaries 6.7 and 6.12 can be replaced by purely deterministic.

6.8. THE DIFFUSION CASE 93

6.8. The Diffusion Case

The main result from Section 6.3 reads much clearer for diffusion processes. In

all applications the generic Itô process Z on (ft,P, (^7i)o<i<oo,lP) given by (6.9) is

rather the solution of a stochastic differential equation

pt dpt

zi,ß = zi,ß + / billt^ Zs) ds + J2 a^x(s, Z„) dWsx, (6.28)Jo

A=1io

for 0 < ß < ni + 1 and 1 < i < K, where b and a are some Borel measurable

mappings from R+ x RN to RN, resp. RNxd.

The coefficients 6 and a could be derived by statistical inference methods from

the daily observations of the diffusion Z made by some central bank. These obser¬

vations are of course made under the objective probability measure. Hence P is not

a martingale measure in applications of this kind.

On the other hand we want a model for pricing interest rate sensitive securities.

Thus the diffusion has to be e-consistent. If we assume that (Pt) = (P^), the last

section applies. To stress the fact that JF^-measurable functions are deterministic,

we denote the initial values of the diffusion in (6.28) with small letters zfjß.Since all reasonable theory for stochastic differential equations requires conti¬

nuity properties of the coefficients, we shall assume in the sequel that b(t, z) and

cr(t,z) are continuous in z. The main result for e-consistent diffusion processes is

divided into the two following theorems. The first one only requires consistencywith EP(K,n).

Theorem 6.14. Let K £ N, n = (n1;... ,nK) £ Nf, (Pt) = (P^), the diffu¬sion Z, b and a as above. If Z is e-consistent with BEP(K,n) or with EP(K,n),then necessarily the exponents are constant

yi,m+ l — _i,nt+lL = Z0 ,

for alll<i<K.

Proof. The significant difference to the proof of Theorem 6.4 is that now the

diffusion matrix a and the drift b depend continuously on z.

First observe that the following sets of singular values

K

M~[]{z£RN\ pi(z) = 0 or m{z) = 0}7=1

M := {zel^l Zi,nt+\ = z3tn]+i + Zfc,nfc+i for some 1 < i,j,k < K)

are contained in a finite union of hyperplanes of R^, see (6.10). Hence (A4 U j\f) CRN has Lebesgue measure zero. Thus the topological closure of G := R^ \ (M. UA")isR^.

Now let Z be the diffusion in (6.28) which is e-consistent either with BEP(K, n)or EP(K,n). A closer look to the proof of Claim 4 shows that the boundedness

assumption z £ Z was not used for the regular case z £ Q, see (6.24). Combiningthis with (6.12), (6.13) and Remark 6.5 we conclude that for any 1 dP-nullset. By the very definition of the

product measure

0= / 1 di ® dP = / R[Nt] dt,

where Nt := {oo \ (t,oo) £ N} £ P. Consequently R[Nt] = 0 for a.e. i G R+. Hence

by continuity of b(t, . ) and a(t, . )

V'ni+1(t, . ) = <ri'ni+1:A(t, . ) = 0 (6.29)

on supp(Zt) n G, for a.e. i G R+. Here supp(Zt) denotes the support of the (n.b.regular) distribution of Zt, which is by definition the smallest closed set A C R^

with R[Zt £ A] = 1. Thus again by continuity of b(t, . ) and a(t, . ) equality (6.29)holds for a.e. i G R+ on the closure of supp(Zj) n G, which is supp(Zt). Hence we

may replace the functions b%,n'+l(t, .) and al'ni+1''x(t, .) by zero for almost every

i without changing the diffusion Z, whence the assertion follows. D

The sum of two real valued diffusion processes with coefficients continuous

in some argument is again a real valued diffusion with coefficients continuous in

that argument. Consequently we may assume that the exponents Zo'n'+ of the

above e-consistent diffusion are mutually distinct. Since otherwise we add the

corresponding polynomials to get in a canonical way an R-^-valued diffusion Z

which is e-consistent with BEP(K,n), resp. EP(K,n), for some K < K, N < N

and some n G N^-. Clearly Z provides the same interest rate model as Z and its

coefficients are continuous in z.

For the second theorem we have to require e-consistency with BEP(K,n).After a re-parametrization if necessary we may thus assume that

0<^'ni+1 <---<z0K'nK+1,

see (6.8). The sequel of Theorem 6.14 reads now

Theorem 6.15. If Z is e-consistent with BEP(K,n), then it is non-trivial

only if there exists a pair of indices 1 y+lLZq — Zq

PROOF. If there is no pair of indices 1 < i < j < K such that 2zl0'ni+1 =

Zq71' ,then V = R+ x ft. But then Z is deterministic by the second part of

Theorem 6.4.

The message of Theorem 6.14 is the following: There is no possibility for mod¬

elling the term structure of interest rates by exponential-polynomial families with

varying exponents driven by diffusion processes. From this point of view there is

no use for daily estimations of the exponents of some exponential-polynomial typefunctions like Gns or Gs- Once the exponents are chosen, they have to be keptconstant. Furthermore there is a strong restriction on this choice by Theorem 6.15.

It will be shown in the next section what this means for Gjvs and Gs in particular.

Remark 6.16. The boundedness assumption in Theorem 6.15 - that is e-con¬

sistency with BEP(K,n) - is essential for the strong (negative) result to be valid. It

can be easily checked, that G(x,z) = zq + Z\X £ EP(1,1) allows for a non-trivial

consistent diffusion process, see [26, Section 7].

6.9. APPLICATIONS 95

Remark 6.17. The choice of an infinite time horizon for traded bonds is not

a restriction, see (6.1). Indeed, we can limit our considerations on bonds P(t,T)which mature within a given finite time interval [0,T*]. Consequently, the HJM

drift condition (6.3) can only be deduced for x £ [0,T* —t], for dt®dR-a.e. (t,oo) £

[0,T*] x ft. But the functions appearing in (6.3) are analytic in x. Hence whenever

t < T*, relation (6.3) extends to all x > 0. All conclusions on e-consistent Itô

processes (Zt)o<t<T* can now be drawn as before.

6.9. Applications

In this section we apply the results on e-consistent diffusion processes to the Nelson-

Siegel and Svensson families.

6.9.1. The Nelson-Siegel family.

GNs(x, z) = zi + (z2 + z3x)e~Z4X

In view of Theorem 6.14 we have z± > 0. Hence it's immediate from Theorem 6.15

that there is no non-trivial e-consistent diffusion. This result has already been

obtained in [25] for e-consistent Itô processes.

6.9.2. The Svensson family.

Gs(x, z) = zi + (z2 + z3x)e~Z5X + ZiXe~Z6X

By Theorems 6.14 and 6.15 there remain the two choices

i) 2z6 = z5>0

ii) 2z5 = z6>0

We shall identify the e-consistent diffusion process Z = (Z1,..., Z6) in both cases.

Let Q be an equivalent martingale measure. Under Q the diffusion Z transforms into

a consistent one. Now we proceed as in the proof of Theorem 6.4. The expansion(6.19) reads as follows

QM + Qê-^ + Qê-^

+ Q4(x)e~2z5X + Q5(x)e~^+Z^x + Q6(x)e~2zeX = 0,

for some polynomials Qi ... ,Qq. Explicitly

Qi(x) = -aitlx + ...

Q2(x) = -olj3x2 + ...

Qz(x) = -ai,4X2 + ..

r\ i \ a3,3 2,

Qi(x)= —-x + ...

Z5

^1 r \ a4,4 2 ,

Qq(x) — X + . . .

ze

where...

denotes terms of lower order in x. Hence 01,1 = 0 in any case. By the

usual arguments (the matrix o is nonnegative definite) the degree of Q2 and Q3reduces to at most 1. Thus in both Cases i) and ii) it follows 03,3 = 04,4 = 0. It


remains

Qi(x) = h

Q2(x) = (b3 + z3z5)x + b2- z3 — + z2z5Z5

(6.30)Q3(x) = (&4 + Z4Z6)x - Zi

/I f \ a2.2Qi(x) =

Zb

while Q5 = Q6 = 0. Since in Case i) it must hold that Q4 = 0, we have 02,2 = 0 and

Z is deterministic. We conclude that there is no non-trivial e-consistent diffusion

in Case i).In Case ii) the condition Q3 + Q4 — 0 leads to

0-2,1 = Z4Z5. (6.31)

Hence a possibility for a non deterministic consistent diffusion Z. We derive from

(6.30) and (6.31)

&i =0

b2 = z3+z4 -z5z2

h = -Z5Z3

64 = —2Z5Z4.

Therefore the dynamics of Z1, Zs,..., Ze are deterministic. In particular

7i _ 1

zt- zo

Z\ = zle~z^ (6.32)

Zl = zê'2^,

while Zf =

Zq and Zf = 2z%. Denoting by W the Girsanov transform of W, we

have under the equivalent martingale measure Q

Zl = 4 + J (*(*) - z%Z2) ds + E / ^2'A(s) dW?, (6.33)

where $(i) and <72,A(i) are deterministic functions in i, namely

$(i):=ê--o*+z4e-2z05tand

A=l

By Levy's characterization theorem, see [40, Theorem (3.6), Chapter IV], the real

valued process

d pt „2,A/„N

w:-.= j:°

{s)z5sdWs\ o<i<oo,

is an (JF^-Brownian motion under Q. Hence the corresponding short rates rt =

G5(0, Zt) =zl + Z2 satisfy

drt = (j>(t) - zln) dt + ä(t) dW*,

6.10 CONCLUSIONS 97

where <j>(t) := $(i) + z\z% and à(t) := \/zoZê^Zot. This is just the generalizedVasicek model. It can be easily given a closed form solution for rt, see [38, p. 293].

Summarizing Case ii) we have found a non-trivial e-consistent diffusion pro¬

cess, which is identified by (6.32) and (6.33). Actually $ has to be replaced bya predictable process é due to the change of measure. Nevertheless this is just a

one factor model. The corresponding interest rate model is the generalized Vasicek

short rate model. This is very unsatisfactory since Svensson type functions Gs(x, z)have six factors Z\,..., z& which are observed. And it is seen that after all just one

of them - that is z2 - can be chosen to be non deterministic.

6.10. Conclusions

Bounded exponential-polynomial families like the Nelson-Siegel or Svensson familymay be well suited for daily estimations of the forward rate curve. They are rather

not to be used for inter-temporal interest rate modelling by diffusion processes.

This is due to the facts that

• the exponents have to be kept constant

• and moreover this choice is very restricted

whenever you want to exclude arbitrage possibilities. It is shown for the Nelson-

Siegel family in particular that there exists no non-trivial diffusion process providingan arbitrage free model. However there is a choice for the Svensson family, but still

a very limited one, since all parameters but one have to be kept either constant or

deterministic.

Blank leaf

CHAPTER 7

Invariant Manifolds for Weak Solutions to

Stochastic Equations

ABSTRACT. Viability and invariance problems related to a stochastic equa¬

tion in a Hilbert space H are studied. Finite dimensional invariant C2 sub¬

manifolds of H are characterized. We derive Nagumo type conditions and

prove a regularity result: Any weak solution, which is viable in a finite di¬

mensional C2 submanifold, is a strong solution.

These results are related to finding finite dimensional realizations for sto¬

chastic equations. There has recently been increased interest in connection

with a model for the stochastic evolution of forward rate curves.

7.1. Introduction

Consider a stochastic equation

'

dXt = (AXt + F(t, Xt)) dt + B(t, Xt) dWt

X0 =x0(7.1)

on a separable Hilbert space H. Here W denotes a Q-Wiener process on some

separable Hilbert space G. The operator A is the infinitesimal generator of a

strongly continuous semigroup in Ü. The random mappings F = F(t,co,x) and

B = B(t,to,x) satisfy appropriate measurability conditions.

This paper studies the stochastic viability and invariance problem related to

equation (7.1) for finite dimensional regular manifolds in Ü. A subset M. of Ü

is called (locally) invariant for (7.1), if for any space-time initial point (io,xo) G

R+ x M any solution X^ta'x°"> to (7.1) is (locally) viable in M, i.e. stays (locally)on M almost surely.

Our purpose is to characterize invariant finite dimensional C2 submanifolds Mof H in terms of A, F and B. Under mild conditions on F and B it is shown

that M. lies necessarily in the domain D(A) of A. The Nagumo type consistencyconditions

ß(t, oo, x) := Ax + F(t, oo,x)--J2 DBJ (*, w> X)BJ (*, oo,x) £ TXM (7.2)3

B3(t,oo,x) £TXM, Vi, (7.3)

equivalent to local invariance of A4, are derived. Here B3 := y<AjBej, where

{Xj,ej} is an eigensequence defined by Q. If moreover M is closed we can prove

that local invariance implies invariance.

The vector fields p(t,oo, . ) and B3(t,oo, .) are shown to be continuous on A4.

It turns out that the stochastic invariance problem related to (7.1) is equivalent to

the set of deterministic invariance problems related to ß(t,oo, .) and B3(t,oo, .).

99

100 7 INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS

By a solution to (7.1) we mean a weak solution, which in contrast to a strong

solution is the natural concept for stochastic equations in Hilbert spaces (see below

for the exact definitions). It is well known that under some Lipschitz and linear

growth conditions on F and B, a classical fixed point argument ensures the existence

of a unique weak solution X to (7.1), see [18]. But there is nothing comparablefor the existence of a strong solution, due to the discontinuity of A. However we

derive the following regularity result: If dimü < oo, any linear operator on H is

continuous and the concepts of a weak and strong solution to (7.1) coincide (atleast if dim G < oo). We show that this is also true in the general case providedthat X is viable in a finite dimensional G2 submanifold of H.

The main idea behind the proofs is the fact that locally any finite dimensional

submanifold A4 of if can be projected diffeomorphically onto a finite dimensional

linear subspace of D(A*), where A* is the adjoint of A. Thus any weak solution

viable in A4 is locally mapped onto a finite dimensional semimartingale, which

can be "pulled back" by Itô's formula (this is why we assumed A4 to be G2).This way equation (7.1) is locally transformed into a finite dimensional stochastic

equation, which in our setting has a unique strong solution. We use implicitly

that A* restricted to a finite dimensional subset of its domain D(A*) is bounded.

Therefore this idea cannot be directly extended to infinite dimensional invariant

submanifolds.

Many phenomena, say in physics or economics, are described by stochastic

equations of the form (7.1). The Hilbert space if is typically a space of functions.

Suppose (7.1) represents a stochastic model. Let G '= {G(-,z) \ z £ Z} be a

parametrized family of functions in H, for some parameter set Z C Rm. Assume

that G is used for statistical estimation of the model coefficients A, F and B in

(7.1). That is, one observes the process Z £ Z which is related to X by

G(;Zt) = Xt. (7.4)

The following problems naturally arise:

i) What are the sufficient and necessary conditions on G such that there exists a

non-trivial 2-valued process Z satisfying (7.4)? We then say that the model

is consistent with G-

ii) If so, what kind of dynamics does Z inherit from X by (7.4)? (Notice that

X is a priori a weak solution to (7.1), hence not a semimartingale in H.)

If G is regular enough, then G is an m-dimensional C2 submanifold in H. Accord¬

ingly our results apply: Problem i) is solved by the consistency conditions (7.2) and

(7.3). These can be expressed in local coordinates involving the first and second

order derivatives of G, making them feasible for applications. By our regularityresult, Z is (locally) an Itô process. This responds to Problem ii).

Similar invariance problems have been studied in [7], [10] and [50]. In [7] and

[10] the consistency problem for HJM models is solved. However they work with

strong solutions in a Hilbert space, which in general is a rather cutting restriction.

In [50] the invariance question for finite dimensional linear subspaces with respect to

Ornstein-Uhlenbeck processes is completely solved and applied to various interest

rate models. But the methods used in [50] have not yet been established for generalequations (7.1). Therefore our results cannot straightly been obtained that way.

Further recent results can be found in [3], [33] for finite dimensional systems,and in [30], [45] for infinite dimensional ones. See also the references therein.

7.2. PRELIMINARIES ON STOCHASTIC EQUATIONS 101

Compared to these studies we put very mild assumptions on the coefficients of

(7.1), which is due to the strong structure of A4.

The paper is organized as follows. In Section 7.2 we give the exact setting for

equation (7.1) and define (local) weak and strong solutions. Some classical results

on stochastic equations are presented. Section 7.3 contains the main results on

stochastic viability and invariance for finite dimensional G2 submanifolds. The

proofs are postponed to Sections 7.4 and 7.5, where we also recall Itô's formula for

our setting. In Section 7.6 we express the consistency conditions (7.2) and (7.3) in

local coordinates, making them feasible for applications. The appendix includes a

detailed discussion on finite dimensional (immersed and regular) submanifolds in a

Banach space. Their crucial properties are deduced.

7.2. Preliminaries on Stochastic Equations

For the stochastic background and notations we refer to [18]. We are given a proba¬

bility space (ft, P, P) together with a normal filtration (Pt)o<t<oo- Let H and G be

separable Hilbert spaces and Q £ L(G) a self-adjoint strictly positive operator. Set

G0 := Q1^2(G), equipped with the scalar product (u,v)aa '= (Q~~1I2u,Q~1I2v)g-We assume W in (7.1) to be a Q-Wiener process on G and Tr Q < oo. Otherwise

there always exists a separable Hilbert space Gi D G on which W has a realization

as a finite trace class Wiener process, see [18, Chapt. 4.3].Denote by L\ := L2(G0;H) the space of Hilbert-Schmidt operators from Go

into H, which itself is a separable Hilbert space. We will focus on the stochastic

equation (7.1) under the following set of (standard) assumptions.

• The operator A is the infinitesimal generator of a strongly continuous semi¬

group in Ü.

• The mappings F and B are measurable from (R+ x ft x H, V ® B(H)) into

(H,B(H)), resp. (L°,B(Ll)).• The initial value is a non random point xq £ H.

Two processes Y and Z are indistinguishable if R[Yt = Zt, Vi < oo] = 1. We

will not distinguish between them.

There is an equivalent way of looking at equation (7.1) which will be used

here. Let {e^} be an orthonormal basis of eigenvectors of Q, such that Qej = Xje3for a bounded sequence of strictly positive real numbers Xj. Then {\/Xje3} is an

orthonormal basis of Go and W has the expansion

w = E y/\ßs^ (7-5)3

where {ß3} is a sequence of real independent (J7t)-Brownian motions. This series

is convergent in the space of G-valued continuous square integrable martingales

M2T(G), for all T < oo.

Recall the following classical result on stochastic integration, see [18, Chapt. 4].

Proposition 7.1. Let E be a separable Hilbert space and a\ be E-valued pre¬

dictable processes, such that J^. \\a3t (w)|||; < oo for all (t,oo) £ R+ x ft and

¥[ [ zZâs\\2Eds< <x^=l, Vi<oo.

102 7. INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS

Then the series of stochastic integrals

Mt -?£••j dM

converges uniformly on compact time intervals in probability. Moreover Mt is an

E-valued continuous local martingale and for any bounded stopping time r

ptAr pt

MtAT = J2 ** dß* = E / ai Vt-i (*) dßi.3

J°3

J°

Set B3(t,oo,x) := ^/XjB(t,oo,x)e3 £ H and take into account the identity

\\B(t,oo,x)||2o = E WBJ(*>w> *)\\h< oo- (7-6)

(7.1')

Then equation (7.1) may be equivalently rewritten in the form

dXt = (AXt + F(t, Xt)) dt + J2 BJ (*> Xt) dßi

X0 = x0.

DEFINITION 7.2. An H-valued predictable process X is a local weak solution

to (7.1), resp. (7.1'), if there exists a stopping time r > 0, called the lifetime of X,such that

ptAr

< 00 = 1, Vi < oo

'

jT

(||*s||ff + \\F(8,XS)\\H + \\B(s,Xs)\\2Lo)dsand for any i < oo and £ G D(A*), P-a.s.

(C, XtAr) = (C, xo) + IT

((A*C, X„) + (C, F(s, Xs))) ds

ptAT+ / (C,B(s,Xs)dWs),

Jo

(7.7)

where

ptAr ptAr/ (C,B(s,Xs)dWs)=J2 (C,Bi(s,Xs))dßi.Jo

;Jo

A local weak solution X is a local strong solution to (7.1), resp. (7.1 '), if in addition

XtAr £ D(A), dt ® dP-a.s.

tAT

\\AXs\\Hds< oo = 1, Vi < oo (7-

and for any t < oo, P-a.s.

/tArptAr

[AXs + F(s,Xs))ds+ B(s,Xs)dWs, (7.9)

where

ptAr ptAr/ B(s,Xs) dWs=J2 BJ(s,Xs) dß3.Jo

jJo

If t = oo we just refer to X as weak, resp. strong solution.

7.2. PRELIMINARIES ON STOCHASTIC EQUATIONS 103

This definition may be straightly extended to random ./"o-measurable initial values

xo- Notice that the lifetime r is by no means maximal.

Lemma 7.3. Let X be a weak solution to (7.1) and r be a bounded stopping

time. Then XT+t is a weak solution to

f dYt = (AYt + F(t +1, Yt)) dt + B(t + t, Yt) dWt

[ Yq = XT

with respect to the filtration Pt := PT+t, where Wt := WT+t — WT is a Q-Wiener

process with respect to Pt. Moreover (7.5) induces the following expansion

^ = Ev/\7^, (7-11)3

where ß\ := ßT+t ~ ßT «s « sequence of real independent (Pt)-Brownian motions.

Proof. First we show that IF is a G-Wiener process with respect to Pt- Ob¬

viously Wo — 0 and W is continuous and .Ft-adapted. Now we proceed as in

the proof of [40, Theorem (3.6), Chapter IV]. Let h £ G. Define the function

/ G G°°(R+ x G; C) as follows

f(t,x) :=ex.p(i(h,x)G + -{Qh,h)G).Identify C = R2, then compute

ft(t,x) = ^f(t,x)(Qh,h)G £ L(R;C)

fx(t,x)=if(t,x)(h,.)G£L(H;C)

fxx(t,x) = -f(t,x)(h, .)G(h, .)G £ L(H x H;C)

and Itô's formula, see [18, Chapt 4.5], gives

f(t, wt) = l + i [ f(s, Ws) d(h, WS)G.Jo

Since f(t,Wt) is uniformly bounded on compact time intervals, Mt :— f(t,Wt)is a complex martingale (its real and imaginary parts are martingales). By the

optional stopping theorem MT+t is a nowhere vanishing complex ^-martingale.For 0 < s < i < oo

"[b£i*] = 1-

Whence for A £ Ps we get

E[UeMi(h,Wt-Ws)G)} = R[A]exp(-^(t-s)(Qh,h)G).Since this is true for any h £ G, the increment Wt — Ws is independent of Ps

and has a Gaussian distribution with covariance operator (t — s)Q. Hence IF is a

Q-Wiener process with respect to Pt-

Next we claim that

T $sdWs = J $T+sdWs, (7.12)


for any predictable L^-vedued process 4> with

rt

1, Vi < oo./ ||$s||ro ds < oo

Jo

If $ is elementary and t a simple stopping time, then (7.12) holds by inspec¬tion. The general case follows by a well known localization procedure, see [18,Lemma 4.9].

Since A is a weak solution to (7.1), for ( £ D(A*) we thus have

(C,XT+t) = (C, XT) + J' ((A*C, X„) + (C, F(s, X8)j) ds

+ JT ((,B(s,Xs)dWs)

= ((, XT) + J ({A*(, XT+S) + ((, F(T + S, XT+s))) ds

+ f (C,B(T + s,XT+s)dWs).Jo

Hence it follows that XT+t is a weak solution to (7.10). D

In view of Lemma 7.3 the following definition makes sense.

Definition 7.4. For (t0,x0) £ R+ x H we denote by X(t°'Xo') any (local) weak

solution to

f dXt = (AXt + F(t0 + t, Xt)) dt + B(to + t, Xt) dWt{to)\ A0 = x0

with respect to the filtration p\ c':= Pta+t- Here W} := Wt0+t — Wt0 is a

Q-Wienerprocess with respect to Pj: °.

All results in this paper for local weak solutions X = X(0'x°ï to (7.1) will straightlyapply for local weak solutions X^to'x°^ to (7.13). Hence they are stated only for the

case io = 0.

In the sequel we assume X to be a (local) weak solution to (7.1). The main

results of the next section follow under some regularity assumptions, which for

clarity are presented here in a collected form.

(Al): (continuity) There exists a continuous version of X, still denoted by X.

(A2): (differentiability) B(t,oo, .) £ G1 (#;£!]) for a11 (*,w) £»+xf!.This implies B3(t,oo, .) £ G^üjü), and since

DB3(t,oo,x)B3(t,oo,x) = (DB(t,oo,x)B3(t,oo,x))^/X~je3hence

z2\\DB3(t,to,x)B3(t,oo,x)\\2H < \\DB(t,oo,x)\\2L(H.LOßB(t,oo,x)\\2L% < oo

3

and £j DB3(t,oo, .)B3(t,oo, .) £ C°(H;H), for all (t,oo) £ R+ x ft.

(A3): (integrability) For any t < oo we have

rt

E (\\X,\\h + \\F(s,Xs)\\H + \\B(s,Xs)\\2Lo)ds < 00.

7 3 INVARIANT MANIFOLDS 105

(A4): (local Lipschitz) For all T,R < oo there exists a real constant G =

C(T,R) suchthat

\\F(t,oo,x) - F(t,oj,y)\\H + \\B(t,oo,x) - B(t,to,y)\\Lo < C\\x - y\\H

for all (t,to) £ [0,T] x ft and x,y £ H with ||x||ff < R and \\y\\H < R.

(A5): (right continuity) The mappings F(t,oo,x) and B(t,oo,x) are right con¬

tinuous in i, for all x G H and w G ft.

Finally we present a classical existence and uniqueness result for local weak

solutions to equation (7.1).

LEMMA 7.5. Assume (A4). Then for any xo there exists a local weak solution

to (7.1) satisfying (Al). Moreover any weak solution to (7.1) satisfying (Al) is

unique.

Proof. Let x0 G H. Set I := 2||x0||h and define

- =ÏF(t,u,x), H\\x\\H<l{^Xh

\F(t,uo,^x), il\\x\\H>l

and similarly B(t,oo,x). Then F and B are bounded and globally Lipschitz in x on

[0, T] x ft x H, for all T < oo. Hence by [18, Theorems 7.4 and 6.5] there exists a

unique continuous weak solution X to (7.1), when replacing F by F and B by B.

Define the stopping time r := inf{i > 0 [ ||A||ff > l}- Then r > 0 and Xt := Amtis a local weak solution to (7.1) with lifetime r and satisfies (Al).

If A is a continuous weak solution to (7.1) then by the above arguments it is

unique on [0,t„] for n > 2, where rn := inf{i > 0 | ||A||jy > n||xo||.ff}. Now use

that Tn "f oo. D

7.3. Invariant Manifolds

Let A4 denote an m-dimensional G2 submanifold of H, m > 1. See the appendix for

the definition and properties of a finite dimensional submanifold in a Hilbert space.

The concept of a submanifold is used in more than one sense in the literature, see

[11]. We therefore point out that M. represents what is also called a regular or

embedded submanifold, i.e. the topology and differentiable structure on A4 is the

one induced by H. This allows for deriving global regularity and invariance results,see Theorems 7.6 and 7.7 below.

Essential for our needs is the following observation. Since H is separable, byLindelöf's Lemma [1, p. 4] there exists a countable open covering (t4)&eN of A4

and for each k a parametrization cf>k : Vk C Rm -?- Uk D A4, where tftk £ C$(Rm; H),see Remark A.8. Using the fact that D(A*) is dense in H, by Proposition A.10 we

can assume that for each k there exists a linearly independent set {(k,i,.. , Cfc.m}in D(A*) such that

M(Ck,i,x),...,{Ck,m,x))=x, \/x£UknM. (7.14)

To simplify notation, we write (Çk,x) instead of (((k,i,x),..., (Çk,m,x)) and skipthe index k whenever there is no ambiguity.

For the sake of readability all proofs are postponed to the following sections.

First we state a (global) regularity result.


Theorem 7.6. Let X be a local weak solution to (7.1) with lifetime t, assume

(Al) and

XtAT G A4, Vi < oo. (7.15)

Then there exists a stopping time 0 < t' < r, such that X is a local strong solution

to (7.1) with lifetime r'.

If in addition r = oo and (A3) holds, then X is a strong solution to (7.1).

We can give sufficient conditions for a weak solution X to be viable in A4.

Notice that ß(t,oo,x) in (7.2) is well defined by assuming (A2).

Theorem 7.7. Let X be a weak solution to (7.1) with X0 £ A4. Assume

(Al), (A2) and (A4). If M is closed, lies in D(A) and satisfies the consistencyconditions (7.2) and (7.3) for dt ® dP-a.e. (t,oo) G K+ x ft, for all x £ M. Then

(7.15) holds for t = oo.

Moreover A is continuous on A4. Consequently (7.2) and (7.3) hold for all

x G A4, for dt <g> dP-a.e. (i, oo) £ R+ x ft.

It turns out that the viability condition (7.15) is too weak to imply (7.2) and

(7.3). Neither does it imply that M lies in D(A). As a link between Theorems 7.6

and 7.7 we shall formulate equivalent conditions for (7.2) and (7.3). Rather than

just asking for the assumptions of Theorem 7.6 we have to require that (7.15) holds

for any space-time starting point (io,xo).

Definition 7.8. The submanifold M is called locally invariant for (7.1), if

for all (to,xo) £ R+ x M there exists a continuous local weak solution X^ta,x°^ to

(7.13) with lifetime r = r(io,xo), such that

X?°;xo) G A4, Vi < oo.

Observe that this definition involves local existence.

THEOREM 7.9. Assume (A2), (A4) and (A5). Then the following conditions

are equivalent:

i) A4 is locally invariant for (7.1)ii) M C D(A) and (7.2), (7.3) hold for dt ® dR-a.e. (t,oo) £ R+ x ft, for all

x G A4,

iii) A4 C D(A) and (7.2), (7.3) hold for all (t,x) G R+ x A4, for R-a.e. to £ ft.

We finally mention the important case where A4 is a linear submanifold. Then

the above results are true under much weaker assumptions.

Theorem 7.10. If M. is an m-dimensional linear submanifold, then Theorems

7.7 and 7.9 remain valid without assuming (A2) and with ß(t,oo,x) in (7.2) re¬

placed by

v(t,oo,x) := Ax + F(t,oo,x).

7.4. Proof of Theorem 7.6

We recall Itô's formula for our setting.

Lemma 7.11 (Itô's formula). Assume that we are given the W^-valued contin¬

uous adapted process

Yt = Yo+ f 6sds +W vldßi,Jo Jo

7.4. PROOF OF THEOREM 7.6 107

where b and a3 are Rm-valued predictable processes such that

E \\4(")\\l- < <*>, V(i,W) G R+ x fi (7.16)3

j (II&.IIk- + E II^Hk-)^ < oo] = 1, Vi < oo (7.17)/0

i

and F0 is P0-measurable. Let cf> £ G62(Rm;if). Then Jt := | £\ D24>(Yt)(ai,ai) is

an H-valued predictable process,

+ Ç \\D<$>(Ys)oi\\2H)ds < oo] = 1, Vt < oo, (7.18)

and (j)oY is a continuous H-valued adapted process given by

(<$> o Y)t = (<£ o V)o + j (p<j>{Y8)ba + Js)ds + Y/f D^(Ys)o-i dß3s. (7.19)

Proof. Point-wise convergence of the series defining Jt follows from (7.16).Hence Jt is predictable. Integrability (7.18) is a direct consequence of (7.17). De¬

note the right hand side of (7.19) with It. Clearly by (7.18) it is a well defined

continuous adapted process in Ü, see Proposition 7.1. Choose an orthonormal ba¬

sis {ej} in H. It is enough to show equality (7.19) for each component with respect

to {et}. Define <j>i(y) := (e», 4>(y)). Then </>* £ G2(Rm; R) and Itô's formula applies,

see [18, Theorem 4.17]. It follows

(eh (<f> o Y)t) = (4>i oY)o + J (DUYs)bs + \ E^'W^'^'))^

E/ DUYsHdßi„•

ô

(euh)-

D

Proof of Theorem 7.6. Assume first that X is a weak solution, that is r =

oo, and that (A3) holds. Fix T > 0. Following [22, Lemma (3.5)] there exists an

increasing sequence of predictable stopping times 0 = To < Ti < T2 < • • < T with

sup„ Tn = T, such that on each of the intervals

Pn,Tn+l] H {Tn+i > Tn}

the process X takes values in some Ua(n) (here [0, T] x {Tn+1 > Tn} is identified

with {T„+i > Tn}). This holds due to (Al).Now let n G N0 and k = a(n). We will analyse X locally on Uk. For simplicity

we shall skip the index k for <f>k, Uk, Vk and (k in what follows. Define the Rm-valued

predictable processes b and a3 as

bt:=(A*Ç,Xt) + (C,F(t,Xt))

a{:=(C,B3(t,Xt)),


where logically (A*(,Xt) stands for ((A*Ci,Xt),..., (A*Cm, Xt)). Then 6 and a3'

satisfy (7.16) and (7.17), see (7.6), and by the very definition of X

Yt := (C,Xt) = (C,x0) + [ bsds + J2 [ H dßi-Jo Jo

Hence Itô's formula (Lemma 7.11) for <f> o Y applies. Since moreover Y(Tn+t)ATn+1takes values in V on {Tn+i > Tn}

X{Tn+t)ATn+1 = (4> ° Y){Tn+t)ATn+1, on {Tn+l > Tn}. (7.20)

Now using (7.7), (7.19) and (7.20) we have for £ G D(A*)

r(Tn+t)ATn+10

it,

= ((A%Xs) + (C,F(s,Xs))

- (C,D4>(Ys)bs+l-Y,D2^Ys)(^sM)))ds (721)3

r(Tn+t)ATn+1+ Ey [(C,B3(s,Xs)) - (C,D<f>(Ys)a3s))dßi.

Notice that the series in the first integral converges point-wise and defines an H-

valued predictable process, see Lemma 7.11. By Proposition 7.1 the last term

in (7.21) is a continuous local martingale with respect to the filtration (pT„+t)-Therefore

(A*£,Xt) = {C,-F(t,Xt) + D<f>(Yt)h + \YJD2<j>(Yt)(o-i,4)) (7-22)3

on [Tn,Tn+i]n{Tn+1 > Tn}, di(g>dP-a.s. (see the proof of [25, Proposition 3.2]). Byseparability of H there exists a countable dense subset T> c D(A*) such that (7.22)holds simultaneously for all Ç G V and for all (t,oo) £ [T„,Tn+1] n {Tn+i > Tn}outside an exceptional dt ® dP-nullset. Let (t, oo) be such a point of validity. Then

(A*(-,Xt(oo)) is a linear functional on D(A*), which is bounded on V by (7.22).Hence it is bounded and therefore continuous on D(A*). Since A** = A, see [41,Theorem 13.12], we conclude that Xt(oo) £ D(A) and - using full notation again -

AXt + F(t, Xt) = D4>(Yt) (<A*£ Xt) + (C, F(t, Xt)))+ \YJD2^Yt)((C,Bj(t,Xt)),(C,Bi(t,Xt))) (7"23)

3

on [T„, Tn+i] n {Tn+1 > Tn}, dt ® dP-a.s.

Similarly one shows that (compute the quadratic covariation of (7.21) with ß3)

B3\t,Xt)=D$(Yt)(C,B3(t,Xt)), Vj, (7.24)

on [Tn,Tn+i] D {Tn+i > Tn}, dt ® dP-a.s.

7.4. PROOF OF THEOREM 7.6 109

In view of (A3)

rT

E

< GiE

<G2E

(WD^YMh + \ J2 \\D2é.(Ys)(ai,ai)\\H + £ \\Dct>(Ys)cri\\2H)ds3 3

^T(||6s||Rm + ElKHM-)^

(||AS||H + \\F(s, X„)\\H + \\B(s,Xs)\\2Lo)ds < 00,

(7.25)

where Gi and G2 depend only on n.

Summing up over 0<n<A — lwe get from (7.23) and (7.25)

Er f'^N/ \\AXa\\H

lJods < 00, VA G N.

Since limjv \TN =T

[|AAs||tfds<co =1, Vt<T,

and by the identities (7.23) and (7.24)

Xt = xo + f (AXS + F(s, X,)) ds + V [ B3(s,Xs) dß3s.Jo

jJo

Since T was arbitrary the second part of the theorem is proved.Now assume A to be a local weak solution with lifetime r. Let <f> : V -> U n A4

be a parametrization in Ao satisfying (7.14). By (Al) there exists a stopping time

Ti > 0, such that XtAr takes values in U on [0, Ii]. Set r' := r A T\.The rest of the proof runs as before, only estimation (7.25) has to be replaced.

We give a sketch. Define

bt := (<A*C,XtAT,) + (C,-F(t,XMr')>)l[0,r'](t)4 := (Ç,B*(t,XtAr.))l[0,r'](t).

Then

where

XtAr' >°Y)t, Vi<oo (7.26)

Yt := (C,Xt„r') = (Cxo)+ [ bsds + Y] f o-i dß{. (7.27)^0 • Jo

Instead of (7.25) we now have (7.17) by the very definition of X. Using (7.26) and

Lemma 7.11 we get the relations (7.23) and (7.24) for X on [0,7-'], dt ® dP-a.s. In

particular XtAT' G D(A), di®dP-a.s. Hence from (7.18) and (7.23) we derive (7.8)and (7.9). D

110 7 INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS

7.5. Proof of Theorems 7.7, 7.9 and 7.10

A key step in proving Theorems 7.7, 7.9 and 7.10 consists m the following property.

Lemma 7.12. Assume (A2) and (A4). Let 4> : V -> U D A4 be a parametriza¬

tion satisfying (7.14). Suppose thatUnM C D(A). Then (7.2) and (7.3) hold fordt <g> dF-a.e. (t, oo) G R+ x fi, for all x £U n A4 if and only if

Ax + F(t, to, x) = D<f>{y) ((A*<, x) + (£, F(t, w, x)))+ \ E ^V(î/)((C, SJ (*, w, x)), (C, ßJ (i,oo, x))) (7.28)

^(i,o;,x) = ^(2/)(C,^(i,a;,x)), Vj, (7.29)

w/iere y = ((, x), /or a// x G U n A4, /or dt ® dP-a.e (i, w)el+x fi.

Consequently A is contmous on U fl A4.

Proof Point-wise equivalence of (7.3) and (7.29), and of (7.2) and the relation

Ax + F(t, cj,x)--^2 DB3 (t, oo, x)B3 (t, oo, x)3

= D<j>(y) (C, Ax + F(t, oo,x)-^Y,DB' (*> ^ X)B' (*> w>x)> (7-30)

follows from Lemma A.11. Assume (7.2) and (7 3) - and hence (7.29) and (7.30) -

hold for dt ® dP-a.e. (i, oo) G R+ x fi, for all x G U fl A4 But both sides of (7.29)are continuous in x by (A2). Hence there exists a di ® dP-nullset A" C R+ x fi,such that (7.29) is true simoultaneously for all x £ U n M and (t,oo) £ j\fc.

Fix (i,w) G j\fc. In view of (A2) Proposition A.12 applies and the last Sum¬

mand in (7.28) equals

\ J] DB3 (t, to, x)B3 (t, oo, x) - l-D<t>(y)(C, E DB3 (t,oo,x)B3(t, oo, x)).3 3

Hence (7 28) holds for (i, uo, x) £ JVC x (U n A4) if and only if (7.30) does so.

It remains to show validity of (7.28) for all x G U n M, for dt ® dP-a.e. (t,oo).We abbreviate the right hand side of (7.28) to R(t,oo,x). Then it reads

Ax = -F(t,u,x)+R(t,u,x) (7 31)

for dt ® dP-a.e. (t, oo), for all x G U C\ M- But due to (A4) the right hand side of

(7.31) is continuous in x, see (7.6). Hence for any (countable) sequence xn —V x in

U fl A4 we have Axn -+ Ax, by the closedness of A. In other words A restricted to

U fl A4 is continuous. This establishes the lemma. D

Proof of Theorem 7.7. Define the stopping time r0 := inf{i > 0 | Xt $A4}. By closedness of A4 and (Al) we have XTo G A4 on {r0 < oo}. We claim

r0 = oo.

Assume P[to < K] > 0 for a number K G N By countability of the cov¬

ering (Uk) there exists a parametrization 0 .V -> U fl A4, satisfying (7.14) and

PLYro G U and tq < K] > 0. Define the bounded stopping time T\ := tq A K and

7.5. PROOF OF THEOREMS 7.7, 7.9 AND 7.10 111

set Yo ~ (C,XT1). Since <f> £ C2(Rm;H), the Revalued, resp. L2(G0;Rra)-valuedmappings

= (A*C, <t>(y)) + (C, F(n (oo) + t,u, <f>(y)))= (C,B(T1(u)+t,ooJ(y))(.))

= \[Xjo(t,uo,y)ej = (C,B3(tx(oo) + i, to,<f)(y)))

b(t,oo,y)

a(t,uo,y)

a3(t,oo,y)

are bounded and globally Lipschitz in y on [0,T] x fi x Rm for all T < oo, which is

due to (A4). By Lemma 7.3 the process Wt ~ WTl+t — WTl is a Q-Wiener process

expanded by {ß3t} given by (7.11). Hence the stochastic differential equation in Rm

Yt = Y0+ [ b(s, Ya) ds + T f a3(s, Ys) d~ß{ (7.32)Jo Jo

has a unique continuous strong solution Y, see [18, Theorem 7.4].Now P[F0 G V | r0 < K] R[t0 < K] = R[Y0 £ V and r0 < K] > 0 and the dis¬

tribution of Y0 under P[. | ro < K] is regular. Hence by normality of Rm there exists

two open sets V0, Vx such that V0 C Vx C Vi C V and R[Y0 £ V0 and r0 < K] > 0.

Define the (.^-stopping time

T2finf{i>0|Yt£Fi}, ify0GFoandr0<A

[0, otherwise.

Then by continuity of Y we have P[r2 > 0] = P[Yb £ V0 and r0 < K) > 0 and

F G U on [0, r2] D {r2 > 0}. (7.34)

From the boundedness of b and a we derive

E J (\\b(s,Ys)\\mm+Y,\WJ(s,Ys)\\lm)ds]<oo, Vi<oo. (7.35)10

3

Set X := (j) o Y. The assumptions of Lemma 7.11 are satisfied, hence

Xt = XT1 + J (D(j>(Ys)b(s,Ys) + ±Y,I)2<t)(Ys)(o-j(s,Ys),o-3(s,Ys)))ds

+ Y, [ D<f>(Ys)a3(s, Ys) dß3s, on [0, r2] n {r2 > 0},. Jo

(7.36)

where the series in the first integral converges point-wise and defines an if-valued

predictable process. Moreover by (7,34) we have

X £(UnM)cD(A) on[0,r2]n{T2 > 0}. (7.37)

Hence due to (A2) and (A4) Lemma 7.12 applies and

AXt + F(Tl+t,Xt) = D<t>{Yt)b(t,Yt) + ^D20(rt)(^(i,y4),^(i,Ft))3

B3(n + t,Xt) = D<P(Yt)a3(t,Yt), Vj,


on [0,t2] n {t2 > 0}, dt <g> dP-a.s. This together with (7.35) and (7.36) implies

rtAr2

/Jo

||AAg||jj ds < oo 1, Vi < oo

A4Ar2 =XT1+ J [AXS +F(r1+s,Xs))ds + J2j BJ(n + s,Xs) dß3,

on {r2 > 0}. On the other hand we know by Lemmas 7.3 and 7.5 that XT1+t is the

unique continuous weak solution to

dZt = (AZt + F(n + i, Zt)) dt + }2 ßi(ri + *» zt) dßjt3

Zo — XT1

Whence Xt = XTl+t on [0,T2] fl {t2 > 0}. Because of (7.37) in particular XTl+T2 £

A4. But by construction {ro < K} = {to < K} H {r0 = n} and

0 < P[r2 > 0] < P[{n + t2 > n} n {r0 < K}} < P[n + r2 > r0],

which is absurd. Hence t0 = oo.

The last statement of the theorem follows from Lemma 7.12 and (A2).

Proof of Theorem 7.9. i)=îi): Fix (io,x0) G R+ x A4 and denote by X =

X(f°'xo) a continuous local weak solution to (7.13) with lifetime r. We proceed

as in the proof of Theorem 7.6 and adapt the notation. Define the P^ "^-stoppingtime Ti > 0 with the property that AtAr' takes values in Ua(0) for r' := r A Ti.

Analogously to (7.22) we derive the equality

(A% Xt) = (t, - F(t0 + i, xt) + D<P(Yt) ((A*C, Xt) + (C, F(*o + *, Xt)))+1 ]TiJ20(yt)((c,i^(io + i,xt)),(Cßîo +1,^)»),

(?'38)

i

for all ^ G D(A*) on [0,r'], di®dP-a.s. Due to (A4) and (A5) both sides of (7.38)are right continuous in i. Hence for P-a.e. oo the limit i J. 0 exists. By (7.6) we can

interchange the limiting i 4- 0 with the summation over j. Arguing as for (7.23) we

conclude that xq G D(A) and

Axo + F(t0, to, xo) = D(j)(yo) ((A*C, x0) + (C, F(t0,to, x0))^+ l/ZD2(t}(yo)((C,Bi(to,oo,xo)),(C,BJ(to,oo,xo))), P-a.s.

3

Similarly

B3(to,oo,x0)=Dcf>(yo)(C,BJ(to,u,xo)), Vi, P-a.s.

Since (to,xo) was arbitrary, Lemma 7.12 yields condition ii).ii)=î): Let (to, xq) £ R+ x M and let (j> : V —> U D M be a parametrization in

xq satisfying (7.14). As in the proof of Theorem 7.7 (setting ro = t\ = 0) we get a

7.6. CONSISTENCY CONDITIONS IN LOCAL COORDINATES 113

unique continuous strong solution Y to

Yt = (C, xo) + j ((A*C, <KYS)) + (C,F(t0 + s, MY,)))) ds

] f (C,B3(to + s,<l)(Ys)))dß^Jo

Here ß\ °''J:= ßlQ+t ~ ßt0 *s tne se9uence of (!F\ °')-Brownian motions related to

Wt by (7-11)• The stopping time t2 given by (7.33) is strictly positive and YiAT2 G

V. Analysis similar to that in the proof of Theorem 7.7 shows that X\ °':=

(d> o Y)t/\T2 is a continuous local strong solution to (7.13) with lifetime r2 and

x(to,xo) ef/nA4, foralli<oo.

ii)<£>iii): This is a direct consequence of Lemma 7.12 and (A5).

Proof of Theorem 7.10. In view of Proposition A.10 we can choose any

parametrization <f>k in (7.14) such that D2<f>k = 0 on Vk. By straightforward in¬

spection of the proofs we see that Lemma 7.12 and Theorems 7.7 and 7.9 remain

valid under the conditions stated in the theorem.

7.6. Consistency Conditions in Local Coordinates

In this section we express the consistency conditions (7.2) and (7.3) in local co¬

ordinates and identify the underlying coordinate process Y. From now on the

assumptions of Theorem 7.9 are in force and A4 is locally invariant for (7.1).Let (j) : V ^ U f\ M be a, parametrization. We assume that (A.l) and (A.2)

hold for all y £ V (otherwise replace V by V n V). Consequently

WDM^Wht^mv) < l|£*(M))ll, Vy G V (7.39)

and the latter is locally bounded since $ is G2. Theorem 7.9 says that

ß(t, oo, cj)(y)) = D(ß(y)b(t, oo, y) (7.40)

B3(t,oo,4>(y))=D(f)(y)p3(t,oo,y), Vj, (7.41)

for all (t,oo,y) G R+ x ft xV (after removing a P-nullset from fi), where b and

p3 are uniquely specified mappings measurable from (R+ x fi x V, V ® B(V)) into

(Rm,B(Rm)), see (A.3). By (7.39)

p(t,to,y) :=D^(y)-lB(t,oo,<P(y))(-)£L2(Go;Rm)-

To simplify notation, we suppress the argument oo whenever there is no ambiguity.In view of (A2) and (A.2), (P(t,-) £ G^VjR).

Fix (i, y) G V. It is shown at the end of the appendix that

DB3(t,<t>(y))B3(t,4>(y)) = ^-sB3(t,c(s))is=owhere c(s) := <p(y + sp^(i, j/)). On the other band, by (7.41)

^B3(t,c(s))[s=0 = ^-s{D4>(y + sp3(t,y))p3(t,y + sp3(t,y)))ls=0= D2<f>(y)(p3(t,y),p3(t,y))+D<f>(y){Dp3\t,y)p3\t,y)),


compare with (A.7). Inserting this in (7.40) yields

A<l>(y)+F(t,(ß(y))-^D2tf>(y)(p'(t,y),^(t,y))=D<l>(y)b{t,y), (7.42)

3

where b(t,oo,y) := b(t,oo,y) + \YJ3Dp3(t,oo,y)p3\t,oo,y). By (7.6), (7.39) and

(A2), the series converge absolutely and are continuous in y.

We now show how equation (7.13) can be written in local coordinates. We

claim that b(t, oo, y) G Rm and p(t, oo, y) are locally Lipschitz in y on [0, T] x fi x V,

for all T < oo. Indeed, the right hand side of (7.28) is locally Lipschitz in y by (A4)and hence also the left hand side of (7.42), see (7.6) and (7.39). Solving (7.42) for

o yields the claim.

Consequently, for any (io,2/o) G R+ x V there exists by [18, Theorem 7.4] a

unique continuous local strong solution y(*°>2«>) g y to

[ dYt = b(t0 + t, Yt) dt + Y^p3(t0 +1, Yt) dß?o)'J\ i (7-43)

I Y0=y0

with lifetime r(io,j/o), where ß\ °'':I:= ßJto+t ~~ ßl0-> see (TAI). Using Itô's formula

(Lemma 7.11) and proceeding as in the proof of Theorem 7.7, we see that

x[t^):=(4>oY^ÛT{t0,ya)is the unique continuous local strong solution to (7.13) with lifetime r(io, J/o), where

xo = 4>(yo)-Summarizing, we have proved

Theorem 7.13. Under the above assumptions, the consistency conditions (7.2)and (7.3) hold for all (t,oo,x) £ R+ x ft x (Un M) if and only if (7.42) and (7.41)hold for all (t, oo, y) G R+ x fi x V.

The coefficients b and p are locally Lipschitz continuous in y £ V: For any

T,R £ R-|_ there exists a real constant C = C(T, R) such that

\\b(t,uo,y) - b(t,oo,z)\\®m + \\p(t,oo,y) - p(t,oo,z)\\L2(Go;igi) < C\\y - z||Rm (7.44)

for all (t, oo) £ [0, T] x ft and y, z £ V n BR(Rm).Moreover equation (7.13) transforms locally into equation (7-43). That is,

X^ta'x°> is a continuous local strong solution to (7.13) in U n A4 if and only if

so is (j)'1 o x(io'x°ï to (7.43) in V, where y0 - ^(xo).If (j) is linear (P>2<f> = 0), then the same holds true without assuming (A2).

Appendix: Finite Dimensional Submanifolds in

Banach Spaces

For the convenience of the reader we discuss and prove some crucial propertiesof finite dimensional submanifolds in Banach spaces. The main tool from infinite

dimensional analysis is the inverse mapping theorem, see [1, Theorem 2.5.2], which

does not rely on a scalar product. Therefore we chose this somewhat oversized

framework, although in our setting stochastic equations take place in Hilbert spaces.

Let E denote a Banach space, E1 it's dual space. We write (e1, e) for the dualitypairing. For a direct sum decomposition E = E1 ® E2 we denote by HÊ2ie^ the

induced projection onto E±.

Let k,m G N. We begin with an important corollary of the inverse mappingtheorem, see [1, Theorem 2.5.12].

Proposition A.l. Let <f> G Ck(V;E), for some open set V C Rm. SupposeD(j)(yo) is one to one for some yo £ V. Then D<j>(yo)Rm is m-dimensional and

complemented in E

E = D<t>(yo)Rm®E2.

Moreover, there exist two open neighborhoods V of (yo, 0) mVxE2 and U of(f>(y0)in E, and a Ck diffeomorphism $ : U —> V such that

9otj>(y) = (y,Q), VyGV'n(Rmx{0}). (A.l)

Furthermore, D<f>(y) is one to one and

D^y)-1 = D*(<Ky))\D4(v)1tm, Vy G V n (Rm x {0}). (A.2)

Proof. Let {eu ..., em} be a basis of D(f>(y0)Rm. Every z £ D4>(y0)Rm has

a unique representation z = z\ei + .. .zmem. Set az(z) := z%. Each a.% extends to

a member e[ of E' by the Hahn-Banach theorem. Define E2 as the intersection

of the null spaces of all e[. Then E = D(j)(y0)Rm © E2 and ïl(B2,Dcf>(yo)^m) =

(e[,-)ei + ...(e'm,-)em.Define $ : V x E2 - E by $(y,z) := <}>(y) + z. Then $ G Ck(V x E2;E)

and D$(y,z)(vi,v2) = D(f>(y)vx + v2 for (t>i,7j2) G Rro x E2, see [1, Prop. 2.4.12].Now D&(yo,0) : Rm X E2 -4- E is linear, bijective and continuous, hence a linear

isomorphism by the open mapping theorem. Accordingly, the inverse mappingtheorem yields the existence of U and V as claimed such that $ : V -> U is a Gfc

diffeomorphism with inverse $.

Consequently, * o </>(y) = * o $(y,0) = (y,0) for y G V n (Rm x {0}). The

chain rule gives D^(4>(y))D4>(y) = Id^m. Hence Dcj)(y) is one to one and (A.2)holds. D

115

116 APPENDIX: FINITE DIMENSIONAL SUBMANIFOLDS IN BANACH SPACES

Definition A.2. The mapping (fi from Proposition A.l is called a Gk immer¬

sion at yo- If (fi is a Ck immersion at each y0 £ V, we just say (fi is a Ck immersion.

If (f> :V —> E is an infective Ck immersion, V C Rm open, we call A4 := <fi(V)an m-dimensional immersed Ck submanifold of E.

The next definition straightly extends the concept of a regular surface in R3.

Definition A.3. A subset A4 C E is an m-dimensional (regular) Ck subman¬

ifold of E, if for all x £ A4 there is a neighborhood U in E, an open set V C Rm

and a Ck map (fi : V —> E such that

i) (fi : V -^ U n Ai is a homeomorphism

ii) D<fi(y) is one to one for all y G V.

The map (fi is called a parametrization in x.

A4 is a linear submanifold iffor allx G At there exists a linear parametrization

of the form (fi(y) = x + YjT=i J/*e» in x-

Let (fi : V -4 E be an injective Ck immersion, V C Rm open, whence A4 = (fi(V)an immersed submanifold. In the notation of Proposition A.l we have by (A.l)that (fi : V n (Rm x {0}) ->• </>(V C (Rm x {0})) is a homeomorphism, and hence

<fi(V' n (Rm x {0})) is a regular submanifold. Yet in general (fi : V -?• M is not a

homeomorphism. In other words, being an immersed submanifold does not imply

being a regular submanifold. This is the crucial difference between an immersion

and an embedding. For examples and more details see [1, Section 3.5].

In what follows, Af denotes an m-dimensional Ck submanifold of E. As an

immediate consequence of Proposition A.l and Definition A.3 we show that A4

shares in fact the characterizing property of a Ck manifold:

Lemma A.4. Let (fii \Vt -4 UiDM, i = 1,2, be two parametrizations such that

W := U\ n U2 n A4 ^ 0. Then the change of parameters

(fi^c<f)2:(fi-\W)^4>î1{W)

is a Ck diffeomorphism.

Proof. Of course, (fi^1 o (fi2 : (fi21(W) -4 ^(W) is a homeomorphism.Now let y G (fi2x(W) and r = cfiT1 o <f>2(y). Then E = Dfa^R © E2 by

Proposition A.l. Moreover there exist two open neighborhoods V( C V\ x E2 of

(r,0) and U' C E of 4>x(r), and a Ck diffeomorphism #:£/'-> V{ such that

^ = *-! on V{ n (Rm x {0}).Now again we use the fact that ^ is a homeomorphism. Hence there exists

an open set U" C E with (fii(V{ n (Rm x {0})) = U" n A4. Set U := U' n U".

Then (fiT1 — * on U n A4. By continuity of <fi2 there is an open neighborhood

V2' C (fi2l(W) of y with (fi2(V^) c U n A4. But then (fir1 o cfi2\V, = * o (fi2^ is Ck

inU2'.Since y was arbitrary, <fi11 ° (fi2 is Gfc in <fi2

1(W). By symmetry the assertion

follows.

Definition A.5. For x G A4 the tangent space to A4 at x is the subspace

TxM:=D<fi(y)Rm, y = (fi-l(x),

where (fi :V C Rm —>• A4 is a parametrization in x.

APPENDIX: FINITE DIMENSIONAL SUBMANIFOLDS IN BANACH SPACES 117

By Lemma A.4, the definition of TXA4 is independent of the choice of the para¬

metrization.

Definition A.6. A vector field X on A4 is a mapping assigning to each pointx of A4 an element X(x) ofTxA4.

A vector field X can be represented locally as

X(x) = D<fi(y)a(y), y = 0"1(x), VxGf/nA4, (A.3)

where (fi : V —> U n A4 is a. parametrization and a is an Rm-valued vector field on

V (uniquely determined by (fi). This is how smoothness of vector fields on Af can

be defined in terms of smoothness of RTO-valued vector fields.

Definition A.7. The vector field X is of class Cr, 0 < r < k, if for any

parametrization (fi the corresponding Rm-valued vector field a in (A.3) is of class

Cr.

Again by Lemma A.4 this is a well defined concept.

It will be useful to extend a parametrization to the whole of Rm. Let x G A4

and </>:V->ornA4bea parametrization in x. Set y = (fi~~l(x). Since V is a

neighborhood of y there exists e > 0 such that the open ball

B2e(y):={v£Rm\\y-v\<2e}

is contained in V. On B2e(y) one can define a function ip £ G°°(Rm; [0,1]) satisfyingvj = 1 on Be(y) and supp(^>) C B2e(y), see [11, Theorem (5.1), Chapt. II]. Since <f> is

a homeomorphism there exists an open neighborhood U' of x in E with (fi(Be(y)) =U'nAi. Set 4> := ifi(fi. Then (fi £ Cj;(Rm;E) and <f>\B.(y) = <P\Be(y) Be(y) -> U'nM

is a parametrization in x. We have thus shown

Remark A.8. We may and will assume that any parametrization <fi : V —>

U n A4 extends to (fi £ CJ;(Rm; E).

Let x G A4. As shown in Proposition A.l, the tangent space Ta,A4 is com¬

plemented in E by E2 - nr^lN(e'i), where {ei,..., em) is a basis for TXA4 and

(e'i,e3) = 5l3 (Kronecker delta). The following lemma states that, locally in x,

H(b2,txm) is a diffeomorphism on A4 into TXA4.

Lemma A.9. There exists a parametrization 0 : F -> [/ n A4 in x such that

(fi((e'1,z),...,(e'm,z)) = z, ~iz£UnA4.

Proof. Let ifi : V\ —> U\ n A4 be a parametrization in x and write y =

ifi~l(x). Denote by Itxm ' TXA4 -+ Rm the canonical isomorphism It*m '=

((ei, •),..., (e'm, )). Then the map h := Itxm °^-(e2,txm) ° V>: Vi -> ^m is Gk and

Dh(y) = Itxm ° Dip(y) is one to one. By the inverse mapping theorem there exist

open neighborhoods FF C Ui of y and V C Rm of h(y) such that h : W -+ V is a

Gfc diffeomorphism. Since ifi is a homeomorphism there is an open neighborhoodU C E of x with vb(W) -Un Ai. The map (fi := ip o h~x : V -> U C A4 is then the

desired parametrization in x, that is, 0_1 = Itxm ° R(e2,t*m) on Un A4. D

The following result is crucial for our discussion on weak solutions to stochastic

equations viable in A4.

118 APPENDIX: FINITE DIMENSIONAL SUBMANIFOLDS IN BANACH SPACES

Proposition A. 10. Let D c E' be a dense subset. Then for any x G A4 there

exist elements f[,...,f'm in D and a parametrization (fi : V —> U n A4 in x such

that

(fi((fi,z),...,(fin,z)) = z, \/z£UnA4.

If A4 is linear, then (fi is linear: (fi(v) = e0 + YaLiCHî NijVj)ei, for v £ V.

Proof. The idea is to find a decomposition E = Fl © F2, dim.Fi = rn, such

that Fi is "not too far" from TXA4 and such that n^^) = (f[,-)fi + +

(f'm, -)/m, with /{..., f'm £ D. Thereby the expression "not too far" means that

^{f2,f1)\txm : TXM -» Fi is an isomorphism.Let {ei,..., em} be a basis for TxAi and tfi : Vi - Ux fl A4 the parametrization

in x given by Lemma A.9. Set y = ifi~1(x). We may assume that ||ej||.E = 1, for

1 < i < m. Since D is dense in E', there exist elements f[,...,f'm in D with

Wf'i — e'i\\E' < 2~m. Notice that by definition (e^,e,) = 5l3. Hence

\(fi,ej)\ < \(e'i,ej)\ + \(f[ - ej,e,)| < 2", if i± j

and

|(/;,e7)|>|«,ei)|-|(/;-<,ei)|>l-2-mThe matrix M := ((fî,e3))i<i,j<m is therefore invertible, which follows from the

theorem of Gerschgorin, see [43]. Consequently, the family {f{,..., f'm} is linearlyindependent in E'. Since E is reflexive, by the Hahn-Banach theorem there exists

a linearly independent set {/i,..., fm} in E such that (f[, fj) = o\,-. Write F\ :=

span{/i,..., /m}. The / and fi induce then a decomposition E = F\ © F2 with

projection n(F2iFl) = (/{, -)/i H + (f'm, -)fm. Since

m

n{F2,Fi)e3 = E Mji^' for l<J<mi7=1

we see that ^-(f2,f1)\txm ' TXA4 —> Fl is an isomorphism. Let i^x : Fi -> Rm

denote the canonical isomorphism Ip1 := ((/{, ),..., (f'm, )). Then the map h :=

IFt ° ^(f2,f1) ° ^ : Vi -> Km is Ck and Dh(y) — Ifx ° n^^) o Difi(y) is one to

one. Now proceed as in the proof of Lemma A.9 to get the desired parametrization

(fi in x with (fi~1 = i^ o n^j^) on U fl A4, for some neighborhood U of x.

If Af is linear, we choose ip(u) = x + 52 i wiei- An easy computation shows

that (fi(v) = (x - (H^f^m)-1 ° n(ir2,Fl)x) + E£i(E^=i ^SK SettinSN := M_1 and eo := x — (n^îÂ-f)-1 ° ^(f2,f1)X completes the proof. D

Lemma A.9 asserts the existence of a particular parametrization (fi : V —>

U n A4 in x G A4 related to the decomposition E = TxAi © F2. Actually, E2 is

complemented in E simultaneously by all tangent spaces TZA4, z £ U fl A4.

Lemma A.ll. Let cfi :V ~ï UnA4 be a parametrization. If there exist elements

e[,...,e'm in E' with the property that

<fi((e'1,x),...,(e'm,x))=x, VxGC/nA4,

then e\,...,e'm are linearly independent in E' and

E = TXA4®E2, \/x£UnAi,

where E2 := H^Li A/"(e^). Moreover, the induced projections are given by

II(e2,t*m) = D(fi(y)((e[, •),..., (e^,-)), y = (fi~l(x), VxG/7nA4. (A.4)

APPENDIX: FINITE DIMENSIONAL SUBMANIFOLDS IN BANACH SPACES 119

PROOF. Define A := ((ei, •),..., (e'm, )) £ L(E; Rm). Fix x £ U D A4 and let

y = (fi~x(x). By assumption A o <f>(v) = v on V, therefore A o D<fi(y) = Id®. It

follows that A^m : TXA4 —> Rm is an isomorphism. Consequently, e[,..., e'm are

linearly independent in E'.

Another consequence is that A : E -> Rm is onto. Now the map n := D(fi(y)oAsatisfies (i) II G L(E), (ii) A/(n) = Af(A) = E2, (iii) 7l(W) = TxAi and (iv) n2 = n,hence E — TXA4 ©F2 and n is the corresponding projection. Since x £ Un A4 was

arbitrary, this completes the proof. D

Finally, we derive a result which will prove crucial for Itô calculus on subman¬

ifolds. In the remainder of this section we assume A4 to be a G2 submanifold. Let

B £ G1 (E) have the property

B(x) £ TxAi, Vx G A4,

i.e. B\m is a G1 vector field on A4.

Fix x G A4 and let (fi : V -> U n M be a parametrization in x. Set y = (fir1 (x).For ö > 0 small enough, the curve

c(t) :=(fi(y + tD(fi(yy1B(x)), t£(-S,S),

is well defined and satisfies

c:(-6,6) ->UnA4, c£C1((-S,6);E), c(0) = x, c'(0) = B(x). (A.5)

Hence

^B(c(t))]t=o = DB(x)B(x). (A.6)

In terms of differential geometry B(x) can be identified with the equivalence class

[c] of all curves satisfying (A.5). By the very definition of DB(x) as a mapping from

TXA4 into TBÊ = E, the vector DB(x)B(x) is in the same manner identified

with [Boc], see [1, Section 3.3]. This is just (A.6) and means that the differentiable

structure on A4 is the one induced by E.

Suppose now that (fi satisfies the assumptions of Lemma A.ll. To shorten

notation, we write {e1, ) instead of ((e^, •),.--, (e'm, )). By (A.4)

B(c(t)) = D(fi((e',c(t)))(e',B(c(t))), Vte(-6,6).

Differentiation with respect to i gives

jtB(c(t))lt=0=D2(fi(y)((e',B(x)),(e',B(x)))+D4>(y)(e',DB(x)B(x)). (A.7)

Combining (A.4), (A.6) and (A.7) we get the following proposition.

Proposition A.12. Let A4, B and (fi : V -+ U n Af be as above. Then

DB(x)B(x) = D(fi(y)(e',DB(x)B(x)) + D2fi(y)({e',B(x)), (e',B(x))),

y = (fi~l (x), is the decomposition according to E = TxAi © E2, for all x £ U n A4.

Bibliography

1. R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications,

Springer-Verlag, Berlin-Heidelberg-New York, 1988.

2. H. Amann, Ordinary differential equations, Walter de Gruyter & Co., Berlin, 1990, An intro¬

duction to nonlinear analysis, Translated from the German by Gerhard Metzen.

3. M. Bardi and P. Goatin, Invariant sets for controlled degenerate diffusions: a viscositysolutions approach, Stochastic analysis, control, optimization and applications, Birkhäuser

Boston, Boston, MA, 1999, pp. 191-208.

4. H. Bauer, Maß- und Integrationstheorie, second ed., Walter de Gruyter & Co., Berlin, 1992.

5. B. M. Bibby and M. S0rensen, On estimation for discretely observed diffusions: A review,

Research Report No. 334, Department of Theoretical Statistics, Institute of Mathematics,

University of Aarhus, 1995.

6. BIS, Zero-coupon yield curves: Technical documentation, Bank for International Settlements,

Basle, March 1999.

7. T. Björk and B. J. Christensen, Interest rate dynamics and consistent forward rate curves,

Math. Finance 9 (1999), 323-348.

8. T. Björk and A. Gombani, Minimal realization of forward rates, Finance and Stochastics 3

(1999), no. 4, 413-432.

9. T. Björk, Y. Kabanov, and W. Runggaldier, Bond market structure in the presence of marked

point processes, Math. Finance 7 (1997), no. 2, 211-239.

10. T. Björk and L. Svensson, On the existence of finite dimensional realizations for nonlinear

forward rate models, Working paper, Stockholm School of Economics, submitted, 1997.

11. W. M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, sec¬

ond ed., Academic Press, 1986.

12. A. Brace, D. Gatarek, and M. Musiela, The market model of interest rate dynamics, Math.

Finance 7 (1997), no. 2, 127-155.

13. A. Brace and M. Musiela, A multifactor Gauss Markov implementation of Heath, Jarrow,

and Morton, Math. Finance 4 (1994), 259-283.

14. H. Brézis, Analyse fonctionnelle, fourth ed., Masson, Paris, 1993, Théorie et applications.

[Theory and applications].15. A. Chojnowska-Michalik, Stochastic differential equations in Hilbert spaces, Probability the¬

ory (Papers, Vllth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1976), PWN,Warsaw, 1979, pp. 53-74.

16. R. Cont, Modeling term structure dynamics, an infinite dimensional approach, Working pa¬

per, Ecole Polytechnique Paris, 1998.

17. J. Cox, J. Ingersoll, and S. Ross, A theory of the term structure of interest rates, Econometrica

53 (1985), 385-408.

18. G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge Univer¬

sity Press, 1992.

19. P. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset

pricing, Math. Ann. 300 (1994), 463-520.

20.,The no-arbitrage property under a change of numéraire, Stochastics Stochastics Rep.

53 (1995), no. 3-4, 213-226.

21. D. Duffle and R. Kan, A yield-factor model of interest rates, Math. Finance 6 (1996), 379-406.

22. M. Emery, Stochastic calculus in manifolds, Springer-Verlag, Berlin-Heidelberg-New York,1989.

23. D. Filipovic, Exponential-polynomial families and the term structure of interest rates, to

appear in BERNOULLI, Chapter 6 in this thesis.

121

r

122 BIBLIOGRAPHY

24.,Invariant manifolds for weak solutions to stochastic equations, to appear in Probab.

Theor. Relat. Fields, Chapter 7 in this thesis.

25.,A note on the Nelson-Siegel family, Math. Finance 9 (1999), 349-359, Chapter 5 in

this thesis.

26.,A general characterization of affine term structure models, Working paper, ETH

Zurich, 2000.

27. B. Goldys and M. Musiela, Infinite dimensional diffusions, Kolmogorov equations and interest

rate models, Tech. report, University of New South Wales, 1998.

28. D. Heath, R. Jarrow, and A. Morton, Bond pricing and the term structure of interest rates:

A new methodology for contingent claims valuation, Econometrica 60 (1992), 77-105.

29. N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, Noth-

Holland, Amsterdam, 1981.

30. W. Jachimiak, Stochastic invariance %n infinite dimension, Working paper, Polish Academyof Sciences, 1998.

31. J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes, Grundlehren der math¬

ematischen Wissenschaften, vol. 288, Springer-Verlag, Berlin-Heidelberg-New York, 1987.

32. I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, second ed., Springer-

Verlag, New York, 1991.

33. A. Milian, Invariance for stochastic equations with regular coefficients, Stochastic Anal. Appl.15 (1997), no. 1, 91-101.

34., Comparison theorems for stochastic evolution equations, Working paper, University

of Technology Krakow, 1999.

35. K. R. Miltersen, An arbitrage theory of the term structure of interest rates, Ann. Appl.Probab. 4 (1994), no. 4, 953-967.

36. A. J. Morton, Arbitrage and martingales, Ph.D. thesis, Cornell University, 1989.

37. M. Musiela, Stochastic PDEs and term structure models, Journées Internationales de Finance,IGR-AFFI, La Baule, 1993.

38. M. Musiela and M. Rutkowski, Martingale methods in financial modelling, Applications of

Mathematics, vol. 36, Springer-Verlag, Berlin-Heidelberg, 1987.

39. C. Nelson and A. Siegel, Parsimonious modeling of yield curves, J. of Business 60 (1987),473-489.

40. D. Revuz and M. Yor, Continuous martingales and Brownian motion, Grundlehren der math¬

ematischen Wissenschaften, vol. 293, Springer-Verlag, Berlin-Heidelberg-New York, 1994.

41. W. Rudin, Functional analysis, second ed., McGiaw-Hill, 1991.

42. H. R. Schwarz, Numerische Mathematik, second ed., Teubner, Stuttgart, 1986.

43. J. Stoer and R. Bulirsch, Einführung in die Numerische Mathematik, second ed., vol. II,Springer-Verlag, Berlin-Heidelberg-New York, 1978.

44. L. E. O. Svensson, Estimating and interpreting forward interest rates: Sweden 1992-1994,IMF Working Paper No. 114, September 1994.

45. G. Tessitore and J. Zabczyk, Comments on transition semigroups and stochastic invariance,Preprint, 1998.

46. T. Vargiolu, Calibration of the Gaussian Musiela model using the Karhunen-Loeve expansion,

Working paper, Scuola Normale Superiore Pisa, 1998.

47.,Invariant measures for the Musiela equation with deterministic diffusion term, Fi¬

nance Stochast. 3 (1999), no. 4, 483-492.

48. O. Vasicek, An equilibrium characterization of the term structure, J. Finan. Econom. 5 (1977),177-188.

49. M. Yor, Existence et unicité de diffusions à valeurs dans un espace de Hubert, Ann. Inst. H.

Poincaré Sect. B (N.S.) 10 (1974), 55-88.

50. J. Zabczyk, Stochastic invariance and consistency of financial models, Preprint, Polish Acad¬

emy of Sciences, Warsaw, 1999.

Curriculum Vitae

Personal Data

Name:

Date of Birth:

Nationality:

Damir Filipovic

26.3.1970

Swiss

Education

Oct 1995 - Apr 2000:

Apr 1995:

Oct 1990 - Apr 1995:

Jan 1990:

Ph.D. student in Mathematical Finance at ETH Zürich

Supervisor: Prof. F. Delbaen

Basic subject: Interest Rate Theory

Diploma with medal in Mathematics, ETH Zürich

Studies in Mathematics and Physics at ETH Zürich

Matura (Typus C), Gymnasium St.Gallen

Employment

Apr 1995 - now:

Apr 1992 - Mar 1994:

Assistant at ETH Zürich

Part-time Teacher at Gymnasium St.Gallen

Research Interests

• Mathematical finance, in particular interest rate theory

• Stochastic equations in infinite dimensions

• Markov processes

123

Documents

In Copyright - Non-Commercial Use Permitted Rights / License: …23396/eth-23396-02.pdf · Stochastic Equations in Infinite Dimensions 9 1.1. Infinite DimensionalBrownianMotion 9