Giulia Di Nunno Time-change in modelling, stochastic ...josepr23/2019/ICIAM2019v3...Time-change is a powerful modelling technique. Main idea: the representation of a complicated stochastic

Giulia Di Nunno

Time-change in modelling, stochastic calculusand controlICIAM 2019, Valencia

Presentation based also on joint works with:

Inga B. Eide, Michele Giordano, Hannes Haferkorn, Asma Kheder, Farai J. Mhlanga, Michèle Vanmaele,

Bernt Øksendal, Amine Oussama, Frank Proske, Yuri A. Rozanov, Steffen Sjursen.

STORM –– Stochastics for Time-Space Risk ModelsFunded by the Research Council of Norway, project no. 274410, and the University of Oslo

Agenda

1. Time-change

2. Information and stochastic calculus

3. BSDEs driven by time-changed noises

4. Stochastic control: maximum principle

Giulia Di Nunno 1 / 39

1. Time-changeTime-change is a powerful modelling technique.

Main idea: the representation of a complicated stochastic structure Xt , t ≥ 0,by a process of well-known structure Lt , t ≥ 0, with a changed time-line, byanother stochastic process Λt , t ≥ 0:

Xt = LΛt .

Interpretation: A way to change the velocity when moving along thetrajectories of L.

Definition. A general time-change process is a non-decreasing càdlàgstochastic process, starting at 0.

Ref: There is quite a history about time-change, particularly w.r.t. the representation of noise, simulation, and modelling e.g.turbulence and volatility. Barndorff-Nielsen and Shiryaev (2015), Swishchuk (2016), Veraart and Winkel (2010).


In our work time-changed processes are the models for the driving noises inthe dynamics. This allows e.g. for the inclusion of clustering effects.

Example: It is more likely to witness a financial price change in the fewseconds following a price change that has just occurred.Consider intraday data on the exchange rate between US Dollars and Euro (EURUSD) on June 4th 2018 (Bloomberg). The dataconsists of 1440 observations (one per minute) and provides: the mid price at the start of the minute and the number of “ticks” inthe one-minute interval. A “tick” corresponds to a change in the bid price, ask price, last trade price or volume. The tick grid forthe EURUSD exchange rate is in 0.0001 Euros. There is an average of 780.6 ticks in each interval, with a minimum of 422 and amaximum of 2564.

Price change

Den

sity

−6 −4 −2 0 2 4 6

0.0

0.1

0.2

0.3

0.4

0 5 10 15 20 25 30

0.00

0.05

0.10

0.15

0.20

Lag

AC

F

Figure: Left: Histogram of EURUSD price changes over one-minute time intervals on June 4th 2018. Right: Autocorrelationof EURUSD price changes. Source: Bloomberg. By Saavitne (2018)


1.1 Representation of the noise: some classics.

Theorem [Dambis 1965, Dubin-Schwarts 1965]. Let X be a continuous localmartingale with X0 = 0 and 〈X 〉∞ =∞ a.s.Then there exists a Brownian motion W such that

Xt = W〈X〉t .

Corollary. Consider the integral Xt =∫ t

0 σsdws, t ≥ 0, with respect to aBrownian motion w . Then

Xt = W〈X〉t , 〈X 〉t =

∫ t

0σ2

sds.

Theorem [Rosinski and Woyczynski (1986)]. Consider Xt =∫ t

0 σsdL(α)s , where L(α) is an α-stable Lévy processes with

α ∈ (0, 2]. Then

Xt = L(α)Λt,

where L(α) is another α-stable Lévy process and Λt =∫ t

0 σαs ds.

Remark. α-stable Lévy processes have the scaling property: σL(α)t

d= L(α)

t σα .


1.2 Time-change and embedding.

The problem: Given a stochastic process Xt , find an increasing family of stopping times Λt and a Brownian motion W, such that

Xtd= WΛt .

Theorem [Monroe (1978)]. Every càdlàg semimartingale Xt , t ≥ 0, can be written as a time-changed Brownian motion W for afamily of stopping times Λt , t ≥ 0, on a suitably extended probability space.

This results generalises the first study of Skorokhod (1964) studying the sums of independent random variables with mean nulland finite variance. Further developments, see e.g. the survey Obloj (2012).

Application to stochastic volatility/clustering.

dS(t) = μS(t)dt + S(t)σt dWt , S(0) >0.

dS(t) = μS(t)dt + S(t) dXt , S(0) >0.

See Dupire (1994), Carr, Geman, Madan, and Yor (2003), Carr and Wu (2004), Swishchuk (2016), Barndorff-Nielsen andShiryaev (2015)...

1.3 Time-change as a distortion of the space-time.

This is a non-negative random measure that is extending the concept of subordination on a Lévy type random field.See Barndorff-Nielsen, Pedersen (2010).

1.4 Time-change in infinite dimensions.

Here the idea is to consider Hilbert space valued time-changes to fit into an infinite dimensional concept of stochastic volatility.See e.g. Mandrekar, Salehi (1970), Albeverio (2003), Benth, Krühner (2015).


1.5 Choice of time-change.

Subordinators. These are non-decreasing Lévy processes.Remark.• Aa subordinated Lévy process is again a Lévy process.• In view of the simplicity (see also simulation potentials), the subordinated Brownian motion is particularly appealing:

Variance Gamma process is a Brownian motion subordinated by a Gamma processNormal inverse Gaussian, which is a Brownian motion subordinated by an Inverse Gaussian subordinatorin general, Normal tempered stable process is a Brownian motion subordinated by a tempered stable process

Absolutely continuous time-change. The time-change process is given by:

Λt =

∫ t

0λs ds,

where λt is a non-negative, integrable process.

1.6 Recovery of time-change. This is the problem of finding the time-changeprocess Λ given the observations X and the base process, e.g. W , L.See Winkel (2001), Sauri, Veraart (2017).


The noise: time-space random fieldsThe field is represented by

X := [0,T ]× Rd = [0,T ]× Z = XB t XH

whereXB := [0,T ]× 0 × Rd−1 = [0,T ]× ZB

XH := [0,T ]× R0 × Rd−1 = [0,T ]× ZH

with R0 := R \ 0.The random field μ is given as the mixture of two components:

μ(ω,∆) := B(ω,∆ ∩ XB

)+ H

(ω,∆ ∩ XH

), ω ∈ Ω,∆ ⊆ X

All stochastic elements are related to the complete probability space (Ω,F ,P).


About the distribution of the noise.

Let Λ be a non-negative, σ-finite, random measure on X such that:

Λ(0 × Rd ) = 0

E[erΛ(∆)

]<∞ for all ∆ ∈ Bc

X, r ∈ R.The σ-algebra generated by Λ is denoted by FΛ.

Naturally, we can consider:

Λ(∆) := ΛB(∆ ∩ XB) + ΛH(∆ ∩ XH), ∆ ⊆ X,

where ΛB ,ΛH are the restrictions of Λ.


Definition.B is a conditional Gaussian measure (cGm), i.e. random measure on XBsuch that

A1) P(

B(∆) ≤ x∣∣∣FΛ

)= P

(B(∆) ≤ x

∣∣∣ΛB(∆))

= Φ( x√

ΛB(∆)

), x ∈ R,

∆ ⊆ XB,

A2) B(∆1) and B(∆2) are conditionally independent given FΛ whenever∆1 and ∆2 are disjoint sets.

H is a conditional Poisson measure (cPm), i.e. a random measure on XHsuch that

A3) P(

H(∆) = k∣∣∣FΛ

)= P

(H(∆) = k

∣∣∣ΛH(∆))

= ΛH (∆)k

k! e−ΛH (∆), k ∈ N,∆ ⊆ XH ,

A4) H(∆1) and H(∆2) are conditionally independent given FΛ whenever∆1 and ∆2 are disjoint sets.

Furthermore we assume that

A5) B and H are conditionally independent given FΛ.

Φ is the standard normal cumulative distribution function.


Definition. The centred conditional Poisson measure (ccPm) is the signedrandom measure on BXH :

H(∆) := H(∆)− ΛH(∆), ∆ ∈ XH .

Among other properties, we have:

V (∆) := Var(μ(∆)) = E[μ(∆)2] = E[Λ(∆)].

Ref: The literature around doubly stochastic Poisson processes and its many applications is particularly extended. Cox (1955),Grandell (1976), Brémaud (1981), Cox and Isham (1980), Daley and Vere-Jones (2008), Carr, Geman, Madan, and Yor (2003),Carr and Wu (2004), Dassions and Jang (2003), Lando (1988), Grandell 1991), Klüppelberg and Mikosch (1995).


Connection with time-changed Lévy noises.

Theorem [Serfoso 72]. Denote Λt := Λ([0, t ]× Z). Let Wt , t ∈ [0,T ] be aBrownian motion and Nt , t ∈ [0,T ] be a centred pure jump Lévy process withLévy measure ν. Assume W , N are independent of Λ.

Then B is a conditional Gaussian random measure if and only if

Bt := B((0, t ]× ZB)d= WΛB

t,

and H is a conditional Poisson process if and only if

ηt :=

∫ t

0

∫ZH

zH(dsdz)d= NΛH

t.

See also Grigelionis (1975) for processes with conditionally independent increments.


2. Information and stochastic calculusInformation is modelled via filtrations, e.g. growing families of σ-algebras.

We focus on two information flows:

Fμ = Fμt , t ∈ [0,T ] where Fμt := σμ(∆) : ∆ ∈ [0, t ]× Z

We can show thatFμt = FB

t ∨ FHt ∨ FΛ

t .

We work with the right-continuous version F of Fμ, i.e. Ft :=⋂

r>t Fμr , and

G = Gt , t ∈ [0,T ] where Gt := Fμt ∨ FΛT .

Note that:

Ft ⊆ Gt , FT = GT

F0 is trivial, but G0 = FΛT

Correspondingly, we have FB ,GB ,FH ,GH .


The noise as integratorWe regard the noises as martingale random fields. See DN, Eide (2010):

Additivity. For pairwise disjoint sets ∆1 . . .∆K : V (∆k ) <∞:

μ( K⊔

k=1

∆k ) =K∑

k=1

μ(∆k )

μ is adapted to F (and G).Namely, for any ∆ ∈ [0, t ]× Z, μ(∆) is Ft -measurable

Martingale property. Consider ∆ ∈ (t ,T ]× Z, then E[μ(∆)|Ft ] = 0.In fact

E[B(∆)

∣∣∣Ft

]= E

[E[B(∆)

∣∣Gt] ∣∣∣Ft

]= 0

E[H(∆)

∣∣∣Ft

]= E

[E[H(∆)

∣∣Gt]− Λ(∆)

∣∣∣Ft

]= E

[E[H(∆)

∣∣FΛT]− Λ(∆)

∣∣∣Ft

]= 0


Also μ has conditional orthogonal values : i.e.

for any two disjoint sets ∆1,∆2 ∈ B(t ,T ]×Z, then

E[μ(∆1)μ(∆2)

∣∣∣Ft

]= E

[E[μ(∆1)

∣∣Gt]E[μ(∆2)

∣∣Gt] ∣∣∣Ft

]= 0

See also Cairoli and Walsh (1975) for martingale difference measure.

Definition. The predictable σ-algebras are given by:

PF := σ

F × (s,u]× B : F ∈ Fs , s <u, B ∈ BZ

PG := σ

F × (s,u]× B : F ∈ Gs , s <u, B ∈ BZ


Predictable compensator of μFor H = G,F.

Theorem. For a martingale random field μ with orthogonal values on a givenfiltration H, the set-function defined by:

PM(F ×∆) = E[μ(∆)21F ], F ×∆ = F × (s,u]× B ∈ PH

admits a unique extension (P-a.s.) as a random measure on PH. Also itadmits representation

PM(dω,dt ,dz) = P(dω)M(ω,dt ,dz)

See DN, Eide (2010).

Remark. The random measure Λ represents the PG-compensator of μ, but itis not necessarily the PF-compensator.Example. Consider μ = H and correspondingly assume E [Λ(t0 × Z)] >0. Then Λ(t0 × Z) is not F H

t0−-measurable.

Hence Λ is not the PF-compensator and there does not exist a modification of Λ that could be the compensator.

Example. Λ(ω,∆) =∫

∆ λ(ω, t , z)ν(t , dz)dt and λ is PF-measurable.

See DN, Sjursen (2014). In Brémaud (1981) there is a study on martingale point processes addressing the characterisation of theintensity as compensator in the case of FH .


Itô non-anticipating (NA) integrationThe NA-integral is defined as a linear isometric operator:

I : L2(Ω× X,PH,P ×M) =⇒ L2(Ω,FT ,P)

For the simple integrands:

φ(t , z) =Kn∑

k=1

φn,k 1∆n,k (t , z)

with ∆n,k = (sn,k ,un,k ]× Bn,k elements of a dissecting system of X and φn,kbounded Hsn,k -measurable, it is defined as

Iφ =

∫Xφ(t , z)μ(dt ,dz) :=

Kn∑k=1

φn,kμ(∆n,k ).

Then it is extended to all integrands φ = limn→∞ φn,k , by Itô isometry:

E[( ∫

Xφ(t , z)μ(dt ,dz)

)2]= E

[ ∫Xφ2(t , z)M(dt ,dz)

]In the isometry one che use Λ instead of M. Also one can be extended even further beyond the L2-setting.


(Non-Anticipating) NA-derivative

Definition. The NA-derivative is the linear operator

Dμ : L2(Ω,F ,P) =⇒ L2(Ω× X,PH)

Such thatDμ·,·ξ := lim

n→∞ϕμn(·, ·) in L2(PH),

where

ϕμn(t , z) :=

Kn∑k=1

E[ξ μ(∆n,k )

E[M(∆n,k )|Hsn,k ]

∣∣∣Hsn,k

]1∆n,k (t , z).

Proposition. The NA-derivative Dμ is the dual of the Itô integral I with respectto μ.

See: DN (2002, 2003); DN, Rozanov (1999, 2007); DN, Eide (2010); DN, Mhlanga (2011) for elements on non-anticipatingdifferentiation.


Computations: example

Proposition (case μ = H, H = GH ).

Consider the Λ-multilinear form ξ = β∏pj=1 H(∆j ), note β ∈ G0. Then

DHt ,zξ = β

p∑i=1

1∆i (t , z)∏j 6=i

H(∆j ∩ [0, t)× Z)

Remark.

If ξ = limn→∞ ξn in L2(P), then Dξ = limn→∞Dξn in L2(PGH).

The Λ-multilinear forms generate the space L2(Ω,FT ,P).


Integral representationsFor H = G,FTheorem: explicit integral representation.

For any ξ ∈ L2(Ω,FT ,P), there exists the representation:

ξ = ξ0 +

∫ T

0

∫Z

Dμt ,zξ μ(dtdz),

where the summands are orthogonal and Dξ0 ≡ 0.

See: DN (2002, 2003); DN, Rozanov (1999, 2007); DN, Eide (2010), for elements on non-anticipating differentiation.

Remark.

Existence of an integrand for the representation is guaranteed by Kunita-Watanabe result. Here the characterisationof the integrand is the added value.

When the integrator μ is a Brownian motion or a centred Poisson random measure, then a similar characterisationcan be given in terms of the Clark-Ocone formula via the Malliavin derivative. Ref. e.g. DN, Øksendal, Proske(2009). In this case, the Domain of the Malliavin derivative is a restriction!


In particular, under filtration G, a lot more can be said:

Theorem: integral representation. For any ξ ∈ L2(Ω,F ,P), we can write

ξ = ξ0 +

∫[0,T ]×Z

Dμt ,zξ μ(dt ,dz)

= E[ξ|FΛ] +

∫[0,T ]×ZB

DBt ,zξ B(dtdz) +

∫[0,T ]×ZH

DHt ,zξ H(dtdz).

The NA-derivatives are given by:

DB·,·ξ := lim

n→∞

Kn∑k=1

E[ξ B(∆n,k ∩ XB)

∣∣Gsn,k

]Λ(∆n,k ∩ XB)

1∆n,k (t , z).

and

DH·,·ξ := lim

n→∞

Kn∑k=1

E[ξ H(∆n,k ∩ XH)

∣∣Gsn,k

]Λ(∆n,k ∩ XH)

1∆n,k (t , z).


3. BSDEs driven by time-changed noisesTo ease notation: X = [0,T ]× R.We are interested in BSDEs of the type:

Yt = ξ +

T∫t

g(s, λs ,Ys , φs

)ds −

T∫t

∫R

φs(z)μ(ds,dz)

= ξ +

T∫t

g(s, λs ,Ys , φs

)ds −

T∫t

φs(0) dBs −T∫

t

∫R0

φs(z) H(ds,dz)

where ξ ∈ L2(Ω,FT ,P). Recall that FT = GT .

The solution of a BSDE is given by the couple (Y , φ) of adapted processes.

Question: But adapted to what filtration?

We can study the solution under both filtrations, but to obtain the most out ofthe knowledge of the noises for the applications to stochastic control, it is bestto work under G.


Definition.

(ξ,g) are standard parameters when ξ ∈ L2(Ω,F ,P

)and

g : [0,T ]× [0,∞)2 × R× R× Ω→ R

is a G-predictable function such that

g(·, λ,Y , φ) is G-adapted for all λ,Y , φg(·, λ,0,0) ∈ L2(PG) for all λfor all (λB , λH) ∈ [0,∞)2, y1, y2 ∈ R, φ(1), φ(2) ∈ Φ∣∣g(t , (λB , λH), y1, φ(1)

)− g

(t , (λB , λH), y2, φ(2)

)∣∣ ≤ Kg

(∣∣y1 − y2∣∣

+∣∣φ(1)(0)− φ(2)(0)

∣∣√λB +

√√√√∫R0

(φ(1) − φ(2))2(z)ν(dz)√λH).

• Let S be the space of G-adapted stochastic processes Y (t , ω), t ∈ [0, T ], ω ∈ Ω such that

‖Y‖2S := E

[sup

0≤t≤T|Yt |

2]<∞,

• Let Φ be the space of deterministic functions φ : R→ R such that ‖φ‖2Φ := |φ(0)|2 +

∫R0

φ(z)2 ν(dz) <∞


Theorem: solution

Let (g, ξ) be standard parameters.Then there exists unique Y ∈ S and φ ∈ L2(PG) such that

Yt = ξ +

T∫t

g(s, λs ,Ys , φs

)ds −

T∫t

φs(0) dBs +

T∫t

∫R0

φs(z) H(ds,dz)

Remark.The initial point Y0 is in general stochastic and FΛ-measurable:

Y0 = E[ξ +

∫ T

0g(s, λs ,Ys , φs) ds

∣∣∣FΛT

].

Remark.The existence of φ is obtained by application of the integral representationtheorems.Ref: DN, Sjursen (2014).Other forms of BSDEs are present in the literature: Jianming (2000), Carbone, Ferrario, Santacroce (2008), Jeanblanc, Mania,Santacroce, Schweizer (2011).


Linear BSDEs - explicit solution.

Theorem. Assume we have BSDE satisfying

−dYt =[AtYt + Ct + Et (0)φt (0)

√λB

t +

∫R0

Et (z)φt (z) ν(dz)√λH

t

]dt

− φt (0) dBt −∫R0

φt (z) H(dt ,dz), YT = ξ

under some conditions on the coefficients, then

Yt = E[ξΓT (t) +

∫ T

tΓs(t)Cs ds

∣∣∣GBt

]where

Γs(t) := exp∫ s

tAu −

12

Eu(0)21λu 6=0 du +

∫ s

tEu(0)

1λu 6=0√λB

u

dBu

+

∫ s

t

∫R0

[ln(1 + Eu(z)

1λHu 6=0√λH

u

)− Eu(z)

1λHu 6=0√λH

u

]ν(dz)λH

u du

+

∫ s

t

∫R0

ln(1 + Eu(z)

1λHu 6=0√λH

u

)H(du,dz)

Ref: DN, Sjursen (2014).Giulia Di Nunno 24 / 39

4. Stochastic control: maximum principleThe problem. Find u, so that

J(u) = maxu∈A

E[ T∫

0

ft (λt ,ut ,Xt ) dt + g(XT )].

for A set of admissible controls u in U ⊆ R convex.The state process is given by:

dXt = bt (λt ,ut ,Xt )dt +

∫R

κt (z , λt ,ut ,Xt )μ(dt ,dz), X0 ∈ R.

where bt (λ,u, x) and κt (z , λ,u, x), are real functions differentiable in x .

We assume that

g(x), x ∈ R, is a real concave differentiable function

ft (λ,u, x) is a real function differentiable in x .


We focus on absolutely continuous time-changes:

Λ(∆) = ΛB(∆ ∩ XB) + ΛH(∆ ∩ XH)

withΛB(∆) =

∫ T

01∆(t ,0) λB

t dt ΛH(∆) =

∫ T

0

∫R0

1∆(t , z) ν(dz)λHt dt

where ν is a σ-finite measure on R0 s.t.∫R0

z2ν(dz) <∞

Remarks. There are a number of optimal control problems of apparentlysimilar form. However, in the present case, we are dealing with:

1 control dynamics that are not jump-diffusions in general

2 the noise is not Markovian in general

For the characterisation of the optimal value process for a general time-change is in progress see DN, Haferkorn, Khedher,Vanmaele (2019).

For the optimal control problem in a mean-field framework see DN, Haferkorn (2017).

For the optimisation problem associated to Volterra-type dynamics driven by μ see DN, Giordano (2019).


Definition. The admissible controls are càglàd processes u : [0,T ]× Ω→ U ,such that

the dynamics of X have a unique strong solution, plus technicalconditions on the interplay with the time change

integrability conditions

E[ T∫

0

|ft (λt ,ut ,Xt )|2 dt + |g(XT )|+ |∂xg(XT )|2]<∞,

The admissible controls are either G-predictable or F-predictable and wedenote these sets as AG and AF respectively. Naturally AF ⊂ AG.

The couple (u,X ) is called an admissible pair.


Maximum principle approach.

We define the Hamiltonian, H : [0,T ]× [0,∞)2 × U × R× R× Φ→ R

Ht (λ,u, x , y , φ) = ft (λ,u, x) + bt (λ,u, x)y + κt (0, λ,u, x)φ(0)λB

+

∫R0

κt (z , λ,u, x)φ(z) λH ν(dz).

Corresponding to the admissible pair (u,X ) is the solution (Y , φ) of the linearBSDE:

Yt = ∂xg(XT ) +

T∫t

∂xHs(λ,us ,Xs ,Ys , φs) ds −T∫

t

∫R

φs(z)μ(ds,dz),

Here ∂xHt = ∂∂xHt (λ, u, x , y , φ).


Theorem (sufficient condition), filtration G.

Let u ∈ AG. Denote the corresponding state process as X and the solution ofthe adjoint equation as (Y , φ). If

ht (x) := maxu∈UHt (λt ,u, x , Yt , φt )

exists and is a concave function in x for all t ∈ [0,T ], and

Ht (λt , ut , Xt , Yt , φt ) = supu∈UHt (λt ,u, Xt , Yt , φt )

for all t ∈ [0,T ], then u is optimal in AG and (u, X ) is an optimal pair.Remark. In line with Framstad, Økesendal, Sulem (2004).


Using partial/restricted information: F ⊆ G, we have:

Theorem (sufficiency), filtration F. Let u ∈ AF. Denote the correspondingstate process as X with solution (Y , φ) of the adjoint equation. Set

HFt (λt , ut , Xt , Yt , φt ) := sup

u∈UE[Ht (λt ,u, Xt , Yt , φt )

∣∣Ft

]= sup

u∈U

ft (λt ,u, Xt ) + bt (λt ,u,Xt )E

[Yt∣∣Ft]

+ κt (0, λt ,u, Xt )E[φt (0)

∣∣Ft]

+

∫R0

κt (z , λt ,u, Xt )E[φt (z)

∣∣Ft]λH

t ν(dz)

for all t ∈ [0,T ]. IfhF

t (x) := maxu∈UHF

t(λt ,u, x , Yt , φt

)exists and is a concave function in x for all t ∈ [0,T ], then u is optimal (u, X )is an optimal pair.

Comments. The optimisation problem for time-changed Lévy processes is here treated in full generality. Other studies in thisdirections, for the power utility are due to Kallsen and Muhle-Karbe (2010).For optimal control under partial information via Malliavin calculus, see DN, Øksendal (2009).


Example: Optimal portfolio selection.

Consider two assets:

dRt = ρtRt− dt , R0 = 1,

dSt = αtSt− dt + St−

∫R

ψt (z)μ(dt ,dz), S0 >0.

The value process of a self-financing portfolio where u is the amount ofstocks, is given by:

dXt =[ρtXt + (αt − ρt )ut

]dt + ut

∫R

ψt (z)μ(dt ,dz).

( αt >ρt deterministic)The solution is

Xt = e∫ t0 ρr dr

(X0 +

t∫0

e−∫ s0 ρr dr

(αs − ρs)us ds +

t∫0

∫R

e−∫ s0 ρr dr usψs(z) μ(ds, dz)

).


We discuss mean-variance portfolio selection starting from an initial wealthx ∈ R and controls in U = R:

infu:E [XT ]=k

E[(XT − E [XT ])2] = sup

u∈AE[− 1

2(XT − k

)2]

=: J(u).

With the maximum principle under G we obtain that:

uGt =

−(αt − ρt

)(At Xt + Ct

)At(|ψt (0)|2λB

t +∫R0|ψt (z)|2 λH

t ν(dz)) .

With the maximum principle under F we have:

uFt = −

(αt − ρt )(E[At∣∣Ft ]Xt + E

[Ct∣∣Ft ]

)E[At∣∣Ft](|ψt (0)|2λB

t +∫R0|ψt (z)|2 λH

t ν(dz)) .

Here above:

At = − exp−

T∫t

(αs − ρs)2

|ψs(0)|2λBs +

∫R0|ψs(z)|2 λH

s ν(dz)− 2ρs ds

Ct = k exp−

T∫t

(αs − ρs)2

|ψs(0)|2λBs +

∫R0|ψs(z)|2 λH

s ν(dz)− ρs ds

.


Maximum principle (necessary)

To find an appropriate guess for u ∈ AF (or u ∈ AG), we have to study criticalpoints for J(u):

∂∂u

J(u + uβ)∣∣∣y=0

= 0

Substantially, we are showing that:

Result. If the control u in AF is locally optimal, then

E[∂Ht

∂u(u, Xt )|Ft

]= 0, dt × dP− a.e.

Remark.

Here we exploit the duality between the NA-derivative and the Itô integral. To achieve this result, only themartingality of μ was used, so very general!

In the cases of a Brownian motion and centred Poisson random measure one could also use the duality Malliavinderivative/Skorohod integral see e.g. Øksendal and Sulem (2007), DN and Øksendal (2009). Issues: domain of theoperators!

For this time-changed framework, we can use a version of conditional Malliavin calculus, see Yablonski (2005), DNand Sjursen (2011). Issues: domain of the operators!

To avoid problems with the domain of the operators, one can work in a white noise framework. See Agram,Øksendal (2018) for the Brownian motion and centred Poisson random measure. See DN, Draouil, Ouerdiane(2019) for the time-changed framework.


The techniques are based on the application of a perturbation argument.So that euristically,

0 =∂∂u

J(u + uβ)|y=0 = E[ ∫ T

0

∂fs∂x

Ys +∂fs∂uβsds + l ′(XT )YT

]where

Yt :=∂X u+yβ

∂y|y=0 =

∫ T

0

(∂bs

∂xYs+

∂bs

∂uβs)ds+

∫ T

0

∫R

(∂κs

∂xYs+

∂κs

∂uβs)μ(ds,dz).

Then we have the duality:

E[l ′(XT )YT

]= E

[ ∫ T

0l ′(XT )

(∂bs

∂xYs +

∂bs

∂uβs)ds

+

∫ T

0

∫R

(Ds,z l ′(XT ))(∂κs

∂xYs +

∂κs

∂uβs)Λ(ds,dz)

].


THANK [email protected]


References

O.E. Barndorff-Nielsen and A. Shiryaev (2015): Change of time and change of measure. World Scientific.

O. E. Barndorff-Nielsen, J. Pedersen (2011), Meta-times and extended subordination, Theory Probab. Appl., 56:2 (2011),319–327.

R. Boel, P. Varaiya and E. Wong (1975): Martingales on jump processes. I. Representation results. SIAM Journal on control,13,999-1021.

P. Brémaud, Point Processes and Queues. Martingale Dynamics, Springer 1981.

R. Cairoli and J. Walsh (1975): Stochastic integrals in the plane. Acta Mathematica. 134, 111-183.

R. Carbone, B. Ferrario, and M. Santacroce (2008): Backward stochastic differential equations driven by càdlàg martingales,Theory Probab. Appl. 52, 304-314.

P. Carr, H. Geman, D.B. Madan, and M. Yor (2003): Stochastic volatility for Lévy processes. Mathematical Finance, 13, 345-382.

P. Carr and L. Wu (2004): Time-changed Lévy processes and option pricing. Journal of Financial Economics, 71, 113-141.

D.R. Cox (1955): Some statistical methods related with series of events. J.R. Statist. Soc. B, 17, 129-164.

D.R. Cox and V. Isham (1980): Point Processes, Chapman & Hall.

D.J. Daley and D. Vere-Jones (2008): An Introduction to the Theory of Point Processes. 2nd Edition. Springer.

A. Dassios and J. Jang (2003): Pricing of catastrophe reinsurance & derivatives using the Cox process with shot noise intensity.Finance & Stochastics, 7, 73-95


References IIG. Di Nunno (2002): Stochastic integral representation, stochastic derivatives and minimal variance hedging. Stochastics andStochastics Reports, 73, 181-198.

G. Di Nunno (2003): Random Fields Evolution: non-anticipating integration and differentiation. Theory of Probability andMathematical Statistics (2002), 66, 82-94; AMS (2003), 66, 91-104.

G. Di Nunno (2007): Random Fields: non-anticipating derivative and differentiation formulas. Infin. Dimens. Anal. QuantumProbab. Relat. Top. 10, 465-481.

G. Di Nunno and I.B. Eide (2010): Minimal variance hedging in large financial markets: random fields approach. StochasticAnalysis and Application, 28, 54-85.

Di Nunno, G., Haferkorn, H. (2017): A maximum principle for mean-field SDEs with time change. Applied Mathematics andOptimization. 76, 137-176.

G. Di Nunno and S. Sjursen (2014): BSDEs driven by time-changed Lévy noises and optimal control. Stochastic Processes andtheir Applications, 124, 1679 -1709.

G. Di Nunno and B. Øksendal (2009): Optimal portfolio, partial information and Malliavin calculus. Stochastics 81, 303-322.

N.C. Framstad, B. Øksendal, and A. Sulem (2004): Sufficient stochastic maximum principle for the optimal control of jumpdiffusions and applications to finance, J. Optim. Theory Appl. 121, 77-98.

J. Grandell (1976): Doubly stochastic Poisson processes. Lecture Notes in Mathematics, Vol. 529. Springer-Verlag.

J. Grandell (1991): Aspects of risk theory. Springer-Verlag

B. Grigelionis (1975): Characterization of stochastic processes with conditionally independent increments. LithuanianMathematical Journal, 15, 562-567.

J. Jacod (1975): Multivariate point processes: predictable projection, Radon–Nikodym derivatives, representation of martingales,Probab. Theory Related Fields 31, 235-253.


References IIIM. Jeanblanc, M. Mania, M. Santacroce, and M. Schweizer (2012): Mean-variance hedging via stochastic control and BSDEs forgeneral semimartingales, Ann. Appl. Probab. 22, 2388-2428.

X. Jianming (2000): Backward stochastic differential equation with random measures. Acta Mathematicae Applicatae Sinica(English Series), 16, 225-234.

J. Kallsen (2006): A didactic note on affine stochastic volatility models, in Y. Kabanov, R. Liptser and J. Stoyanov, eds, ‘FromStochastic Calculus to Mathematical Finance’, Springer, 343-368.

J. Kallsen and J. Muhle-Karbe (2010): Utility maximization in affine stochastic volatility models, Int. J. Theor. Appl. Finance 13,459-477.

J. Kallsen and J. Muhle-Karbe (2010: Utility maximization in models with conditionally independent increments, Ann. Appl.Probab. 20, 2162-2177.

J. Kallsen and A. Shiryaev (2002): Time change representation of stochastic integrals, Theory of Probability and Its Applications46, 522-528.

C. Kluppelberg and T. Mikosch (1995): Explosive Poisson shot noise processes with applications to risk reserves, Bernoulli, 1,125-147.

A. Lim (2005): Mean-variance hedging when there are jumps. Siam J. Control Optim., 44, 1893–1922.

D. Lando (1988): On Cox Processes and Credit Risky Securities, Review of Derivatives Research, 2, 99-120.

I. Monroe (1978): Processes that can be embedded in Brownian motion, The Annals of Probability 6, 42-56.

J. Obloj (2012): On some aspects of Skorokhod Embedding Problem and its applications in Mathematical Finance. Lecture NotesSummer School in Mathematical Finance. Online.


References IV

B. Øksendal and A. Sulem (2007): Applied Stochastic Control of Jump Diffusions, Springer.

J. Rosinski and W.A. Woyczynski (1986): On Ito stochastic integration with respect to p-stable motion: inner clock, integrability ofsample paths, double and multiple integrals, Ann. Probab. 14, 271-286.

Sauri and A. Veraart (2017): On the class of distributions of subordinated Lévy processes and bases. SPA, 127, 475–496.

R. F. Serfozo (1972): Processes with conditional stationary independent increments. Journal of applied probability, 9

S. Sjursen (2013): Maximum principles for martingale random fields via non-anticipating stochastic derivatives. Preprint, UiO.

A. Swishchuk (2016): Change of time methods in quantitative finance. Springer, Briefs in Mathematics.

A. Veraart and M. Winkel (2010): Time change. Encyclopedia of Quantitative Finance, ed. R. Cont, Wiley.

M. Winkel (2001). The recovery problem for time-schanged Lévy processes. Research Report MaPhySto 2001-37.

A. Yablonski (2007): The Malliavin calculus for processes with conditionally independent increments. In G. Di Nunno and B.Øksendal (Eds.) Stochastic Analysis and Applications, Springer, Pages 641-678.


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