Integral representations and BSDEsdriven by doubly stochastic Poisson processes

Giulia Di Nunno

Controlled Deterministic and Stochastic SystemsIasi, 2-7 July 2012

——————Based on works in progress with:

Steffen Sjursen (CMA, Oslo)

Doubly stochastic Poisson random measures

The doubly stochastic Poisson process, also known as Cox process, wasintroduced in [Cox ’55] as a generalization of the Poisson process in thesense that the intensity is stochastic. These processes are largely studiedwithin the theory of point processes, see e.g. [Bremaud ’81].

Within mathematical finance, models based on DSPP appear in risktheory, in the study of ruin probabilities in insurance and insurance-linkedsecurity pricing and also in stochastic volatility models and optionpricing. See e.g. [Carr, Geman, Madan, and Yor ’03], [Carr and Wu ’04],[Dassions and Jang ’03], [Lando ’88], [Grandell ’91], [Kluppelberg andMikosch ’95].

We are interested in control problems in presence of a possibly exogenous

source of risk. However, here we present topics of stochastic calculus

with respect to the centered doubly stochastic Poisson random measure

(cDSPRM).

Outlines

1. Doubly stochastic Poisson random fields

2. Multilinear forms and polynomials

3. Non-anticipating integration and differentiation

4. About BSDEs for time-changed Levy noises

References

1. Doubly stochastic Poisson random fields

Let X be a locally compact, second countable, Hausdorff topologicalspace - in particular this implies that X =

⋃∞n=1 Xn with compact Xn’s

and that the topology on X has a countable basis consisting ofprecompact sets, ie sets with compact closure.We denote BX the Borel σ-algebra of X and Bc

x the precompacts of BX .

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

Let α be a random measure on X , σ-finite and non-atomic P-a.s.Moreover, we assume that α satisfies:

(1) E[ecα(∆)

]<∞ for all ∆ ∈ Bc

X , c ∈ R

Let us defineV (∆) := E[α(∆)], ∆ ⊆ BX ,

which is a non-atomic, σ-finite measure, finite on all precompact sets.The σ-algebra generated by α will be denoted Fα.

Let H be a random measure on X and let FH∆ denote the σ-algebra

generated by H(∆′), ∆′ ∈ BX : ∆′ ⊆ ∆ (with ∆ ∈ BX ).

Definition. The random measure H is doubly stochastic Poisson if

A1. P(

H(∆) = k∣∣∣α(∆)

)= α(∆)k

k! e−α(∆)

A2. FH∆1

and FH∆2

are conditionally independent given Fα,whenever ∆1 and ∆2 are disjoint sets.

Definition. The centered doubly stochastic Poisson random measure(cDSPRM) is the signed random measure

H(∆) := H(∆)− α(∆), ∆ ∈ BX .

We denote F H the σ-algebra generated by H(∆), ∆ ∈ BX .

See e.g. [Grandell ’76]. See e.g.[Bremaud ’81], [Cox and Isham ’80], [Daley and Vere-Jones ’08] for a presentationin the context of point processes.

Some properties. Naturally for any ∆ ∈ BX : V (∆) <∞, we have

E[H(∆)|Fα

]= 0

E[H(∆)2|Fα

]= α(∆) E

[H(∆)2

]= V (∆)

E[H(∆)3

∣∣Fα] = α(∆)

and, in general, we can prove by induction that:

E[H(∆)n+1

∣∣Fα] = α(∆) + α(∆)n−1∑k=2

(n

k

)E[H(∆)k

∣∣∣Fα], n ≥ 3.

This is obtained as adaptation of some computations in [Privault ’11].

Hence, we have that, for any n ≥ 3,

E[H(∆)n

]<∞ ⇐⇒ E

[α(∆)n−2

]<∞.

2. Multilinear forms and polynomialsWe recall that X =

⋃∞n=1 Xn with Xn growing sequence of compacts,

hence V (Xn) <∞ and α(Xn) <∞ a.s.

Being V non-atomic, for every n and εn > 0, there exists a finite partitionof Xn, i.e.

(2) ∆n,1, ...,∆n,Kn ∈ BcX : Xn =

Kn⊔k=1

∆n,k

such that supk=1,...,KnV (∆n,k) ≤ εn. Consider εn ↓ 0, n→∞.

Definition. A dissecting system of X is the sequence of partitions of X

(3) ∆n,1, ...,∆n,Kn ,∆n,Kn+1, n = 1, 2, ...

with⊔Kn

k=1 ∆n,k = Xn from (2) and ∆n,Kn+1 := X \ Xn, satisfying thenesting property:

(4) ∆n,k ∩∆n+1,j = ∆n+1,j or ∅, ∀k , j

Naturally, we have: supk=1,...,KnV (∆n,k) ≤ εn → 0, n→∞.

The dissecting systems are defined from the properties of V . However,the random measure α plays a crucial role and the following technicalresults is fundamental.

We recall that α is non-atomic P-a.s.

Proposition. With reference to (2)-(4), for any ∆ ∈ BX such thatα(∆) <∞ P-a.s. we have that

supk=1,...,Kn

α(∆ ∩∆n,k) −→ 0, n→∞, P− a.s.

We remark that all the sets in a dissecting system constitute a semi-ringof elements of X .

We can refer to e.g. [Kallenberg ’86], [Daley and Vere-Jones ’08] for more information of dissecting systems andpartitions related to measures such as V .

Hereafter we construct an orthogonal system based on multilinear formsof values of H and we show how it describes the intrinsic structure ofL2(Ω,F H ,P).

First we clarify the relationship between F H and FH ∨ Fα.

While it is easy to see that F H ⊆ FH ∨Fα. The converse is not obvious.

Theorem. The following equality holds:

F H = FH ∨ Fα.Proof. For n large enough we have α(∆n,k ) < 1 P-a.s. for any semi-ring set ∆n,k and

ceil“H(∆n,k )− α(∆n,k )

”= H(∆n,k ).

Here ceil(y) is the smallest integer greater than y .The results depends on the proposition before.

Definition. For any group of disjoint sets Λ1, . . .Λp ∈ BcX , an

α-multilinear form of order p is a random variable ξ of type:

(5) ξ := β

p∏j=1

H(Λj), p ≥ 1.

where β is an Fα-measurable random variable such that E[βn] <∞,n = 1, 2, ...The 0-order α-multilinear forms are the Fα-measurable random variableswith finite moments of all orders.

Remark. For any p, any α-multilinear form ξ belongs to L2(Ω,F H ,P).

In fact, for p ≥ 1, Eˆξ2˜ = E

ˆβ2 Qp

j=1 Eˆβ2H(Λj )2|Fα

˜˜= E

ˆQpj=1 α(Λj )

˜, which is a finite quantity by (1).

Comment. The use of measure based multilinear forms (without multipliers) in stochastic calculus is introduced in[dN ’07] for Levy random fields. The structure of independence was there heavily exploited.

In the sequel we consider multilinear forms on the dissecting system of X .

Definition. We denote by Hp the subspace of L2(Ω,F H ,P) generated bythe linear combinations of α-multilinear forms:

(6)∑

i

(βi

p∏j=1

H(∆i,j)).

Note that, in particular, H0 are all the square integrable Fα-measurablerandom variables.

Proposition. The subspaces Hp, p = 0, 1, 2, ..., are orthogonal.

Theorem: chaos expansion. The following representation holds:

L2(Ω,F H ,P) =∞∑

p=0

⊕Hp

Namely, for every ξ ∈ L2(Ω,F H ,P), there exists ξp ∈ Hp for p = 0, 1, ...such that ξ =

∑∞p=0 ξp.

This theorem is based on the following result:

Theorem. All the polynomials with degree less or equal to q of the valuesof H on the sets of the dissecting system of X are random variables thatbelong to

∑qp=0⊕Hp.

Comment. Note that if we considered the p-order multilinear forms oftype ξ =

∏pi=1 H(∆i ) and Mp the corresponding space generated by their

linear combinations, then

L2(Ω,F H ,P) 6=∞∑

p=0

⊕Mp.

In fact we can consider ∆1,∆2 ∈ BcX : ∆1 ∩ ∆2 = ∅ and a random measure α such that

E[α(∆1)|Fα∆2] 6= α(∆1). Then the element

ξ =“α(∆1)− E[α(∆1)|Fα∆2

]”H(∆2)

belongs to L2(Ω,F H , P), but it is orthogonal toP∞

p=0 ⊕Mp .

N.B. This observation has impact on the discussion about integral representation of elements of L2(Ω,F H , P).

Theorem [dN ’07]. If µ on X is a Levy random field (i.e. homogeneous and independent values) in L2, then theequality

L2(Ω,Fµ, P) =∞Xp=0

⊕Mp =∞Xp=0

⊕Hp

holds if and only if the µ is either a Gaussian or centered Poisson random field.

We also have that:If µ is a random field with independent values (i.e. drop homogeneous), then the equality holds if and only if µ isthe mixture of a Gaussian and centered Poisson random fields.

3. Non-anticipating integration and differentiationWe consider X = [0,T ]× Z where the ordering is given by time.We choose the dissecting system of X to be of form:

(7) ∆n,k = (sn,k , un,k ]× Bn,k , where sn,k ≤ un,k , Bn,k ∈ BcZ .

We assume that

(8) α(0 × Z ) = 0 a.s. or equiv. V (0 × Z ) = 0

We can consider the two filtrations

FH = F Ht , t ∈ [0,T ] where F H

t := σH(∆) : ∆ ∈ B[0,t]×Z

FH,α = FH,αt , t ∈ [0,T ] where FH,α

t := FHt ∨ FαT .

Note that:

I F Ht ⊆ F

H,αt

I F H0 is trivial, but FH,α

0 = FαT

I F HT = FH,α

T

Here Fα := Fαt , t ∈ [0,T ] where Fαt := σα(∆) : ∆ ∈ B[0,t]×Z.

Following [dN and Eide ’10] we can see that H is a martingale randomfield:

1. Additivity. For pairwise disjoint sets Λ1 . . .ΛK : V (Λk) <∞:

H( K⊔

k=1

Λk) =K∑

k=1

H(Λk)

2. H is adapted to FH and FH,α. Namely, for any ∆ ∈ B[0,t]×Z , H(∆)

is F Ht -measurable

3. Martingale property. Consider ∆ ∈ B(t,T ]×Z , then

E[H(∆)

∣∣∣F Ht

]= E

[E[H(∆)

∣∣F Ht ∨ FαT

]− α(∆)

∣∣∣F Ht

]= 0

4. Orthogonal values. Consider any two disjoint sets∆1,∆2 ∈ B(t,T ]×Z , then

E[H(∆1)H(∆2)

∣∣∣F Ht

]= E

[E[H(∆1)

∣∣F Ht ∨ FαT

]E[H(∆2)

∣∣F Ht ∨ FαT

] ∣∣∣F Ht

]= 0

Compare [Cairoli and Walsh ’75] for martingale difference measures.

Hence we can apply an Ito-type non-anticipating integration scheme with

respect to both FH and FH,α based on the following:Definition. The predictable σ-algebras are given by:

PH := σ

F × (s, u]× B, F ∈ F Hs , s < u, B ∈ BZ

PH,α := σ

F × (s, u]× B, F ∈ FH,α

s , s < u, B ∈ BZ

Consideration. The random measure α represents the PH,α-compensator

of H, but it is not necessarily the PH -compensator.

Example. Assume V (t0 × Z) > 0. Then α(t0 × Z) is not F Ht0−

-measurable. Hence α is not the

PH -compensator and there does not exist a modification of α that could be the compensator.

On the other side there exist situations in which α is also thePH -compensator.

Example. α(ω,∆) =R

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

In [Bremaud ’81] there is a study on martingale point processes addressing the characterization of the intensity as

compensator in the case of FH .

Theorem: integral representation. For any ξ ∈ L2(Ω,F HT ,P) there exists

a unique φH,α ∈ L2(PH,α) such that

(9) ξ = ξH,α0 +

∫ T

0

∫Z

φH,α(t, z) H(dt, dz).

The integrand is given by the non-anticipating derivative DH,αξ withrespect to H under FH,α:

(10) φH,α(·, ·) = DH,α·,· ξ := lim

n→∞ϕH,α

n (·, ·) in L2(PH,α),

where

(11) ϕn(t, z) :=Kn∑

k=1

E[ξ H(∆n,k)

∣∣FH,αsn,k

]α(∆n,k)

1∆n,k(t, z).

Moreover,

(12) ξH,α0 ∈ L2(Ω,FH,α,P) : DH,αξH,α

0 ≡ 0.

Furthermore, ξH0 = E[ξ|FαT ].

See also [dN ’02], [dN ’03] for elements on non-anticipating differentiation.

Comments.

I The non-anticipating integration scheme can be equivalently set up

with respect to the filtration F H . An explicit integral representationis also proved in that context. However much less information ispossible to obtain on the stochastic orthogonal remain.

I Stochastic integral representations for the DSPRM have beeninvestigated by, e.g., [Boel, Varaiya, and Wong ’75], [Grandell ’76],[Jacod ’75], [Bremaud ’81]. We consider the compensated DSPRM.This implies that we are dealing with substantially differentfiltrations.

I Once working in a martingale random fields setting, we canrecognize representation (9) (in what concerns existence of anintegrand φ) as a result in line with the Kunita-Watanabedecomposition theorem for orthogonal martingales.

Anticipating derivatives

A stochastic anticipating derivative. We consider an operatorDc : Dc → L2(Ω× X ), where Dc ⊆ L2(Ω,F ,P) defined as follows. Firsttake

Dcs,zξ(n) :=

Kn∑k=1

E[ξ

H(∆n,k)

α(∆n,k)

∣∣∣FH,α∆c

n,k

]1∆n,k

(s, z).

where FH,α∆c

n,k= FH

∆cn,k∨ Fα. Note that Dc

s,zξ(n) ∈ L2(Ω× X ).

Then consider the limit

Dcξ = limn→∞

Dcξ(n).

Hence ξ ∈ Dc whenever the limit exists in L2(Ω× X ).

We remark that for any β ∈ H0, β ∈ Dc and Dcβ = 0.

Proposition For p ≥ 1, let ξ be a p-order α-multilinear form, i.e. we haveξ = β

∏pj=1 H(∆j). Then

(13) Dcs,zξ = β

p∑i=1

1∆i (s, z)∏j 6=i

H(∆j),

and

Dcs,zξ(n) =

p∑i=1

Kn∑k=1

α(∆i ∩∆n,k)

α(∆n,k)

∏j 6=i

H(∆j)1∆n,k∩∆j =∅(k, j) 1∆n,k(s, z).

Furthermore‖Dcξ‖L2(Ω×X ) =

√p‖ξ‖L2(Ω,F,P).

Remark. Hence, Dc is the space of linear combinations of ξ =∑∞

p=0 ξpsuch that

∑∞p=0 p‖ξp‖2

L2(Ω,F,P) <∞, where ξp are p-order α-multilinearforms.

Malliavin type derivative. Basically developed in [Yablonski ’07] based onchaos expansions via iterated Ito integrals which are related to the chaosexpansions via α-multilinear forms.

Indeed one can define a Malliavin type derivative DMall on the domaindenoted D1,2 ⊂ L2(Ω).

Theorem. The operators Dc and DMall coincide, i.e. Dc = D1,2 ⊂ L2(Ω)and

Dcξ = DMallξ in L2(Ω× X )

Comment. Hence we can also interpret the operator Dc as andalternative approach to describe the Malliavin derivative which shows theanticipative dependence of the operator on the information in a muchmore structural and explicit way than the classical approach via chaosexpansions of iterated integrals.

Going back to integral representations.

Theorem. For any ξ ∈ Dc we have

E[Dc

s,zξ∣∣∣FH,α

s−

]= E

[DMall

s,z ξ∣∣∣FH,α

s−

]= DH,α

s,z ξ P× α a.e.

Corollary. For any ξ ∈ L2(Ω,F ,P) there exists a sequence ξq ∈ Dc ,q = 1, . . . such that ξq → ξ in L2(Ω,F ,P) and

DH,αξq = E[Dcξq|FH,α

]−→ DH,αξ as q →∞,

in L2(Ω× X ). Thus

ξ = E[ξ|Fα] + limq→∞

T∫0

∫Z

E[Dc

s,zξq|FH,αs−]

H(ds, dz)

with convergence in L2(Ω,F ,P).

Example: take ξq ∈∑q

p=0 Hp.

4. About BSDEs for time-changed Levy noises

The aim is to study a BSDE 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)

whereµ(∆) := B

(∆ ∩ X0

)+ H

(∆ ∩ X0

), ∆ ⊆ X

Here:

X := [0,T ]× R, X0 := [0,T ]× 0, X0 := [0,T ]× R0

.

To be more specific:

The rate λ := (λB , λH) is a two dimensionial stochastic process such thatλk , k = B,H satisfy

1. λkt ≥ 0 P-a.s. for all t ∈ [0,T ],

2. limh→0 P(∣∣λk

t+h−λkt

∣∣ ≥ ε) = 0 for all ε > 0 and almost all t ∈ [0,T,

3. E[ ∫ T

0λk

t dt]<∞,

and the time-change/intensity is:

Λ(∆) :=

T∫0

1(t,0)∈∆(t)λBt dt +

T∫0

∫R0

1(t,z)∈∆)(t, z) ν(dz)λHt dt,

Then the noises are given by:

Definition.B is a random measure on the Borel sets of X0 satisfying

A1). P(

B(∆) ≤ x∣∣∣FΛ

)= P

(B(∆) ≤ x

∣∣∣ΛB(∆))

= Φ(

x√ΛB (∆)

), x ∈ R,

∆ ⊆ X0,

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

H is a random measure on the Borel sets of X0 satisfying

A3). P(

H(∆) = k∣∣∣FΛ

)= P

(H(∆) = k

∣∣∣ΛH(∆))

= ΛH (∆)k

k! e−ΛH (∆),

k ∈ N, ∆ ⊆ X0,

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.

Connection with time-changed Levy noises.

Theorem [Serfoso 72]. Let Wt , t ∈ [0,T ] be a Brownian motion and Nt ,t ∈ [0,T ] be a centered pure jump Levy process with Levy measure ν.Assume that with both W and N are independent of Λ. Then B is aconditional Gaussian random measure as above if and only if, for any t,

Btd= WΛB

t,

and η is a conditional Poisson process if and only i,f for any t,

ηtd= NΛH

t.

Existence and uniqueness.

The study of the BSDEs is carried through in a classical fashion.

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

‖Y‖S :=r

Eˆ

sup0≤t≤T

|Yt |2˜<∞,

HB,H,Λ2 the space of FH,α-predictable stochastic processes f (t, ω), t ∈ [0,T ], ω ∈ Ω such that

Eh TZ

0

f (s, ·)2 dsi<∞.

Denote Z the space of deterministic functions φ : R→ R such thatZR0

φ(z)2ν(dz) + |φ(0)| ≤ ∞

Definition. We say that (ξ, g) are standard parameters when ξ ∈ L2`Ω,F, P´

and g is a FH,α-predictable

function g : Ω× [0,T ]× [0,∞)2 × R× L2(Z)→ R such that g satisfies

g(·, λ·, 0, 0) ∈ HB,H,Λ2 ,(14) ˛

g`t, (λB

, λH ), y1, φ

(1)´− g`t, (λB

, λH ), y2, φ

(2)´˛ ≤ Cg

“˛y1 − y2

˛+˛φ

(1)(0)− φ(2)(0)˛pλB +

vuutZR0

(φ(1) − φ(2))2(z)ν(dz)pλH”.

(15)

Theorem. Let (g, ξ) be standard parameters. Then there exists unique Y ∈ S and φ ∈ L2(Ω× X ; PB,H,α)such that

Yt = ξ +

TZt

g`s, λs , Ys , φs

´ds −

TZt

ZR

φs (z)µ(ds, dz)

= ξ +

TZt

g`s, λs , Ys , φs

´ds −

TZt

φs (0) dBs +

TZt

ZR0

φs (z) H(ds, dz)(16)

Remark. The initial point Y0 of the solution Y is in general not a real number, indeed it is stochastic. We see that

Y0 is a square integrable FΛ-measurable random variable. To be specific we have:

Y0 = Ehξ +

TZ0

g(s, λs , Ys , φs ) ds˛FΛ

T

i.

Linear BSDEs - explicit solution.

Theorem Assume we have BSDE satisfying

−dYt =hAtYt + Ct + Et (0)φt (0)

qλB

t +

ZR0

Et (z)φt (z) ν(dz)qλH

t

idt

− φt (0) dBt −ZR0

φt (z) H(dt, dz), Y (T ) = ξ

where the coefficients satisfy

1. A is a bounded stochastic process,

2. C ∈ HB,H,Λ2 ,

3. E ∈ L2(Ω× X ; PB,H,Λ),

4. There exists CE > 0 such that 0 ≤ Et (z)| < CE z for x ∈ R0 and |Et (0)| < CE for all t ∈ [0,T ].

Then there exists a solution (Y , φ) and Y has representation

Yt = EhξΓt

T +

TZt

ΓtsCs ds

˛FH,α

t

i

where

Γts = exp

n sZt

A(u)−1

2φu(0)2

1λBu 6=0qλB

u

du +

sZt

φu(0)1λu 6=0q

λBu

dBu

+

sZt

ZR0

hln`

1 + Eu(z)1λH

u 6=0qλH

u

´− Eu(z)

1λHu 6=0qλH

u

iν(dz)λH

u du

+

sZt

ZR0

ln`

1 + Eu(z)1λH

u 6=0qλH

u

´H(du, dz)

o

Comparison of BSDEs.

Theorem. Let (g1, ξ1) and (g2, ξ2) be two sets of parameters with solutions denoted by (Y (1),φ(1)and

Y (2), φ(2). Assume that

g2(t, λ, y, φ) = f“t, y, φ(0)κt (0)

pλB ,

ZR0

φ(z)κt (z) ν(dz)pλH”

where κ ∈ L2(Ω× X ; PB,H,Λ) satisfies the conditions 4. from the theorem above.Furthermore let f be a function f : Ω× [0,T ]× R× R× R→ R satisfying

|f (t, y, b, h)− f (t, y′, b′, h′)| ≤ Ch

“|y − y′| + |b − b′| + |h − h′|

”

Eh TZ

0

|f (t, 0, 0, 0)|2 dti<∞.

If ξ1 ≤ ξ2 a.s. and g1(t, λt , Y(1)t , φ

(1)t ) ≤ g2(t, λt , Y

(1)t , φ

(1)t ) (t, ω)-a.e. then

Y (1) ≤ Y (2) (ω, t) a.e.

References

The talk is based on:

G. Di Nunno and S. Sjursen (2012): On chaos representation andorthogonal polynomials for the doubly stochastic Poisson process. Eprintin Pure Maths 1, UiO. Submitted.

G. Di Nunno and S. Sjursen (2012): BSDEs driven by time-changed Levynoises. Manuscript.

References

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

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

P. Carr and L. Wu (2004): Time-changed Lvy 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.

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