Backward Stochastic Evolution EquationsBackward Stochastic Evolution Equations A Wiener chaos approach A. N. Yannacopoulos Department of Statistics Athens University of Economics and

Backward Stochastic Evolution EquationsA Wiener chaos approach

A. N. YannacopoulosDepartment of Statistics

Athens University of Economics and BusinessAthens – Greece

June, 2011

A. N. Yannacopoulos (AUEB) Crete 2011 – ACMAC SPDE Workshop June, 2011 1 / 41

Outline

1 Introduction

2 The Wiener chaos expansion

3 A modification of the Cameron-Martin basis

4 Weighted Wiener chaos spaces

5 Backward SEEs and their solutions

6 Wiener chaos and backward stochastic evolution equations

7 Conclusions


Outline

In collaboration with:

Professor N. E Frangos, AUEB

Professor I. Karatzas, Columbia University

YFK, Wiener chaos solutions for linear backward stochastic evolutionequations, SIAM Math. Analysis, 2011


Introduction

Introduction

The study of backward stochastic evolution equations (BSEEs) is animportant field of stochastic analysis.In this talk we will present an approach on how the theory of the Wienerchaos expansion can be used for the construction of solutions of backwardstochastic evolution equations.The Wiener chaos expansion reduces a stochastic evolution equation toan infinite hierarchy of deterministic evolution equations (the propagator).The solution of the stochastic evolution equation can be reconstructedfrom knowledge of the solution of this deterministic system.This allows not only the representation of known solutions in a convenientform, but also the construction of novel solutions (under weakerassumptions) the weighted Wiener chaos solutions.This permits a better understanding of the structure of solutions, the useof deterministic numerical techniques for their implementation, and theconstruction of generalized solutions to problems that may not admitsolutions under the standard theory of stochastic differential equations.


The Wiener chaos expansion


Let Y be a separable Hilbert space with inner product(⋅, ⋅)Y

andorthonormal basis {yk}k∈ℕ , and (Ω,ℱ ,ℙ, {ℱt}0≤t≤T ) a complete, filteredprobability space

Definition

A cylindrical Brownian motion, W = {Wy (t)}0≤t≤T , y∈Y is a Gaussianfamily of random variables with the following properties: For each y ∈ Y ,the stochastic process t 7→Wy (t) is adapted to the filtrationF = {ℱt}0≤t≤T ; and for every (y , z) ∈ Y 2 , (s, t) ∈ [0,T ]2 we have

E[Wy (t)] = 0 , E[Wy (t)Wz(s)] = (s ∧ t)(y , z)Y.

This cylindrical Brownian motion can be considered as a collection ofreal-valued and independent Brownian motions – in the sense that it canbe represented in terms of standard, real-valued and independent Wienerprocesses wk(⋅) := Wyk (⋅), k ∈ ℕ as Wy (t) =

∑k∈ℕ

(y , yk

)Y

wk(t) ,(t, y) ∈ [0,T ]× Y .



Let

Hn(x) := (−1)n e x2/2 dn

dxne −x

2/2 , x ∈ ℝ ,

be the Hermite polynomial of order n ∈ ℕ0

Let ℳ = {mi (⋅)}i∈ℕ be an orthonormal basis of L2([0,T ]) such thatmi (⋅) ∈ L∞([0,T ]), i = 1, 2, ⋅ ⋅ ⋅ (e.g. the Fourier basis)

Define,

�� :=1√�!

∏(i ,k)∈ℕ2

H�ki(�ik), where �ik :=

∫ T

0mi (t)dwk(t) , � ∈ J .

Here the multi-index � denotes an element of

J :={(�ki

)(i ,k)∈ℕ2 : �k

i ∈ ℕ0 , ∣�∣ :=∑i ,k

�ki <∞

}, and �! : =

∏i ,k

�ki ! .

The set J is countable and, for each � ∈ J , only finitely many �ki are

non-zero.



The {��}�∈J are orthonormal,

E[��] = ��,�

The Hilbert space consisting of all ℱT−measurable and ℙ−squareintegrable random variables L2

T (W) , called Wiener Chaos space, can beconstructed using the Cameron-Martin basis



Theorem (Wiener-Cameron-Martin)

Any X ∈ L2T (W) admits a representation in the series

X =∑�∈J

x� ��

where x� = E[X ��] deterministic.The following Parseval identity holds:

E[X 2] =∑�∈J

x2�,

Conversely if

X =∑�∈J

x� ��

for x� ∈ ℝ such that∑

�∈J x2� <∞ holds then X ∈ L2

T (W).



This construction can be generalized (e.g. Rosovskii - Lototsky Ann. Prob.2006) for square integrable random variables taking values in a generalHilbert space X, where now

X =∑�∈J

x� ��

where x� = E[X ��] ∈ X deterministic.The following Parseval identity holds:

E[∣∣X ∣∣2X] =∑�∈J∣∣x�∣∣2X,

and criterion for the convergence of the series∑�∈J

x� ��

is∑

�∈J ∣∣x�∣∣2X <∞.



Suppose now that instead of characterizing a single random variablewe want to characterize a whole family X (t), t ∈ [0,T ], a stochasticprocess.

If the process is such that X (t) for all t ∈ [0,T ], is measurable withrespect to the �−algebra ℱT = �(W (r), r ≤ T ), then we applyCameron-Martin theorem for every t and obtain the seriesrepresentation

X (t) =∑�∈J

x�(t)��, x�(t) = E[X (t) ��]



Related research

Rosovskii and Lototsky in a series of seminal works (e.g. RL, Ann.Prob. 2006) have studied the problem of whether the expansioncoefficients {u�(t)}�∈J can be obtained directly from the evolutionequation and gave a number of interesting results for the case oflinear parabolic stochastic evolution equations of the form

du = (Au + f ) dt +∑k∈ℕ

Mku + gk)dwk ,

where we consider W (t) =∑

k∈ℕ wk(t)yk , {yk}k , basis of Y.

Their finding is that the deterministic amplitudes {u�}�∈J are thesolutions of an infinite hierarchy of deterministic evolution equationsof the form

u′� = A u� + f� +∑i ,k

√aki (Mkua−(i ,k) + gk)) mi , � ∈ J



This system is lower triangular.

If the solution of this system, the propagator, satisfies∑�∈J ∣∣u�∣∣2X <∞, then the random field

∑�∈J u�(t) �� coincides

with the “traditional” square integrable solution of the stochasticevolution equation.

If the solution of the propagator satisfies∑

�∈J r2�∣∣u�∣∣2X <∞, forsome weight functions r�, then the random field

∑�∈J u�(t) ��

corresponds to new types of solutions not derivable by the standardtheory of stochastic evolution equations.

These solutions are called Wiener chaos solutions and correspond tosolutions of the stochastic evolution equation under weakerassumptions on the data of the problem.



What is to be done?

The Wiener Chaos approach is a beautiful construction leading todeep understanding of the solutions of SPDEs and to a wealth of newsolutions.

Can it be extended to other types of stochastic evolution equationsrather than forward SPDEs?

▶ Backward stochastic evolution equations


A modification of the Cameron-Martin basis


However, in most cases of interest in stochastic evolution equations weneed to make sure that for all t ∈ [0,T ], X (t) is measurable with respectto the �−algebra ℱt = �(W (r), r ≤ t), and not ℱT = �(W (r), r ≤ T ),which is larger!

The intuition behind that is that the values of the process up to time tmust be known only by the evolution of the noise up to this time and notby the future values of time!

Adapted vs. anticipative processes — Causality issues.

The Wiener-Cameron-Martin expansion always guarantees thatX (t) ∈ m −ℱT , ∀ t ∈ [0,T ] – However, this is not enough we need tomake sure that X (t) ∈ m −ℱt , ∀ t ∈ [0,T ]

▶ We need a new basis!



A candidate for this basis is {��(t)}�∈J := {E[�� ∣ ℱt ]}�∈J .

This set is adapted, but unfortunately no longer orthonormal!

Lemma

(i) The quantities ��,�(t) := E[��(t)��(t)], 0 ≤ t ≤ T satisfy thedeterministic system of backward ordinary differential equations

d��,�(t) =∑i ,j ,k

√�ki

√�kj ��−(i ,k), �−(j ,k)(t) mi (t) mj(t) dt , ��,�(T ) = ��

for (�, �) ∈ J 2. Here by �−(i , k) we denote the matrix

(�−(i , k)

)ℓj

=

{max

(�ki − 1, 0

), if i = j and k = ℓ

�ℓj , otherwise

and mi (⋅) are the elements of the othornormal basis ℳ for L2([0,T ]) chosenin the construction of the Cameron-Martin basis {��}�∈J .(ii) The quantities ��,�(t) satisfy the bound ��,�(t) ≤ 1 for all t ≤ T .



However eventhough this is new “basis” is not orthogonal, we have thefollowing:

Lemma

Suppose u(⋅) : [0,T ]× Ω→ ℝ is an F−adapted process such thatu(t) ∈ L2

t (W;X) for every t ∈ [0,T ] .Then u(t) admits the Wiener chaos expansion

u(t) =∑�∈J

u�(t)��(t) with u�(t) = E[u(t)��(t)] , ��(t) := E[��∣ℱt ] (1)

and the Parseval-type identity E[ ∣∣(u(t))∣∣2X ] =∑

�∈J(∣∣E[ u(t)��(t) ]∣∣X

)2holds.Conversely, if we define the stochastic process

u(t) =∑�∈J

u�(t) ��(t)

and the {u�}�∈J are such that∑

�∈J ∣∣u�∣∣2X <∞ then the stochastic processu(⋅) ∈ L2

t (W;X)



Furthermore,

Lemma

For every t ∈ [0,T ] , the family {��(t)}�∈J is total in L2t (W); to wit, if

E[X ��(t)] = 0 , ∀ � ∈ J holds for some ℱt−measurable random variableX ∈ L2

t (W), then X = 0 a.e.

Lemma

For every F−adapted, square-integrable stochastic process u(⋅) withWiener chaos expansion u(⋅) =

∑�∈J u�(⋅)��, the deterministic

coefficients u�(⋅) satisfy the system of equations

u�(t) =∑�∈J

u�(t)��,�(t), 0 ≤ t ≤ T for all � ∈ J . (2)



Remark

Because the {��(t)}�∈J are not necessarily linearly independent, theremay exist several expansions of the process u(⋅) , sayu(t) =

∑�∈J u�(t)��(t) with u�(t) ∕= u�(t), in addition to the

expansion of (1).In general we shall have u(t) ∕=

∑�∈J u�(t)��, unless the coefficients

u�(t) satisfy the “compatibility condition” (2) of Lemma 6.If this condition is satisfied, then u(t) admits an expansion in the basis{��} and an expansion in the basis {��(t)} with the same expansioncoefficients u�(t).


Weighted Wiener chaos spaces


Consider now the formal “Fourier” series∑�∈J

u�(t) ��, u� ∈ X, X

If the coefficients {u�(t)}�∈J are such that∑

�∈J ∣∣u�∣∣2X <∞ for allt, then this series defines for every fixed t ∈ [0,T ] a square integrable(with respect to the probability measure) random element in X, andvarying t over the interval [0,T ] we obtain an X-valued stochasticprocess.



What if the numerical series∑

�∈J ∣∣u�∣∣2X does not converge, butrather

∑�∈J r2� ∣∣u�∣∣2X <∞ for a {r�}�∈J , r� < 1?

Then we obtain generalized stochastic processes, “living” in weakerspaces than the space of square integrable random elements.

If on the other hand,∑

�∈J r2� ∣∣u�∣∣2X <∞ for a {r�}�∈J , r� > 1,then we obtain generalized stochastic processes which are betterbehaved than square integrable random elements.

▶ These two types of spaces are called weighted Wiener chaos spaces.


Backward SEEs and their solutions


Definition

A linear backward stochastic evolution equation (BSEE) for the pair ofF− progressively measurable stochastic processes (u(⋅),Z (⋅)) is anintegral equation

Λ− u(t) =

∫ T

t

(A u(s) + B Z(s) + f (s)

)ds +

∫ T

t

Z(s)dW (s) , 0 ≤ t ≤ T (3)

with u(t) ∈ V and Z (t) ∈ L2(K ; H) .

∫ t

0

Z(s)dW (s) =∑k∈ℕ

∫ t

0

Z(s; ek)dwk(s) :=∑k∈ℕ

∫ t

0

zk(s)dwk(s),

where {ek} is an orthonormal basis of the Hilbert space K , and

zk(s) := Z (s; ek) = Z (s)ek , k ∈ ℕ (4)



Definition

We say that the pair of F−progressively measurable stochastic processes(u(⋅),Z(⋅)) ∈ L2

(W, L2([0,T ]; V )

)× L2

(W, L2([0,T ];L2(K ; H))

)is a weak solution of

the backward stochastic evolution equation (3), if the equation (3) holds in V ′, i.e., iffor every v ∈ V we have almost surely:

(Λ, v

)H−(u(t), v

)H

=

∫ T

t

(⟨A u(s), v⟩+ (B Z(s), v)H + ⟨f (s), v⟩

)ds

+∑k∈ℕ

∫ T

t

(zk(s), v

)H

dwk(s) , ∀ 0 ≤ t ≤ T . (5)

Definition

Within the semigroup framework, we say that a weak solution of (3) is a strong

solution, if for every t ∈ [0,T ] we have u(t) ∈ D(A). By the density of D(A) in H, this

implies that (3) is satisfied a.s.


Wiener chaos and backward stochastic evolution equations

Wiener chaos and stochastic evolution equations

Since the Wiener chaos expansion may be used to generate anystochastic process, square integrable or not, it may also be used togenerate solutions of backward stochastic evolution equations.

The question is the following:

If the stochastic process u(⋅), taking values in X is the solution of thestochastic evolution equation

du = (A u + B Z + f ) dt + Z dW (t)

andu(t) =

∑�∈J

u�(t) �� or u(t) =∑�∈J

u�(t) ��(t)

(where at this point this series is understood as a formal object)what can we deduce concerning the amplitudes of the expansion {u�}or {u�}?



The resembles the Faedo-Galerkin expansion which is used in thenumerical or even analytical treatment of deterministic evolutionequations.

This Galerkin expansion “separates” randomness which is included inthe choice of the basis {��}�∈J , or {��(t)}�∈J from thedeterministic part of the solution which is included in the{u�(t)}�∈J ,t∈[0,T ] or the {u�(t)}�∈J ,t∈[0,T ]

The stochastic part {��}�∈J , or {��(t)}�∈J is determined only bythe Wiener process and is thus independent of the stochasticevolution equation at hand!

As we shall see shortly, the part of the expansion that depends on theparticular type of evolution equation are the deterministic coefficients{u�(t)}�∈J ,t∈[0,T ] or {u�(t)}�∈J ,t∈[0,T ]



Representation of solutions using Wiener chaos expansions

In this section assuming the existence of square integrable solutions to BSEEswe investigate their representation in Wiener chaos expansion.We need “regularity” assumptions on the data.

Assumption

The following hold:

A1: The terminal condition Λ admits the Wiener chaos expansion

Λ =∑�∈J

Λ�� (6)

with deterministic coefficients Λ� = E[Λ ��] ∈ H.

A2: The F−progressively measurable process f (⋅) ∈ L2(W, L1([0,T ]; V ′)

)admits the

Wiener chaos expansion

f (t) =∑�∈J

f�(t)�� , ∀ t ∈ [0,T ] , (7)

with deterministic coefficients f�(⋅) = E[f (⋅) ��] ∈ L1([0,T ]; V ′) .

A3 The data of the problem and its solution are sufficiently smooth in terms of Malliavinderivatives.



Proposition (Expansion I)

Suppose that

(i) the pair of F−progressively measurable processes (u(⋅),Z(⋅)) is a weak solution ofthe BSEE, with B = 0, and

(ii) Regularity assumptions hold for the “data” of the BSEE and its solution.

Then each u(t) has a Wiener chaos expansion of the form u(t) =∑�∈J u�(t)�� in

which the deterministic coefficients u�(t) = E[u(t)��] satisfy in the weak sense thesystem of deterministic evolution equations∑�∈J

Λ��,�(t)− u�(t) =∑�∈J

(∫ T

t

(Au�(s) + f�(s)

)ds

)��,�(t) , t ∈ [0,T ] (8)

for each � ∈ J whereas the process Z(⋅) is determined via (4) by the expansion, validfor every k ∈ ℕ and t ∈ [0,T ]:

zk(t) =∑�∈J

(Λ� −

∫ T

t

(Au�(s) + f�(s)

)ds

)∑j∈ℕ

mj(t)√�kj ��−(j,k)(t) . (9)



Proposition (Expansion II)

Suppose that

(i) the BSEE (3) admits a weak solution (u(⋅),Z(⋅)), and

(ii) Regularity assumptions hold.

Then u(⋅) admits the Wiener chaos representation

u(t) =∑�∈J

u�(t) ��(t)

where the u�(⋅) ∈ L2([0,T ]; V ) are non-random functions that satisfy in the weak sensethe infinite system of deterministic backward differential equations

Λ� − u�(t) =

∫ T

t

(Au�(s) + f�(s)

)ds , 0 ≤ t ≤ T (10)

for � ∈ J . Furthermore, each Z(t) has the Wiener chaos expansion

zk(t) := Z(t)ek =∑�∈J

∑i∈ℕ

u�(t)√�ki ��−(i,k)(t) mi (t) , k ∈ ℕ . (11)



Expansion II provides an easier representation for the solution than that ofExpansion I: the u�(t) have a simpler form than u�(t), and the formerexpansion guarantees by construction the adaptivity of the solution, anessential requirement for the definition of BSEEs.However, in general, some care should be taken with expansions of theform

u(t) =∑�

u�(t) ��(t)

1 It is not the case that u�(t) = E[u(t) ��], and there is no uniquenessfor the u�(t) .

2 Consequently, we have in general u(t) ∕=∑

�∈J u�(t) �� for0 ≤ t < T .

3 Furthermore, only the stochastic process∑

�∈J u�(t) ��(t) isadapted, and not

∑�∈J u�(t) �� in general.



The two expansions are related.

Proposition

A solution {u�(⋅)}�∈J of the system (8) may be obtained asu�(⋅) =

∑�∈J u�(⋅)��,�(⋅) , where the {u�(⋅)}�∈J solve the decoupled

deterministic evolution equations (10).



We now give a converse proposition, which shows how to construct a weaksolution to the BSEE (3) starting from (10), (11).

Proposition

Suppose that

(i) the functions u�(⋅) solve the infinite system (10) of deterministic backwarddifferential equations; that

(ii) the series

u(t) =∑�∈J

u�(t)��(t) 0 ≤ t ≤ T (12)

zk(t) =∑�∈J

∑i∈ℕ

u�(t)√�ki ��−(i,k)(t) mi (t) , k ∈ ℕ

converge in L2(W; L2([0,T ]; V )

)and in L2

(W; L2([0,T ];L2(K ,H))

)respectively; and that

(iii) Regularity assumptions hold for the “data” of BSEE (3) and for the stochasticprocess u(⋅) defined in (12).

Then the pair of stochastic processes(u(⋅), Z(⋅)

)defined via (12), (4) constitute a

weak solution of the BSEE (3).A. N. Yannacopoulos (AUEB) Crete 2011 – ACMAC SPDE Workshop June, 2011 30 / 41


The case where B ∕= 0

Theorem

Suppose that the standing assumptions hold, and that the solution of the BSEEadmits a weak solution

(u(⋅),Z (⋅)

), with Wiener chaos expansions∑

�∈J u�(t)��(t).Then the functions {u�(⋅)}�∈J satisfy the system of deterministic backwardequations

Λ� − u�(t) =

∫ T

t

(Au�(s) + f�(s)

)ds

+∑k∈ℕ

∑i∈ℕ

∫ T

t

√�ki + 1 mi (s) Bk u�+(i ,k)(s) ds , � ∈ J

in the weak sense, and

zk(t) =∑�∈J

u�(t)∑i∈ℕ

√�ki ��−(i ,k)(t)mi (t) , 0 ≤ t ≤ T .

We now obtain an upper triangular system.A. N. Yannacopoulos (AUEB) Crete 2011 – ACMAC SPDE Workshop June, 2011 31 / 41


Forward SEE

W0 ⊕W1 ⊕W2 ⊕ ⋅ ⋅ ⋅−→

Backward SEE

W0 ⊕W1 ⊕W2 ⊕ ⋅ ⋅ ⋅←−



Solvability of the propagator and generalized solutions

In this section we

1 Look at the solvability of propagator (the deterministic problems) foru�(t) and thus provide conditions for solvability of BSEEs usingWiener chaos

2 Investigate the existence of generalized solutions for BSEEs using thedeterministic equations for coefficients, in weighted Wiener spaces.



We shall work in terms of the expansion u(⋅) =∑

�∈J u�(⋅)��(⋅) , wherethe expansion coefficients satisfy deterministic backward equations of theform

u�(t) = Λ� −∫ T

t

(Au�(s) + f�(s) + G�(s, u)

)ds , 0 ≤ t ≤ T . (13)

Here G�(t, u), � ∈ J is a family of linear operators of the form

G�(t, u) =∑�∈J

c�,�(t)u�(t) (14)

for appropriately chosen operators c�,�(t) : V → V ′, 0 ≤ t ≤ T ;



Solvability of the deterministic problems

Assumption

Let (V ,H,V ′) be a Gel’fand triple and V × V ∋ (�, ) 7→ A(�, ) ∈ ℝ abilinear form with the following properties:

(i) there exists a constant > 0 such that

∣A(�, )∣ ≤ ∣∣�∣∣V ∣∣ ∣∣V , ∀ (�, ) ∈ V 2 .

(ii) there exist constants k0 ∈ ℝ, � > 0 such that

A(�, �) + k0∣∣�∣∣2H ≥ � ∣∣� ∣∣2V , ∀ � ∈ V .

A linear operator A : V → V ′ is defined then via the “Riesz representation”A(�, ) = ⟨A�, ⟩ .

The general equation arising from the projection of BSEEs onto the Wienerchaos basis, is

u�(�)− Λ� +

∫ �

0Au�(t) dt = −

∫ �

0f�(t) ds +

∑�∈J

∫ �

0c��(t)u�(t) dt (15)

for � ≥ 0, � ∈ J .A. N. Yannacopoulos (AUEB) Crete 2011 – ACMAC SPDE Workshop June, 2011 35 / 41


Proposition

Assume the following:

(i) there exist a double sequence{"(�, �)

}(�,�)∈J 2 of positive numbers, a weight function{

r�}�∈J , and a constant K ∈ (0,∞) such that the coupling coefficients c�,�(t)

satisfy the conditions∑�∈J

∣c�,�(t)∣"(�, �)

≤ K for all � ∈ J , t ∈ [0,T ], (16)

∑�∈J

"(�, �) ∣c�,�(t)∣r 2�r 2�≤ K for all � ∈ J , t ∈ [0,T ] ; (17)

(ii) the forcing term satisfies∑�∈J

∫ T

0r 2� ∣∣f�(s)∣∣2

V′ ds < ∞ ; and

(iii) the terminal condition has Wiener chaos expansion with coefficients Λ� such that∑�∈J r 2� ∣∣Λ�∣∣2H < ∞ .

Then the solutions of the deterministic hierarchy (15) generate Fourier coefficients{u�(⋅)

}�∈J , such that for some positive constant C we have

sup0≤t≤T

∑�∈J

r 2�∥∥u�(t)

∥∥2H≤ C

(∑�∈J

∫ T

0

r 2�∥∥f�(s)

∥∥2V′ ds +

∑�∈J

r 2�∥∥Λ�

∥∥2H

).



If the assumptions of the above Proposition hold, then the ”formal”expansion in (weighted) Wiener chaos converges and thus defines ageneralized solution for the BSEE.

The choice of r� is essential for our purposes.

The choice r� = 1, � ∈ J corresponds to weak solutions for (15) and,using the Wiener chaos representations, to weak solutions for (3).

It can be shown that for the particular case r� = 1 coincide thepropagator is well posed as long as the usual assumptions for squareintegrable solutions of BSEEs hold.

However, other choices for r� can prove useful, either for studying theregularity of weak solutions, or for the definition of even weaker typesof solutions.



Corollary (Malliavin differentiable solutions)

Assume that

(i) there exists a double sequence{"(�, �)

}(�,�)∈J 2 such that conditions (16)-(17)

hold with r� =√∣�∣ ,

(ii) the forcing term satisfies t∑�∈J

∫ T

0∣�∣∥∥f�(t)

∥∥V ′

dt < ∞ , and

(iii) the terminal condition Λ has Wiener chaos expansion with coefficients Λ� such

that∑�∈J ∣�∣

∥∥Λ�∥∥2H< ∞ .

Then there exists a constant C ∈ (0,∞) , such that the solutions of (13) satisfy

max0≤t≤T

∑�∈J

∣�∣∥∥u�(t)

∥∥2H≤ C

(∑�∈J

∫ T

0

∣�∣∥∥f�(s)

∥∥V ′

ds +∑�∈J

∣�∣∥∥Λ�

∥∥2H

)<∞ .

In particular, u(t) =∑�∈J u�(t) ��(t) ∈ D1,2(H) for all t ∈ [0,T ] .



Weighted Wiener chaos solutions

Definition

The stochastic process u(⋅) =∑

�∈J u�(⋅) ��(⋅) is called a QL2(W,H)weighted Wiener chaos solution of the BSEE (3) with weightQ = {q1, q2, ⋅ ⋅ ⋅ }, if the functions u�(⋅), � ∈ J satisfy the hierarchy ofdeterministic evolution equations (13) and

supt∈[0,T ]

∑�∈J (q�)2

∥∥u�(t)∥∥2H<∞ .

Proposition

Assume that

(i) there exists a real valued sequence Q = {q1, q2, ⋅ ⋅ ⋅ } such that conditions

(16)-(17) hold with r� = q� :=∏

i ,k q�ki

k ,

(ii) the forcing term f (⋅) satisfies the condition f (⋅) ∈ L2([0,T ],QL2(W; V′))

(iii) the final condition Λ satisfies the condition Λ ∈ QL2(W; H).

Then, BSEE (3) admits a QL2(W,H) weighted Wiener chaos solution.


Conclusions

Conclusions

We have proposed a construction of solutions of BSEEs using theWiener chaos expansion

We reduce the problem to the solution of an infinite hierarchy ofdeterministic evolution equations which play the role of the Fouriercoefficients of the solution.

Through the solvability of this hierarchy1 We provide existence results and regularity results for solutions of

BSEEs using the relevant theory for deterinistic systems2 We define and provide existence results for a new type of generalized

solutions, the weighted Wiener chaos solutions


Conclusions

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Backward Stochastic Evolution EquationsBackward Stochastic Evolution Equations A Wiener chaos approach A. N. Yannacopoulos Department of Statistics Athens University of Economics and