getdoc0fab. 338KB Jun 04 2011 12:04:05 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 13 (2008), Paper no. 54, pages 1529–1561.

Journal URL

http://www.math.washington.edu/~ejpecp/

Quadratic BSDEs with random terminal time and elliptic

PDEs in infinite dimension.

Philippe Briand

IRMAR, Université Rennes 1, 35042 Rennes Cedex, FRANCE

philippe.briand@univ-rennes1.fr

Fulvia Confortola

Dipartimento di Matematica, Politecnico di Milano

Piazza Leonardo da Vinci 32, 20133 Milano, ITALY

fulvia.confortola@polimi.it

Abstract

In this paper we study one dimensional backward stochastic differential equations (BSDEs) with random terminal time not necessarily bounded or finite when the generator F(t,Y,Z) has a quadratic growth in Z. We provide existence and uniqueness of a bounded solution of such BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained results are then applied to prove existence and uniqueness of a mild solution to elliptic partial differential equations in Hilbert spaces. Finally we show an application to a control problem.

Key words: Backward stochastic differential equations, quadratically growing driver, elliptic partial differential equation, stochastic optimal control.

AMS 2000 Subject Classification:Primary 60H10; 60H3.

1

Introduction

In this paper, we are mainly interested in finding a probabilistic representation for the solution to the following elliptic PDE

Lu(x) +F(x,u(x),∇u(x)σ) =0, x∈H, (1) whereH is a Hilbert space andL is a second order differential operator of type

Lφ(x) = 1

2Trace(σσ

∗_∇2_{φ(x)) +〈Ax,∇φ(x)〉+〈b(x),∇φ(x)〉.}

withAbeing the generator of a strongly continuous semigroup of bounded linear operators(etA)_t≥0 inH andF being a nonlinear function.

It is by now well known that this kind of Feynman-Kac formula involves Markovian backward stochastic differential equations (BSDEs for short in the remaining of the paper) with infinite hori-zon, which, roughly speaking, are equations of the following type

Y_tx = Z ∞

t

F€X_sx,Y_sx,Z_sxŠds−

Z ∞

t

Z_sxdW_s (2)

where{X_tx}t≥0stands for the mild solution to the SDE

d X_tx =AX_txd t+b€X_txŠd t+σdW_t, t≥0, X₀x =x, (3)

W being a cylindrical Wiener process with values in some Hilbert spaceΞ(see Section 2 for details). With these notations, the solutionuto the PDE (1) is given by

∀x ∈H, u(x) =Y₀x, (4)

where(Yx,Zx)is the solution to the previous BSDE. For this infinite dimensional setting, we refer to the article[14]in which the authors deal with functionsF being Lipschitz with respect toz. One of the main objective of this study is to obtain this nonlinear Feynman-Kac formula in the case where the functionF is not Lipschitz continuous with respect tozbut has a quadratic growth with respect to this variable meaning that the PDE is quadratic with respect to the gradient of the solution. In particular, in order to derive this formula in this setting, we will have to solve quadratic BSDEs with infinite horizon.

BSDEs with infinite horizon are a particular class of BSDEs with random terminal time which have been already studied in several paper. Let us recall some basic facts about these equations. Letτbe a stopping time which is not assumed to be bounded orP–a.s. finite. We are looking for a pair of processes(Yt,Zt)t≥0, progressively measurable, which satisfy,∀t≥0,∀T ≥t,

    

Y_t∧τ=Y_T∧τ+ Z T∧τ

t∧τ

F(s,Y_s,Z_s)ds−

Z T∧τ

t∧τ

Z_sdW_s

Yτ=ξon{τ <∞}

(5)

where the terminal conditionξisFτ-measurable,{Ft}t≥0 being the filtration generated byW. As

mentioned before, there exists a wide literature about the problem, mainly when the generator F

has a sublinear growth with respect toz. There are two classical assumptions on the generator F in order to solve such BSDEs, we refer to[7],[23]and[3]:

• F is Lipschitz with respect toz: |F(t,y,z)−F(t,y,z′)| ≤K|z−z′|;

• F is monotone in y:(y−y′) F(t,y,z)−F(t,y′,z)≤ −λ|y−y′|2_.

Of course, one also needs some integrability conditions on the data namely

E

–

eρτ|ξ|2+ Z τ

0

eρs|F(s, 0, 0)|2ds

™

<+∞

for someρ >K2−2λ. Under these assumptions, the BSDE (5) has a unique solution(Y,Z)which satisfies

E

–Z τ

0

eρs€|Ys|2+|Zs|2

Š

ds

™

<∞.

Thus, solving BSDEs with random terminal time requires a “structural” condition on the coefficientF

which links the constant of monotonicity and the Lipschitz constant ofF inz, that isρ >K2−2λ. In particular, ifτ= +∞andF(s, 0, 0)bounded (there is no terminal condition in this case), one needs

λ >K2/2, in order to construct a solution. Let us point out that, under this structural condition, BSDE (5) can be solved when the processY takes its values inRk withk≥1 and also in an infinite dimensional framework (see e.g. [14]).

For real-valued BSDEs, in other words when the process Y takes its values inR, Briand and Hu in [4]and, afterward Royer in[25], improve these results by removing the structural condition on the generatorF. In the real case, they require thatF(t, 0, 0)is bounded and use the Girsanov transform to prove that the equation (5) has unique solution(Y,Z)such that Y is a bounded process as soon asλ >0. The same arguments are handled by Hu and Tessitore in[17]in the case of a cylindrical Wiener process. The main idea which allows to avoid this structural condition is to get rid of the dependence of the generatorFwith respect tozwith a Girsanov transformation. To be more precise, the main point is to write the equation (5) in the following way

Yt∧τ=YT∧τ+

Z T∧τ

t∧τ

F(s,Ys, 0) +〈bs,Zs〉

ds−

Z T∧τ

t∧τ

ZsdWs,

=Y_T∧τ+

Z T∧τ

t∧τ

F(s,Y_s, 0)ds−

Z T∧τ

t∧τ

Z_sdWc_s, (6)

whereWct=Wt−

Rt

0 bsdsand the process bis given by b_s= F(s,Ys,Zs)−F(s,Ys, 0)

|Zs|2

Z_s1_|Zs|>0.

WhenF is Lipschitz with respect toz, the process bis bounded and, for eachT >0,

Et=exp

‚Z t

0

bsdWs−

1 2

Z t

0

|bs|2ds

Œ

, 0≤t≤T,

is a uniformly integrable martingale. If Pstands for the probability under which W is a Wiener process, the probability measureQ_T, whose density with respect to the restriction,P_T, ofPtoF_TW

isET, is equivalent toPT and, underQT,

¦ c

Wt

©

working underQ_T, we see that the dependence of the generator with respect tozhas been removed allowing finally to get rid of the structure condition.

As mentioned before, we are interested in the case where F has a quadratic growth with respect to

z andF is strictly monotone in y without any structure condition. We will assume more precisely

that _¯

¯_F(t,y,z)−F(t,y,z′)¯¯≤C(1+|z|+|z′|)|z−z′|,

and we will apply more or less the same approach we have just presented whenF is Lipschitz with respect to z. In this quadratic setting, the process bwill not be bounded in general. However, we will still be able to prove that{Et}0≤t≤T is a uniformly integrable martingale for each T >0. This

will result from the fact thatnR₀tb_sdW_s

o

0≤t≤T is a BMO-martingale. We refer to[18]for the theory

of BMO-martingales.

Let us also mention that M. Kobylanski in[19]considers also quadratic BSDEs with random terminal time. However, she requires that the stopping time is bounded orP-a.s finite. Her method, based on a Hopf-Cole transformation together with some sharp approximations of the generatorF, do not allow to treat the case we have in mind, precisely the case where the stopping time τ is almost surely equal to+∞.

The results on quadratic BSDEs on infinite horizon that we will obtain in Section 3 will be exploited to study existence and uniqueness of a mild solution (see Section 5 for the definition) to the PDE (1) whereF is a function strictly monotone with respect the second variable and with quadratic growth in the gradient of the solution. Existence and uniqueness of a mild solution of equation (1) in infinite dimensional spaces have been recently studied by several authors employing different techniques (see[6],[15], [11]and[20]). Following several papers (see, for instance[5],[7],[22]for finite dimensional situations, or [14], [17] for infinite dimensional case), we will use a probabilistic approach based on the nonlinear Feynman-Kac formula (4).

The main technical point here will be proving the differentiability of the bounded solution of the backward equation (2) with respect to the initial datum x of the forward equation (3). The proof is based on an a-priori bound for suitable approximations of the equations for the gradient of Y with respect to x and to this end we need to require that the coefficient σ in the forward equation is constant andA+∇b is dissipative. We use arguments based on Girsanov transform that we have previously described. We stress again that doing this way we need only the monotonicity constant ofF to be positive. The same strategy is applied by Hu and Tessitore[17]to solve the equation (1) when the generator has sublinear growth with respect to the gradient.

The mild solutions to (1), together with their probabilistic representation formula, are particularly suitable for applications to optimal control of infinite dimensional nonlinear stochastic systems. In Section 6 we consider a controlled processX solution of

¨

d X_s=AX_sdτ+b(X_s)ds+σr(X_s,u_s)ds+σdW_s,

X₀=x∈H, (7)

whereudenotes the control process, taking values in a given closed subsetU of a Banach spaceU. The control problem consists of minimizing an infinite horizon cost functional of the form

J(x,u) =E

Z ∞

0

Due to the special structure of the control term, the Hamilton-Jacobi-Bellman equation for the value function is of the form (1), provided we set, forx ∈H andz∈Ξ∗,

F(x,y,z) =inf{g(x,u) +z r(x,u):u∈ U } −λy (8) We suppose thatris a function with values inΞ∗with linear growth inuandgis a given real function with quadratic growth inu. λis any positive number. We assume that neitherU norr is bounded. In this way the HamiltonianFhas quadratic growth in the gradient of the solution and consequently the associated BSDE has quadratic growth in the variableZ. Hence the results obtained on equation (1) allow to prove that the value function of the above problem is the unique mild solution of the corresponding Hamilton-Jacobi-Bellman equation. We adapt the same procedure used in [12] in finite dimension to our infinite dimensional framework. We stress that the usual application of the Girsanov technique is not allowed (since the Novikov condition is not guaranteed) and we have to use specific arguments both to prove the fundamental relation and to solve the closed loop equation. The substantial differences, in comparison with the cited paper, consist in the fact that we work on infinite horizon and we are able to characterize the optimal control in terms of a feedback that involves the gradient of that same solution to the Hamilton-Jacobi-Bellman equation. At the end of the paper we provide a meaningful example for this control problem. We wish to mention that application to stochastic control problem is presented here mainly to illustrate the effectiveness of our results on nonlinear Kolmogorov equation.

Such type of control problems are studied by several authors (see[13],[12]). We underline that the particular structure of the control problem permits that no nondegeneracy assumptions are imposed onσ. In[13]the reader can find a model of great interest in mathematical finance, where absence of nondegeneracy assumptions reveals to be essential.

The paper is organized as follows: the next Section is devoted to notations; in Section 3 we deal with quadratic BSDEs with random terminal time; in Section 4 we study the forward backward system on infinite horizon; in Section 5 we show the result about the solution to PDE. The last Section is devoted to the application to the control problem.

Ackwoledgments. The authors would like to thank the anonymous referee for his careful reading

of this manuscript. His remarks and comments allowed to improve this paper.

2

Notations

The norm of an element x of a Banach space Ewill be denoted|x|E or simply|x|, if no confusion is

possible. If F is another Banach space, L(E,F)denotes the space of bounded linear operators from

EtoF, endowed with the usual operator norm.

The letters Ξ, H, U will always denote Hilbert spaces. Scalar product is denoted 〈·,·〉, with a subscript to specify the space, if necessary. All Hilbert spaces are assumed to be real and separable.

L₂(Ξ,U)is the space of Hilbert-Schmidt operators fromΞtoU, i.e.

L2(Ξ,U) =¦T ∈L(Ξ,U):|T|2<+∞©, with |T|2=X

n≥1|Ten| 2

U,

where {en}n≥1 is a orthonormal basis of U. L2(Ξ,U)is a Hilbert space, and the norm |T| defined

L(Ξ,R)of bounded linear operators fromΞtoR. By the Riesz isometry the dual spaceΞ∗=L(Ξ,R) can be identified withΞ.

By a cylindrical Wiener process with values in a Hilbert space Ξ, defined on a probability space (Ω,F,P), we mean a family{W_t, t≥0}of linear mappings fromΞto L2(Ω), denotedξ7→ 〈ξ,W_t〉, such that

(i) for everyξ∈Ξ,{〈ξ,Wt〉, t≥0}is a real (continuous) Wiener process;

(ii) for everyξ1,ξ2∈Ξandt≥0,E(〈ξ1,Wt〉 · 〈ξ2,Wt〉) =〈ξ1,ξ2〉Ξt.

{Ft}t≥0 will denote, the natural filtration of W, augmented with the family of P-null sets. The

filtration {Ft}t≥0 satisfies the usual conditions. All the concepts of measurability for stochastic

processes refer to this filtration. ByB(Λ)we mean the Borelσ-algebra of any topological spaceΛ. We introduce now some classes of stochastic processes with values in a Hilbert spaceK which we use in the sequel.

• Lp€Ω;L2(0,s;K)Š defined fors∈]0,+∞]andp ∈[1,∞), denotes the space of equivalence classes of progressively measurable processesψ:Ω×[0,s[→K, such that

|ψ|p_Lp_(Ω;L2_(0,s;K))=E

–Z s

0

|ψ_r|2_K d r

™p/2

<∞.

Elements of Lp(Ω;L2(0,s;K))are identified up to modification.

• Lp(Ω;C(0,s;K)), defined fors∈]0,+∞[and p∈[1,∞[, denotes the space of progressively measurable processes{ψr,r∈[0,s]}with continuous paths inK, such that the norm

|ψ|p_Lp_{(Ω;C([0,s];K))}=E

– sup

r∈[0,s]

|ψr| p K

™

is finite. Elements ofLp(Ω;C(0,s;K))are identified up to indistinguishability.

• L2_loc(K) denotes the space of equivalence classes of progressively measurable processes ψ : Ω×[0,∞)→K such that

∀t>0, E

–Z t

0

|ψr|2d r

™

<∞.

• Ifǫis a real number, M2,ǫ(K)denotes the set of{Ft}t≥0-progressively measurable processes {ψt}t≥0with values inKsuch that

E

 

Z +∞

0

e−2ǫs|ψs|2ds

 <∞.

We also recall notations and basic facts on a class of differentiable maps acting among Banach spaces, particularly suitable for our purposes (we refer the reader to[13]for details and properties).

Let now X, Z, V denote Banach spaces. We say that a mapping F : X → V belongs to the class

G1(X,V) if it is continuous, Gâteaux differentiable on X, and its Gâteaux derivative ∇F : X → L(X,V)is strongly continuous.

The last requirement is equivalent to the fact that for everyh∈X the map∇F(·)h:X →V is contin-uous. Note that∇F :X →L(X,V)is not continuous in general if L(X,V)is endowed with the norm operator topology; clearly, if this happens then F is Fréchet differentiable on X. It can be proved that if F ∈ G1(X,V)then(x,h) 7→ ∇F(x)his continuous from X×X toV; if, in addition,G is in

G1(V,Z)thenG(F)belongs toG1(X,Z)and the chain rule holds:∇(G(F))(x) =∇G(F(x))∇F(x). When F depends on additional arguments, the previous definitions and properties have obvious generalizations.

3

Quadratic BSDEs with random terminal time

In all this section, let τbe an{F_t}_t≥0 stopping time where{Ft}t≥0 is the filtration generated by

the Wiener process defined in the previous section. We use also the following notation.

Definition 3.1. A couple(ξ,F)is said to be a standard quadratic parameter if:

1. theterminal conditionξis a bounded,Fτ–measurable, real valued random variable;

2. thegenerator Fis a function defined onΩ×[0,∞)×R×Ξ∗with values inR, measurable with respect toP ⊗ B(R)⊗ B(Ξ∗) andB(R) where P stands for theσ-algebra of progressive sets and such that, for some constantC ≥0,P–a.s. and for allt≥0,

(a) (y,z)7−→F(t,y,z)is continuous;

(b) ∀y ∈R,∀z∈Ξ∗,|F(t,y,z)| ≤C€1+|y|+|z|2Š_.

Let us mention that these conditions are the usual ones for studying quadratic BSDEs.

Let(ξ,F)be a standard quadratic parameter. We want to construct an adapted solution(Y_t,Z_t)_t≥0 to the BSDE

−d Yt=1t≤τF(t,Yt,Zt)d t−ZtdWt, Yτ=ξon{τ <∞}. (9)

Let us first recall that by a solution to the equation (9) we mean a pair of progressively measurable processes(Y_t,Z_t)_t≥0 with values inR×Ξ∗such that:

1. Y is a continuous process, P–a.s., for each T > 0, t 7−→ Zt belongs to L2((0,T);Ξ∗) and

t7−→F(t,Y_t,Z_t)∈L1((0,T);R);

2. on the set{τ <∞}, we have, fort≥τ,Y_t=ξandZ_t=0; 3. for each nonnegative real T,∀t∈[0,T],

Y_t=Y_T + Z T

t

1_s≤τF(s,Y_s,Z_s)ds−

Z T

t

Z_sdW_s.

Remark3.2. In the case of a deterministic and finite stopping time, this definition is the usual one except that we define the process(Y,Z)on the whole time axis.

Since the stopping time τ is not assumed to be bounded or P–a.s. finite we will need a further assumption on the generator.

Assumption A1. There exist two constants,C ≥0 andλ >0, such that,P–a.s., for allt ≥0,

(i) for all real y,

∀z∈Ξ∗, ∀z′∈Ξ∗, ¯¯F(t,y,z)−F(t,y,z′)¯¯≤C€1+|z|+¯¯z′¯¯Š ¯¯z−z′¯¯; (ii) F is strictly monotone with respect to y: for allz∈Ξ∗,

∀y∈R, ∀y′∈R, y− y′ F(t,y,z)−F(t,y′,z)≤ −λ¯¯y− y′¯¯2.

Theorem 3.3. Let(ξ,F)be a standard quadratic parameter such that F satisfies A1.

Then, the BSDE (9) has a unique solution(Y,Z) such that Y is a bounded process and Z belongs to

L2_loc(Ξ∗). Moreover, Z∈M2,ǫ(Ξ∗)for allǫ >0.

Before proving this result, let us state a useful lemma.

Lemma 3.4. Let0≤S < T and €ξ1,F1Š, €ξ2,F2Š be two standard quadratic parameters. Let, for

i=1, 2,€Yi,ZiŠbe a solution to the BSDE

Y_ti=ξi1τ≤T+

Z T

t

1_s≤τFi(s,Y_si,Z_si)ds−

Z T

t

Z_sidW_s, (10)

such that Yi is a bounded process and Zi ∈L2((0,T)×Ω).

If A1 holds for F1, ξ1_−ξ2 _{= 0on the set {S < τ} and} ¯_¯F1_−F2¯_¯€_s,Y2

s ,Zs2

Š

≤ ρ(s) where ρ is a deterministic Borelian function then

∀t∈[0,T], ¯¯Y_t1−Y_t2¯¯≤ξ1−ξ2_∞e−λ1(S−t)+₊

Z T

t

e−λ1(s−t)_ρ(s)ds,

whereλ1>0is the constant of monotonicity of F1.

Proof. Let us start with a simple remark. Let i ∈ {1, 2}. Since (ξi_,Fi_{) is a standard quadratic}

parameter, and(Yi,Zi)a solution to (10) withYi bounded and Zi square integrable, it is by know well known (see e.g. [2]) that the martingale

n

N_ti=R₀tZ_sidW_s

o

0≤t≤T has the following property:

there exists a constantγi such that, for each stopping timeσ≤T,

E¯¯N_Ti −N_σi¯¯2 ¯¯Fσ

=E

Z T

σ

|Z_si|2ds

¯ ¯ ¯Fσ

!

≤γ_i.

In other words (we refer to N. Kamazaki[18]for the notion of BMO–martingales), {Ni

t}0≤t≤T is a

With this observation in hands, let us prove our lemma. SinceF1satisfies A1,F1is strictly monotone with respect to y. Let us denoteλ1>0 the constant of monotonicity ofF1end let us fixt ∈[0,T].

We set, fors∈[0,T],

E_s=exp ‚

−λ1

Z s

0

1_u≤τ1u>tdu

Œ

=exp −λ1(τ∧s−τ∧t)+

.

We have, from Itô–Tanaka formula applied toEs

¯ ¯_Y1

s −Y

2

s

¯ ¯, ¯

¯Y_t1−Y_t2¯¯=E_T¯¯ξ1−ξ2¯¯1_τ≤T−

Z T

t

E_ssgn€Y_s1−Y_s2Š €Z_s1−Z_s2ŠdW_s−

Z T

t

d L_s

+ Z T

t

Es1s≤τ

”

λ1

¯ ¯_Y1

s −Y

2

s

¯

¯+sgn€Y_s1−Y_s2Š €F1€s,Y_s1,Z_s1Š−F2€s,Y_s2,Z_s2ŠŠ—ds,

where Lis the local time at 0 of the semimartingaleY1−Y2 and sgn(x) =−1_x≤0+1_x>0. Now, we use the usual decomposition

F1€s,Y_s1,Z_s1Š−F2€s,Y_s2,Z_s2Š=F1€s,Y_s1,Z_s1Š−F1€s,Y_s2,Z_s1Š

+F1€s,Y_s2,Z_s1Š−F1€s,Y_s2,Z_s2Š+€F1−F2Š €s,Y_s2,Z_s2Š. By assumption, we have¯¯F1−F2¯¯€s,Y_s2,Z_s2Š≤ρ(s). Moreover, sinceF1isλ1–monotone,

sgn€Y_s1−Y_s2Š €F1€s,Y_s1,Z_s1Š−F1€s,Y_s2,Z_s1ŠŠ≤ −λ1

¯ ¯_Y1

s −Y

2

s

¯ ¯_. Thus we get,Lbeing nondecreasing,

¯

¯Y_t1−Y_t2¯¯≤E_T¯¯ξ1−ξ2¯¯1τ≤T −

Z T

t

E_ssgn€Y_s1−Y_s2Š €Z_s1−Z_s2ŠdW_s+ Z T

t

E_s1_s≤τρ(s)ds

+ Z T

t

Es1s≤τsgn

€

Y_s1−Y_s2Š €F1€s,Y_s2,Z_s1Š−F1€s,Y_s2,Z_s2ŠŠds.

To go further, let us remark, fors∈[t,T], E_s1_s≤τ=e−λ1(τ∧s−τ∧t)₁

s≤τ ≤e−λ1(s−t). Moreover, since

ξ1−ξ2=0 on the set{S< τ≤T}, we have

E_T¯¯ξ1−ξ2¯¯1_τ≤T =e−λ1(T∧τ−t∧τ)+¯¯_ξ1_−ξ2¯¯₁

S<τ≤T ≤e−λ1(S−t)+kξ1−ξ2k∞, from which we deduce the following inequality

¯ ¯_Y1

t −Y

2

t

¯

¯_≤e−λ1(S−t)+_kξ1_−ξ2_k ∞+

Z T

t

e−λ1(s−t)_ρ(s)ds−

Z T

t

Essgn

€

Y_s1−Y_s2Š €Z_s1−Z_s2ŠdWs

+ Z T

t

E_s1_s≤τsgn€Y_s1−Y_s2Š €F1€s,Y_s2,Z_s1Š−F1€s,Y_s2,Z_s2ŠŠds.

To conclude, the proof of this lemma, le us define the process{bs}0≤s≤T with values inΞ∗, by setting

bs=1s≤τ

F1€s,Y_s2,Z_s1Š−F1€s,Y_s2,Z_s2Š

¯ ¯Z1

s −Zs2

¯ ¯2

€

Z_s1−Z_s2Š1_|Z1

We can rewrite the previous inequality in the following way ¯

¯_Y1

t −Y

2

t

¯

¯_≤e−λ1(S−t)+_kξ1_−ξ2_k ∞+

Z T

t

e−λ1(s−t)_ρ(s)ds

−

Z T

t

E_ssgn€Y_s1−Y_s2Š €Z_s1−Z_s2ŠdW_s+ Z T

t

E_ssgn€Y_s1−Y_s2Š ¬b_s,Z_s1−Z_s2¶ds.

Let us observe that, since F1 satisfies A1.1, we have |b_s| ≤ C€1+¯¯Z_s1¯¯+¯¯Z_s2¯¯Š. Since we know that the stochastic integral (as process on[0,T]) of Z1 and Z2 are BMO–martingales we deduce that nRt

0 bsdWs

o

0≤t≤T is also a BMO–martingale. As a byproduct, see [18, Theorem 2.3], the

exponential martingale,

E_t=exp ‚Z t

0

b_sdW_s−1

2 Z t

0

|b_s|2ds

Œ

, 0≤t ≤T,

is a uniformly integrable martingale. Let us consider the probability measureQ_T on(Ω,FT)whose

density with respect toP_|FT is given byET. ThenQT andP|FT are equivalent on(Ω,FT), and under Q_T, by Girsanov theorem, the processncWt=Wt−

Rt

0 bsds

o

0≤t≤T is a Wiener process.

To conclude, let us write the last inequality in the following way ¯

¯Y_t1−Y_t2¯¯≤e−λ1(S−t)+_kξ1_−ξ2_k ∞+

Z T

t

e−λ1(s−t)_ρ(s)ds−

Z T

t

E_ssgn€Y_s1−Y_s2Š €Z_s1−Z_s2ŠdWc_s;

taking the conditional expectation underQ_T with respect toFt, we obtain the result of the lemma.

Now we can prove the main result of this section, concerning the existence and uniqueness of solutions of BSDE (9).

Proof of Theorem 3.3.

Existence.We adopt the same strategy as in[4]and[25], with some significant modifications.

Let us denote byγa positive constant such that

kξk_∞≤γ, |F(t,y,z)| ≤γ(1+|y|+|z|), |F(t,y,z)−F(t,y,z′)| ≤γ(1+|z|+|z′|)|z−z′|, (11) and byλ >0 the monotonicity constant of F.

Fore each integern, let us denote(Yn,Zn)the unique solution to the BSDE

Y_tn=ξ1τ≤n+

Z n

t

1s≤τF(s,Ysn,Z n s)ds−

Z n

t

Z_sndWs, 0≤t≤n. (12)

We know from results of [19] (these results can be easily generalized to the case of cylindrical Wiener process) that, (ξ,F) being a standard quadratic parameter, the BSDE (12) has a unique bounded solution under A1. Moreover we haveY_tn=Y_tn_∧τ,Z_tn1t>τ=0, see e.g.[25].

We define,(Yn,Zn)on the whole time axis by setting,

∀t>n, Y_tn=Y_nn=ξ1τ≤n, Ztn=0.

First of all we prove, thanks to the assumption of monotonicity A1.2, that Yn is bounded by a constant independent ofn. Let us apply Lemma 3.4, withS=0,T =n,F1=F, F2=0,ξ1_=ξand

ξ2=0. We get, for allt∈[0,n],

|Y_tn| ≤ kξk_∞+γ

Z n

t

e−λ(s−t)ds≤γ

1+ 1

λ

. (13)

In all the remaining of the proof, we will denote C(γ,λ) a constant depending on γandλ which may change from line to line.

Moreover we can show that, for eachε >0,

sup

n≥1

E

–Z ∞

0

e−2εs¯¯Z_sn¯¯2ds

™

<∞. (14)

To obtain this estimate we consider the functionϕ(x) =€e2γx −2γx−1Š/(2γ)2, where γ >0 is the constant defined in (11) which has the following properties: forx ≥0,

ϕ′(x)≥0, ϕ′′(x)−2γϕ′(x) =1.

The functionϕ(|x|)isC2 and the estimate follows directly from the computation of the Itô differ-ential ofe−2εtϕ(|Y_tn|).

Now we study the convergence of the sequence(Yn)_n≥0. By construction we have, for n<m,

Y_tm=ξ1_τ≤m+ Z m

t

1_s≤τF€s,Y_sm,Z_smŠds−

Z m

t

Z_smdWs, 0≤t≤m,

Y_tn=ξ1τ≤n+

Z m

t

1_s≤τFb

€

s,Y_sn,Z_snŠds−

Z m

t

Z_sndW_s, 0≤t≤m,

where Fb(s,y,z) =1_s<nF(s,y,z). Let us apply Lemma 3.4 with T =m,(ξ1,F1) = (ξ,F),(ξ2,F2) = (ξ1τ≤n,Fb). We haveξ−ξ1τ≤n=ξ1τ>n, and

¯

¯_F−Fb¯¯(s,Y_sn,Z_sn) =1_s>n¯¯F(s,Y_sn,Z_sn)¯¯=1_s>n¯¯F(s,ξ1_τ≤n, 0)¯¯≤C(γ,λ)1_s>n. ChoosingS=n, we get, for t∈[0,m],

¯ ¯_Ym

t −Y n t

¯

¯_≤C(γ,λ) ‚

e−λ(n−t)+₊

Z m

t

e−λ(s−t)1s>nds

Œ

≤C(γ,λ)e−λ(n−t)+_.

Since both processesYnandYmare bounded by a constant depending only onγandλ, the previous inequality holds for all nonnegative realt, namely

We deduce immediatly from the previous estimate that the sequence(Yn)_n≥0 converges uniformly on compacts in probability (ucp for short) since, for anya≥0, we have

sup

0≤t≤a

¯

¯Y_tm−Y_tn¯¯≤C(γ,λ)e−λ(n−a),

as soon asa≤n≤m. LetY be the limit of(Yn)_n≥0. Since, for eachn,Ynis continuous and bounded byγ(1+1/λ)the same is true forY, and sendingmto infinity in (15), we get

∀t≥0, ¯¯Y_t−Y_tn¯¯≤C(γ,λ)e−λ(n−t)+_.

It follows that the convergence of(Yn)_n≥0 to Y holds also in M2,ǫ(R) for all ǫ > 0. Indeed, it is enough to prove this convergence for 0< ǫ < λand in this case we have

E

–Z ∞

0

e−2ǫs¯¯Yt−Ytn

¯ ¯2

ds

™

≤C(γ,λ) Z ∞

0

e−2ǫse−2λ(n−s)+_ds

=C(γ,λ)

₁

2(λ−ǫ) €

e−2ǫn−e−2λnŠ+ 1 2ǫe

−2ǫn

.

Let us show that the sequence(Zn)n≥0 is a Cauchy sequence in the space M2,ǫ(Ξ∗), for allǫ >0. Let

ǫ >0, andm>nbe two integers. Applying Ito’s formula to the processe−2ǫt¯¯Y_tm−Y_tn¯¯2we get ¯

¯Y₀m−Y₀n¯¯2+ Z m

0

e−2ǫs¯¯Z_sm−Z_sn¯¯2 ds (16)

= e−2ǫm|ξ|21_n<τ≤m−

Z m

0

2e−2ǫs€Y_sm−Y_snŠ €Z_sm−Z_snŠdW_s

+2 Z m

0 e−2ǫs

h

ǫ¯¯Y_sm−Y_sn¯¯2+€Y_sm−Y_snŠ1_s≤τ€F(s,Y_sm,Z_sm)−Fb(s,Y_sn,Z_sn)Šids. Since Yn andYm are bounded by C(γ,λ), we have in view of the growth assumption onF, for a constantDdepending onγ,λandǫ(and changing from line to line if necessary),

ǫ¯¯Y_sm−Y_sn¯¯2+€Y_sm−Y_snŠ1_s≤τ€F(s,Y_sm,Z_sm)−bF(s,Y_sn,Z_sn)Š

≤D¯¯Y_sm−Y_sn¯¯

1+¯¯Z_sm−Z_sn¯¯2

. Coming back to (16) and taking the expectation, we obtain the inequality, since Z_sm = Z_sn = 0 for

s>mandξis bounded byγ, E

–Z ∞

0

e−2ǫs¯¯Z_sm−Z_sn¯¯2 ds

™

≤γe−2ǫm+DE

–Z ∞

0

e−2ǫs¯¯Y_sm−Y_sn¯¯1+¯¯Z_sm−Z_sn¯¯2ds

™

≤γe−2ǫm+DE

–Z ∞

0

e−2ǫse−λ(n−s)+

1+¯¯Z_sm−Z_sn¯¯2

ds

™ , where we have used (15) to get the last upper bound. We have, finally

E

 

Z n/2

0

e−2ǫse−λ(n−s)+

1+¯¯Z_sm−Z_sn¯¯2

ds



≤e−λn/2E

 

Z n/2

0

e−2ǫs

1+¯¯Z_sm−Z_sn¯¯2

ds  , E   Z ∞

n/2

e−2ǫse−λ(n−s)+

1+¯¯Z_sm−Z_sn¯¯2ds



≤e−ǫn/2E

 

Z ∞

n/2

e−ǫs1+¯¯Z_sm−Z_sn¯¯2ds

 ,

from which we get the result since we have already shown that the sequence(Zn)_n≥0is bounded in M2,ǫ(Ξ∗)for eachǫ >0. We callZthe limit of(Zn)_n≥0.

It remains to show that the process(Y,Z)satisfies the BSDE (9). We already know thatY is contin-uous and bounded andZ belongs to M2,ǫ(Ξ∗). Let us fix 0≤t ≤T. By definition, for each n≥T, we have

Y_tn=Y_Tn+ Z T

t

1s≤τF

€

s,Y_sn,Z_snŠds−

Z T

t

Z_sndWs. (17)

The sequence (Yn)_n≥0 converges to Y ucp and is bounded by γ(1+1/λ) uniformly in n. Thus sup_0≤t≤T¯¯Y_tn−Y_t¯¯converges to 0 in L1. Moreover, from Doob’s inequality, we get

E

   sup

0≤t≤T

¯ ¯ ¯ ¯ ¯ Z t 0 €

Z_sn−Z_sŠdW_s

¯ ¯ ¯ ¯ ¯ 2 

≤4e2λTE

–Z ∞

0

e−2λs¯¯Z_sn−Z_s¯¯2ds

™

→0.

Finally, let us notice thatRT

t 1s≤τF(s,Y n s ,Z

n

s)dsconverges to

RT

t 1s≤τF(s,Ys,Zs)dsin L

1_{. Indeed,}

E   ¯ ¯ ¯ ¯ ¯

Z T∧τ

t∧τ

F(s,Y_sn,Z_sn)ds−

Z T∧τ

t∧τ

F(s,Y_s,Z_s)ds

¯ ¯ ¯ ¯ ¯  ≤E

 

Z T

0

¯

¯F(s,Y_sn,Z_sn)−F(s,Y_s,Z_s)¯¯ds

 ,

and, by the growth assumption on F, the map (Y,Z) → F(·,Y,Z) is continuous from the space

L1(Ω;L1([0,T];R))×L2(Ω;L2([0,T];Ξ∗)to L1(Ω;L1([0,T];R)), by classical result on continuity of evaluation operators, see e.g. [1]. Hence, passing to the limit in the equation (17), we obtain forall tand all T such that 0≤t≤T

Y_t=Y_T+ Z T

t

1_s≤τF(s,Y_s,Z_s)ds−

Z T

t

Z_sdW_s.

So to conclude the proof, it only remains to check that, on the set{τ <+∞}, we have Yτ=ξ. Let

us fixa>0. For each n≥a, we have in view of (15), ¯

¯Y_a−ξ1τ≤n

¯

¯=¯¯Y_a−Y_nn¯¯≤¯¯Y_a−Y_an¯¯+¯¯Y_an−Y_nn¯¯≤C(γ,λ)e−λ(n−a)+¯¯Y_an−Y_nn¯¯.

Let us recall that, for each t, Y_tn = Y_tn_∧τ and Y_t = Y_t∧τ. Hence, on the event {τ ≤ a}, we have,

since n ≥ a, Y_an = Y_nn = Y_τn and Y_a = Y_τ. Thus, we deduce from the previous inequality, that

|Yτ−ξ| ≤C(γ,λ)e−λ(n−a)on the set{τ≤a}. It follows that Yτ=ξP-a.s. on the set{τ <∞}, and

the process(Y,Z)is solution for BSDE (9).

Uniqueness. Suppose that (Y1,Z1)and(Y2,Z2)are both solutions of the BSDE (9) such that Y1

and Y2 are continuous and bounded and Z1 and Z2 belong to L2_loc(Ξ∗). It follows directly from Lemma 3.4 that,P–a.s.,

∀t≥0, Y_t1=Y_t2.

Applying Ito’s formula to¯¯Y_t1−Y_t2¯¯2, we have thatdP⊗d t–a.e. Z_t1=Z_t2. Let us finish this section by the following remark.

Remark3.5. Let{(Y_t,Z_t)}0≤t≤T be a solution, withY bounded andZsquare integrable, to the linear

BSDE

Y_t=ξ+ Z T

t

ψs+asYs+〈bs,Zs〉

ds−

Z T

t

Z_sdW_s, 0≤t≤T,

where the processesψ,aandbare progressively measurable with values inR,RandΞ∗respectively. Let us assume thatξis bounded, for someγ≥0,|ψs| ≤γandas≤ −λforλ >0. Then, arguing as

in the proof of Lemma 3.4, when

Et=exp

‚Z t

0

bsdWs−

1 2|bs|

2_ds

Œ

, 0≤t ≤T,

is a uniformly integrable martingale, we have,P–a.s.,

∀t∈[0,T], |Yt| ≤



kξk_∞+ γ

λ

‹

e−λ(T−t).

4

The forward-backward system on infinite horizon

In this Section we use the previous result to study a forward-backward system on infinite horizon, when the backward equation has quadratic generator.

We consider the Itô stochastic equation for an unknown process{Xs,s≥0}with values in a Hilbert

spaceH:

Xs=esAx+

Z s

0

e(s−r)Ab(Xr)d r+

Z s

0

e(s−r)AσdWr, s≥0. (18)

Our assumptions will be the following:

Assumption A2. (i) The operatorAis the generator of a strongly continuous semigroupetA,t ≥0,

in a Hilbert spaceH. We denote bymandatwo constants such that|etA_{| ≤me}at _{fort ≥0.}

(ii)b:H→H satisfies, for some constant L>0,

|b(x)−b(y)| ≤L|x− y|, x,y ∈H. (iii)σbelongs toL(Ξ,H)such thatetAσ∈L₂(Ξ,H)for everyt>0, and

|etAσ|L2(Ξ,H)≤L t

−γ_eat_,

for some constants L>0 andγ∈[0, 1/2). (iv) b(·)∈ G1(H,H).

(v) OperatorsA+b_x(x)are dissipative:〈Ay,y〉+〈bx(x)y,y〉 ≤0 for allx ∈H and y∈D(A).

Remark4.1. This kind of requests is usual if we wish to study the problem of the regular dependence on the data in a forward-backward system in the degenerate case on infinite horizon (compare with [17]).

We note that the assumptions (i)-(iii) are the classical assumptions to prove existence and unique-ness of the solution of equation (18) (see[9], Theorem 5.3.1, for the theory and §11.2, or[13],[14]

for some typical examples). In general the coefficientσdepends on the processX. We need thatσ

is constant and moreover we have to assume assumption (v) to obtain the following estimate P-a.s. |∇xXtxh| ≤K|h|, ∀t>0.

We stress that the previous inequality is crucial in order to show the regular dependence with respect to x of the processY in the forward-backward system (Theorem 4.6 below). Assumption (iv) is clearly natural to have differentiable dependence onx.

We start by recalling a well known result on solvability of equation (18) on a bounded interval, see e.g.[13].

Proposition 4.2. Under the assumption A2, for every p ∈ [2,∞) and T > 0there exists a unique

process Xx ∈ Lp(Ω;C(0,T;H))solution of (18). Moreover, for all fixed T > 0, the map x → Xx is continuous from H to Lp(Ω;C(0,T;H))and

E

– sup

r∈[0,T]

|X_rx|p

™

≤C(1+|x|)p,

for some constant C depending only on p,γ,T,L,a and m.

We need to state a regularity result on the process X. The proof of the following lemma can be found in[17]. In the sequelXx denotes the unique mild solution to (18) starting fromX0=x.

Lemma 4.3. Under the assumption A2, the map x → Xx is Gâteaux differentiable and belongs to

G(H,Lp(Ω,C(0,T;H)). Moreover denoting by ∇_xXx the partial Gâteaux derivative, then for every direction h∈H, the directional derivative process∇xXxh,t∈R, solves,P−a.s., the equation

∇xXtxh=e tA_h+

Z t

0

esA∇xb(Xsx)∇xXsxh ds, t∈R

+_.

Finally,P-a.s.,|∇xXtxh| ≤K|h|, for all t>0.

The associated BSDE is:

Y_tx =Y_Tx+ Z T

t

F(X_sx,Y_sx,Z_sx)ds−

Z T

t

Z_sxdWs, 0≤t≤T <∞. (19)

HereY is real valued andZ takes values inΞ∗, F:H×R×Ξ∗→Ris a given measurable function. The notationYx and Zx stress the dependence of the processes Y and Z solution to the backward equation by the starting pointx in the forward equation.

We assume the following onF:

Assumption A3. There existC ≥0 andα∈(0, 1)such that

1. |F(x,y,z)| ≤C€1+|y|+|z|2Š; 2. F(·,·,·)isG1,1,1(H×R×Ξ∗;R);

3. ¯¯∇xF(x,y,z)

¯ ¯_≤C; 4. ¯¯∇zF(x,y,z)

¯

¯_≤C(1+|z|); 5. ¯¯∇yF(x,y,z)

¯

¯_≤C(1+|z|)2α;

6. F is monotone in y with constant of monotonicityλ >0 in the following sense:

∀x ∈H,y,y′∈R,z∈Ξ∗, (y−y′)(F(x,y,z)−F(x,y′,z))≤ −λ|y−y′|2.

Remark 4.4. In comparison with the assumptions of the previous section, we add mainly the dif-ferentiability of F A3.2. We use again an approximation procedure in order to prove the regular dependence on the parameter x of the solution to the BSDE (19). Hence to use known results on differentiability for BSDEs with quadratic generator on finite time interval (see[2]) we need A3.5. Finally we use A3.3 to obtain a uniform estimate on∇xY0x (see Theorem 4.6 below).

Applying Theorem 3.3, we obtain:

Proposition 4.5. Let us suppose that Assumptions A2 and A3 hold. Then we have:

(i) For any x ∈ H, there exists a solution (Yx,Zx) to the BSDE (19) such that Yx is a continuous process bounded by C/λ, and Z∈M2,ǫ(Ξ∗)for eachǫ >0. The solution is unique in the class of processes(Y,Z)such that Y is continuous and bounded, and Z belongs toL2_loc(Ξ∗).

(ii) For all T > 0 and p ≥ 1, the map x → (Yx¯¯_[0,T],Zx¯¯_[0,T]) is continuous from H to the space Lp(Ω;C(0,T;R))×Lp(Ω;L2(0,T;Ξ∗)).

Proof. Statement (i) is an immediate consequences of Theorem 3.3. Let us prove (ii). Denoting by (Yn,x,Zn,x)the unique solution of the following BSDE (with finite horizon):

Y_tn,x = Z n

t

F(X_sx,Y_sn,x,Z_sn,x)ds−

Z n

t

Z_sn,xdWs, (20)

then, from Theorem 3.3 again,|Y_tn,x| ≤ C_λ and the following convergence rate holds:

|Y_tn,x−Y_tx| ≤ C

λexp{−λ(n−t)}.

Now, if x′_i →x asi→+∞then

|Yx′i

T −Y x

T| ≤ |Y x′_i T −Y

n,x′_i T |+|Y

n,x T −Y

x T|+|Y

n,x′_i T −Y

n,x T |

≤ 2C

λexp{−λ(n−T)}+|Y

n,x′_i T −Y

n,x T |.

Moreover for fixedn, asi→ ∞, Yn,x

′ i

T →Y n,x T in L

p_(Ω,F

T,P;R)for allp>1, by Proposition 4.2 in

[2]ThusYx

′ i

T →YTx inLp(Ω,FT,P;R).

Now we can notice that(Yx¯¯_[0,T],Zx¯¯_[0,T])is the unique solution of the following BSDE (with finite horizon):

Y_tx =Y_Tx+ Z T

t

F(X_sx,Y_sx,Z_sx)ds−

Z T

t

and the same holds for(Yx′i¯¯

[0,T],Z

x_i′¯_¯

[0,T]). By similar argument as in[2]we have

E

– sup

t∈[0,T]

|Y_tx −Yx

′ i

t |p

™1∧1/p

+E    Z T 0

|Z_sx−Zx

′ i

s |ds

!p/2

 

1∧1/p

≤CE

¯ ¯

¯Y_Tx−Yx

′ i T ¯ ¯ ¯

p+1p+11

+E    Z T 0 ¯ ¯

¯F(s,X_sx,Y_sx,Z_sx)−F(s,Xx

′ i

s ,Y x_i′ s ,Z

x′_i s )

¯ ¯ ¯ds

!p+1

 

1

p+1

and we can conclude that (Yx′i¯¯

[0,T],Z

x′ i¯¯

[0,T]) −→ (Y

x¯_¯

[0,T],Z

x¯_¯

[0,T]) in L

p_{(Ω;C(0,T;R))×}

Lp(Ω;L2(0,T;Ξ∗)).

We need to study the regularity ofYx. More precisely, we would like to show that Y₀x belongs to

G1(H,R).

We are now in position to prove the main result of this section.

Theorem 4.6. Let A2 and A3 hold. The map x→Y₀x belongs toG1_{(H,R). Moreover|Y}x

0|+|∇xY0x| ≤ c, for a suitable constant c.

Proof. Fix n ≥ 1, let us consider the solution (Yn,x,Zn,x) of (20). Then, see [2], Proposi-tion 4.2, the map x → (Yn,x(·),Zn,x(·))is Gâteaux differentiable from H to Lp(Ω, C(0,T;R))× Lp(Ω;L2(0,T;Ξ∗)), ∀p ∈(1,∞). Denoting by(∇xYn,xh,∇xZn,xh)the partial Gâteaux derivatives

with respect to x in the direction h∈H, the processes {∇xYtn,xh,∇xZtn,xh,t ∈[0,n]} solves the

equation,P−a.s.,

∇xY n,x t h =

Z n

t

∇xF(Xsx,Y n,x s ,Z

n,x

s )∇xXsn,xh ds

+ Z n

t

∇yF(Xsx,Y n,x s ,Z

n,x

s )∇xYsn,xh ds (21)

+ Z n

t

∇_zF(X_sx,Y_sn,x,Z_sn,x)∇_xZ_sn,xh ds−

Z n

t

∇_xZ_sn,xh dW_s.

We note that we can write the generator of the previous equation as

φ_sn(u,v) =ψn_s +a_snu+<b_sn,v>

whereψandaare real processes defined respectively by

ψn_s =∇xF(Xsx,Y n,x s ,Z

n,x

s )∇xXsn,xh, a n

s =∇yF(Xsx,Y n,x s ,Z

n,x s )

andbnis given by

bn_s =∇zF(Xsx,Y n,x s ,Z

n,x s ).

bnbelongs to the space L(Ξ∗,R)and by Riesz isometry can be identified with an element ofΞ∗. By Assumption A3 and Lemma 4.3, we have that for allx,h∈Hthe following holdsP-a.s. for alln∈N and alls∈[0,n]:

|ψn_s|= ¯ ¯

¯∇xF(Xsx,Y n,x s ,Z

n,x

s )∇xXsxh

¯ ¯

¯≤C|h|,

a_sn=∇yF(Xsx,Y n,x s ,Z

n,x

s )≤ −λ≤0, |b n s|=

¯ ¯

¯∇zF(Xsx,Y n,x s ,Z

n,x s )

¯ ¯

¯≤C(1+|Z_sn,x|).

As mentionned before,R₀·Z_sn,xdW_sis a BMO–martingale. Hence,

E_t=exp ‚Z t

0

b_sdW_s−1

2 Z t

0

|b_s|2ds

Œ

, 0≤t≤n, R·

0b

n

sdWs is also a is a uniformly integrable martingale. By Remark 3.5, we obtain

sup

t∈[0,n]

|∇xY n,x

t | ≤C|h|, P−a.s.

We recall that (see (14))

sup

n≥1

E( Z ∞

0

e−εs|Z_sn,x|2ds)<∞. (22)

Hence, applying Itô’s formula toe−2λt|∇xYtn,xh|2and arguing as in the proof of Theorem 3.3, thanks

to the (22), we get: E

Z ∞

0

e−2λt(|∇xYtn,xh|2+|∇xZtn,xh|2)d t≤C1|h|2.

Fix x,h∈H, there exists a subsequence of {(∇_xYn,xh,∇_xZn,xh,∇_xY₀n,xh) :n∈N}which we still denote by itself, such that(∇xYn,xh,∇xZn,xh)converges weakly to(U1(x,h),V1(x,h))in M2,λ(Ξ∗)

and∇xY0n,xhconverges toξ(x,h)∈R.

Now we write the equation (21) as follows: t∈[0,n]

∇xYtn,xh = ∇xY0n,xh−

Z t

0

∇xF(Xsx,Y n,x s ,Z

n,x

s )∇xXsxhds

−

Z t

0

∇yF(Xsx,Y n,x s ,Z

n,x

s )∇xYsn,xhds (23)

−

Z t

0

∇zF(Xsx,Y n,x s ,Z

n,x

s )∇xZsn,xhds+

Z t

0

∇xZsn,xhdWs

and we define an other processU_t2(x,h)by

U_t2(x,h) = ξ(x,h)−

Z t

0

∇xF(Xsx,Y x s ,Z

x

s)∇xXsxhds

−

Z t

0

∇yF(Xsx,Y x s ,Z

x s)U

1

s(x,h)ds (24)

−

Z t

0

∇zF(Xsx,Y x s ,Z

x s )V

1

s (x,h)ds+

Z t

0

where(Yx,Zx)is the unique bounded solution to the backward equation (19), see Proposition 4.5. Passing to the limit in the equation (23) it is easy to show that∇xYtn,xhconverges toUt2(x,h)weakly

in L1(Ω)for allt>0.

ThusU_t2(x,h) =U1_t(x,h),P-a.s. for a.e. t∈R+ and|U2_t(x,h)| ≤C|h|. Now consider the following equation on infinite horizon

U_t(x,h) = U₀(x,h)−

Z t

0

∇xF(Xsx,Y x s ,Z

x

s)∇xXsxhds

−

Z t

0

∇yF(Xsx,Y x s ,Z

x

s )Us(x,h)ds (25)

−

Z t

0

∇zF(Xsx,Y x s ,Z

x

s )Vs(x,h)ds+

Z t

0

V_s(x,h)dW_s.

We claim that this equation has a solution.

For eachn∈Nconsider the finite horizon BSDE (with final condition equal to zero):

U_tn(x,h) = Z n

t

∇xF(Xsx,Y x s ,Z

x

s )∇xXsxhds

+ Z n

t

∇yF(Xsx,Y x s ,Z

x s )U

n

s(x,h)ds

+ Z n

t

∇zF(Xsx,Y x s ,Z

x s )V

n

s (x,h)ds−

Z n

t

V_sn(x,h)dW_s,

By the result in [2] we know that this equation has a unique solution (Un(·,x,h),Vn(·,x,h)) ∈

Lp(Ω;C(0,n;R))×Lp(Ω;L2(0,n;Ξ∗)). The generator of this equation can be rewritten as

φt(u,v) =ψt+atu+btv

where ψt = ∇xF(Xtx,Ytx,Ztx)∇xXtx and |ψt| ≤ C|h|, at = ∇yF(Xs,Ys,Zs) ≤ −λ, bt =

∇_zF(Xs,Ys,Zs) and |b_t| ≤ C(1+|Zx

t|). On the interval [0,n] the process

R·

0Z

x

s dW s is a

BMO-martingale. Hence, from Remark 3.5 it follows thatP-a.s.∀n∈N,∀t∈[0,n]|U_tn| ≤ C_λ|h|and as in the proof of existence in the Theorem 3.3, we can conclude that

1. for each t ≥0 U_tn(x,h) is a Cauchy sequence in L∞(Ω)which converges to a processU and P-a.s.,∀t∈[0,n]

|U_tn(x,h)−Ut(x,h)| ≤

C

λ|h|e

−λ(n−t)_{; (26)}

2. V_·n(x,h)is a Cauchy sequence in L2_loc(Ξ∗);

3. The limit processes(U_·(x,h),V_·(x,h)satisfy the BSDE (25). Moreover still from Remark 3.5 we get that the solution is unique.

Coming back to equation (24), we have that(U2(x,h),V1(x,h))is solution inR+ of the equation (25).

E[〈W_t′,ξ′〉_Ξ′〈W ′

s,η

′

〉_Ξ′] =〈J J∗ξ ′

,η′〉_Ξ′(t∧s) for all ξ ′

,η′ ∈Ξ′, t,s≥ 0. This is called a J J∗ -Wiener processes inΞ′ in [10], to which we refer the reader for preliminary material on Wiener processes on Hilbert spaces. Let us denote byG◦ theP◦-completion ofB and byN the family of setsA∈ G◦withP◦(A) =0. LetBt=σ{W

′

s :s∈[0,t]}andFt◦=σ(Bt,N),t ≥0, where as usual σ(·) denotes theσ-algebra inΩgenerated by the indicated collection of sets or random variables. Thus(F_t◦)_t≥0is the Brownian filtration ofW′.

TheΞ-valued cylindrical Wiener process {W_tξ : t ≥ 0,ξ∈Ξ}can now be defined as follows. For

ξ in the image of J∗J we take η such that ξ = J∗Jη and define W_sξ = 〈W_s′,Jη〉_Ξ′. Then we notice that E|W_tξ|2 _{= t|Jη|}2

Ξ′ = t|ξ| 2

Ξ, which shows that the mapping ξ→ W

ξ

s , defined for ξ∈ J∗J(Ξ)⊂Ξwith values in L2(Ω,F,P◦), is an isometry for the norms ofΞand L2(Ω,F,P◦). Consequently, noting thatJ∗J(Ξ)is dense inΞ, it extends to an isometryξ→L2(ω,F,P◦), still denotedξ→W_sξ. An appropriate modification of{W_tξ:t≥0,ξ∈Ξ}gives the required cylindrical Wiener process, which we denote byW◦. We note that the Brownian filtration ofW◦coincides with

(F_t◦)_t≥0.

Now letX ∈L_locp (Ω,C(0,+∞;H))be the mild solution of

¨

d X_sx =AX_sx dτ+b(X_sx)ds+σdW_s◦ s≥0

X0=x

(61) If together with the previous forward equation we consider the backward equation

Y_tx −Y_Tx +

Z T

t

Z_sxdW_s◦=

Z T

t

F(X_sx,Y_sx,Z_sx)ds, 0≤t ≤T <∞, (62)

we know that there exists a unique solution {Xx

t,Ytx,Ztx,t ≥ 0} to the forward-backward system

(61)-(62) and by Proposition 5.2, the function

v(x) =Y₀x. is the solution to the nonlinear Kolmogorov equation:

Lv(x) +F(x,v(x),∇v(x)σ) =0, x∈H. (63) Moreover the following holds:

Y_tx =v(X_tx), Z_tx =∇v(X_tx)σ. (64) We have

E◦

Z ∞

0

e−(λ+ε)t|Z_tx|2d t<∞. (65) and hence, for eachT>0,

E◦

Z T

0

|Z_tx|2d t<∞. (66)

By (34) we have

and by (58),

|u(X_tx)|=|γ(X_tx,∇v(X_tx)σ)| ≤C(1+|∇v(X_tx)σ|) =C(1+|Z_tx|). (68) Let us define, for eachT>0,

MT =exp

Z T

0

〈r(X_sx,u(X_sx)),dW_s◦〉Ξ− 1 2

Z T

0

|r(X_sx,u(X_sx)|2_Ξds

!

. (69)

Now, arguing exactly as in the proof of Proposition 5.2 in[12], we can prove thatE◦M_T =1, and

M is a P◦-martingale. Hence there exists a probabilityPT on FT◦ admitting MT as a density with

respect toP◦, and by the Girsanov Theorem we can conclude that the process{Wt,t∈[0,T]}given

byW_t=W_t◦−R₀tr(Xx

s,u(Xsx))ds is a Wiener process with respect toPT and(Ft◦)t≥0. SinceΞ ′

is a Polish space andP_T+h coincide withP_T on BT, T,h≥0, by known results (see[24], Chapter VIII,

§1, Proposition (1.13)) there exists a probabilityPonB such that the restrictions onBT ofPT and

that ofP coincide, T ≥ 0. LetG be the P-completion of B and F_T be theP-completion of B_T. Moreover, since for all T >0, {Wt : t ∈[0,T]} is aΞ-valued cylindrical Wiener process underPT

and the restriction ofP_T and ofPcoincide onBT modifying{Wt:t ≥0}in a suitable way on aP

-null probability set we can conclude that(Ω,G,{Ft:t≥0},P,{Wt :t≥0},γ(Xx,∇v(Xx)σ))is an

admissible control system. The above construction immediately ensures that, if we choose such an admissible control system, then (59) is satisfied. Indeed if we rewrite (61) in terms of{Wt:t ≥0}

we get _¨

d X_sx =AX_sx+b(X_sx)dτ+σ[r(X_sx,u(X_sx))dτ+dW_s] X₀=x.

It remains to prove (60). Let us introduce, for each integern, the following stopping time

σ_n=inf

½

t≥0 :

Z t

0

e−2λs|Z_sx|2ds≥n

¾

,

with the convention thatσn =∞if the indicated set is empty. Of courseσn ≤σn+1 and by (65), sup_n≥1σn = ∞ P◦–a.s. Let us prove that supn≥1σn = +∞ P–a.s. For each T > 0, since MT is P◦–integrable, the bounded dominated convergence theorem gives

P€sup_n≥1σn≤T

Š

= lim

n→∞P(σn≤T) =n→lim+∞E ◦”₁

σn≤T MT

—

=0. Hence, sup_n≥1σn=∞P–a.s.

Let us fixT>0 andn≥1 ; we setτ=σn∧T. Applying Itô’s formula toe−λtYtx, we get

Y₀x=e−λτY_τx+

Z τ

0

e−λs”F(X_sx,Y_sx,Z_sx) +λY_sx—ds−

Z τ

0

e−λsZ_sxdW_s◦, and coming back to the definition ofW,

Y₀x =e−λτY_τx +

Z τ

0

e−λs”F(X_sx,Y_sx,Z_sx) +λY_sx−Z_sr€X_sx,u(X_sx)Š—ds−

Z τ

0

e−λsZ_sxdW_s. By definition ofu, we have

F(X_sx,Y_sx,Z_sx) +λY_sx−Zsr

€

and hence

Y₀x =e−λτY_τx +

Z τ

0

e−λsg€X_sx,u(X_sx)Š−

Z τ

0

e−λsZ_sxdWs.

Taking the expectation with respect toP(actually with respect toP_T), we get, sinceYx is a bounded process,

E

–Z τ

0

e−λsg€X_sx,u(X_sx)Š

™

=Y₀x−E”e−λτY_τx—≤C,

where C is independent of nand T. Taking into account (36), we finally prove (60) by sendingn

and thenT to infinity.

Corollary 6.8. By Corollary 6.6 it immediately follows that if X is the solution to (59) and we set

e

u_s = u(X_sx), then J(x,eu) = v(x), and consequently Xx is an optimal state, eu_s is an optimal control, and u is an optimal feedback.

Example 6.9. Finally we briefly show that our results can be applied to perform the synthesis of optimal controls for infinite horizon costs when the state equation is a general semilinear heat equation with additive noise. Namely, fort≥0,ξ∈[0, 1]

  

∂

∂tX(t,ξ) = ∂2

∂ ξ2X(t,ξ) +eb(ξ,X(t,ξ)) +σ(ξ)e er(X(t,ξ),u(t,ξ)) +σ(ξ)e

∂

∂tW(t,ξ) X(t, 0) =X(t, 1) =0,

X(0,ξ) =x0(ξ)

(70)

whereW is a space-time white-noise onR+×[0, 1]. Moreover we introduce the cost functional:

J(x₀,u) =E

Z ∞

0

Z 1

0

e−λt[l(ξ,X(t,ξ)) +u2(t,ξ)]dξd t,

that we minimize over all adapted controlsusuch thatER₀∞R₀1e−λt|u(t,ξ)|2dξd t<∞. To fit the assumptions of our abstract results we will suppose that the functionseb,σeander are all measurable and real-valued and moreover:

• ebis defined on[0, 1]×Rand

|eb(ξ,η1)−eb(ξ,η1)| ≤L|η1−η2|,

Z 1

0

|eb(ξ, 0)|2dξ <∞

for a suitable constantL, almost allξ∈[0, 1], and allη1,η2∈R. Moreover for a.a.ξ∈[0, 1],

eb(ξ,·)∈C1(R)with∇ηeb(ξ,η)≤0 for a.a.ξ∈[0, 1]and allη∈R.

• σe is defined on[0, 1]and there exists a constantK such that|σ(ξ)| ≤e Kfor a.a.ξ∈[0, 1].

• eris defined onR×Rand

|er(θ,η)| ≤C(1+|η|) |er(θ₁,η)−er(θ₂,η)| ≤C(1+|η|)|θ₁−θ₂|, for a suitable constantC, for allθ,θ1,θ2∈Rand for allη∈R.

• l is defined on [0, 1]×R and 0 ≤ l(ξ,η) ≤ c₁(ξ) for a.a. ξ ∈ [0, 1] and all η ∈ R with

c1∈L1(0, 1). Moreover for a.a. ξ∈[0, 1]the mapl(ξ,·)∈C1(R,R)and

¯ ¯ ¯ ¯

∂

∂ ηl(ξ,η)

¯ ¯ ¯

¯≤c2(ξ)

witc₂∈L2(0, 1).

Finally we assume hatx0∈L2(0, 1).

To rewrite the above problem in abstract way we set H = Ξ = U = L2[0, 1]. By{Wt : t ≥0}we

denote a cylindrical Wiener process in L2[0, 1]. Moreover we define the operator Awith domain

D(A)by

D(A) =H2[0, 1]∩H1₀[0, 1], (Ay)(ξ) = ∂

2

∂ ξ2y(ξ), ∀y∈D(A) whereH2[0, 1]andH₀1[0, 1]are the usual Sobolev spaces, and we set

b(x)(ξ) =eb(ξ,x(ξ)), (σz)(ξ) =σ(ξ)z(ξ)e , r(x,u) =_er(x(ξ),u(ξ))

g(x,u) =|u|2_U+q(x) =

Z 1

0

[|u(ξ)|2+l(ξ,x(ξ))]dξ

for all x,z,u ∈ L2[0, 1] and a.a. ξ ∈ [0, 1]. Under previous assumptions we know, (see [10], §11.2.1) thatA,b,σverify Assumptions A2. Moreover noticing that

∇xq(x)h=

Z 1

0

∂

∂ ηl(ξ,x(ξ))h(ξ)dξ

and recalling the result in Example 6.4 it can be easily verified that Assumptions A4, A5 and A6 are satisfied.

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