Backward stochastic variational inequalities on random interval
Abstract
The aim of this paper is to study, in the infinite dimensional framework, the existence and uniqueness for the solution of the following multivalued generalized backward stochastic differential equation, considered on a random, possibly infinite, time interval:
where is a stopping time, is a progressively measurable increasing continuous stochastic process and is the subdifferential of the convex lower semicontinuous function .
As applications, we obtain from our main results applied for suitable convex functions, the existence for some backward stochastic partial differential equations with Dirichlet or Neumann boundary conditions.
0 \volume21 \issue2 2015 \firstpage1166 \lastpage1199 \doi10.3150/14BEJ601 \newremarkremarkRemark[section] \newproclaimdefinition[theorem]Definition \newremarkexample[theorem]Example
Backward SVIs on random interval
1,2]\initsL.\fnmsLucian \snmMaticiuc\corref\thanksref1,2,e1label=e1,mark]lucian.maticiuc@uaic.ro and 1,3]\initsA.\fnmsAurel \snmRăşcanu\thanksref1,3,e2label=e2,mark]aurel.rascanu@uaic.ro
backward stochastic differential equations \kwdsubdifferential operators \kwdstochastic variational inequalities \kwdstochastic partial differential equations
1 Introduction
In this paper, we are interested to prove the existence and uniqueness of a triple which is the solution for the following generalized backward stochastic variational inequality (BSVI for short) considered in the Hilbert space framework:
(1) 
where is a cylindrical Wiener process, , are the subdifferentials of a convex lower semicontinuous functions , , is a progressively measurable increasing continuous stochastic process, and is a stopping time.
In fact, we will define and prove the existence of the solution for an equivalent form of (1):
(2) 
with , and adequately defined. The notation means that is a continuous stochastic process and for any continuous stochastic process and any , the bounded variation of on is finite and the following inequality holds:
The study of the backward stochastic differential equations (BSDEs for short) in the finite dimensional case (equation of type (1) with and equal to ) was initiated by Pardoux and Peng [16] (see also Pardoux and Peng [15]). The authors have proved the existence and the uniqueness of the solution for the BSDE on fixed time interval, under the assumption of Lipschitz continuity of with respect to and and square integrability of and . The case of BSDEs on random time interval (possibly infinite), under weaker assumptions on the data, have been treated by Darling and Pardoux [5], where it is obtained, as application, the existence of a continuous viscosity solution to the elliptic partial differential equations (PDEs) with Dirichlet boundary conditions.
The more general case of scalar BSDEs with onesided reflection and associated optimal control problems was considered by El Karoui, Kapoudjian, Pardoux, Peng and Quenez [8] and with twosided reflection associated with stochastic game problem by Cvitanić and Karatzas [4].
When the obstacles are fixed, the reflected BSDE become a particular case of BSVI of type (1), by taking as convex indicator of the interval defined by obstacles. We must mention that the solution of a BSVI belongs to the domain of the operator and it is reflected at the boundary of this.
The standard work on BSVI in the finite dimensional case is that of Pardoux and Răşcanu [17], where it is proved the existence and uniqueness of the solution for BSVI (1) with , under the following assumptions on : monotonicity with respect to (in the sense that ), Lipschitzianity with respect to and a sublinear growth for . Moreover, it is shown that, unlike the forward case, the process is absolute continuous with respect to . In Pardoux and Răşcanu [18], the same authors extend these results to the Hilbert spaces framework. Afterwards, various particular cases of BSVI (1) were the subject of many articles: Maticiuc and Răşcanu [11], Maticiuc, Răşcanu and Zălinescu [12], Maticiuc and Rotenstein [13], Maticiuc and Nie [9] (where the backward equations are studied in the frame of fractional stochastic calculus) and Diomande and Maticiuc [7] (where the generator at the moment is allowed to depend on the past values on of the solution ).
Our paper generalizes the existence and uniqueness results from Pardoux and Răşcanu [18] by considering random time interval and the Lebesgue–Stieltjes integral terms, and by assuming a weaker boundedness condition for the generator (instead of the sublinear growth), that is,
(3) 
We mention that, since is a stopping time, the presence of the process is justified by the possible applications of equation (1) in proving probabilistic interpretation for the solution of elliptic multivalued partial differential equations with Neumann boundary conditions on a domain from . The stochastic approach of the existence problem for finite dimensional multivalued parabolic PDEs, was considered by Maticiuc and Răşcanu [11].
Concerning assumption (3), we recall that, in the case of finite dimensional BSDE, Pardoux [14] has used a similar condition, in order to prove the existence of a solution in . His result was generalized by Briand, Delyon, Hu, Pardoux and Stoica [3], where it is proved the existence in of the solution for BSDEs considered both with fixed and random terminal time. We mention that the assumptions from our paper are, broadly speaking, similar to those of Briand, Delyon, Hu, Pardoux and Stoica [3].
The article is organized as follows: in the next section a brief summary of infinite dimensional stochastic integral and the assumptions are given. Section 3 is devoted to the proof of the existence and uniqueness of a strong solution for (2). In the Section 4, is a new type of solution (called variational weak solution) and it is also proves the existence and uniqueness result. In Section 4 are obtained, as applications, the existence of the solution for various type of backward stochastic partial differential equations with boundary conditions. The Appendix contains, following Pardoux and Răşcanu [19], some results useful throughout the paper.
2 Preliminaries
2.1 Infinite dimensional framework
In the beginning of this subsection, we give a brief exposition of the stochastic integral with respect to a Wiener process defined on a Hilbert space. For a deeper discussion concerning the notion of cylindrical Wiener process and the construction of the stochastic integral, we refer reader to Da Prato and Zabczyk [6].
We consider a complete probability space , the set , a right continuous and complete filtration , and two real separable Hilbert spaces .
Let us denote by , , the complete metric space of continuous progressively measurable stochastic process (p.m.s.p.) with the metric given by
and by the space of p.m.s.p. such that, for all , the restriction . To shorten notation, we continue to write for . Remark that is a Banach space for .
By , , we denote the Banach space of the continuous stochastic processes such that , , a.s., and , a.s. for all . The norm is defined by . If , then is a closed linear subspace of .
Let be a Gaussian family of realvalued random variables with zero mean and the covariance function given by , , . We call a Wiener process if, for all ,

[(ii)]

,

is independent of , for all , .
Let be an orthonormal and complete basis in . We introduce the separable Hilbert space of Hilbert–Schmidt operators from to , that is, the space of linear operators such that . It will cause no confusion if we use to designate the norm in .
The sequence , defines a family of realvalued Wiener processes mutually independent on .
If is finite dimensional space then we have the representation , but, in general case, this series does not converge in , but rather in a larger space such that with the injection being a Hilbert–Schmidt operator. Moreover, .
For , we will denote by , , the space , that is, the complete metric space of progressively measurable stochastic processes with metric of convergence
The space is a Banach space for with norm . From now on, for simplicity of notation, we write instead of (when no confusion can arise).
Let us denote by the space of measurable stochastic processes such that, for all , the restriction .
For any let the stochastic integral , , where is an orthonormal basis in . Note that the introduced stochastic integral does not depend on the choice of the orthonormal basis on . By the standard localization procedure, we can extend this integral as a linear continuous operator , and it has the following properties:
Proposition 2.1
Let . Then

[(iii)]

, , if ,

, if ,

, if (Burkholder–Davis–Gundy inequality),

, if .
From now on, we shall consider that the original filtration is replaced by the filtration generated by the Wiener process. The following Hilbert space version of the martingale representation theorem, extended to a random interval, holds the following proposition.
Proposition 2.2
Let be a stopping time, and be a measurable random variable such that . Then

[3.]

there exists a unique stochastic process such that and , , or equivalently,

there exists a unique pair such that
(4) or equivalently,

there exists a unique pair such that , a.s., and and , .
2.2 Assumptions and definitions

[(A)]

The parameter ;

The random variable is a stopping time;

The random variable is measurable such that and the stochastic process is the unique pair associated to such that we have the martingale representation formula (4);

The process is a progressively measurable increasing continuous stochastic process such that ;

The functions and are such that
and , , a.s., where and .
Moreover, there exist two p.m.s.p. such that and for all a.s., and there exists such that, for all ,
(5)
Let us introduce the function
and let be the a real positive p.m.s.p. (given by Radon–Nikodym’s representation theorem) such that and and .
Let
in which case (2.2) yields
For , let
We can give now some a priori estimates concerning the solution of (1).
Lemma 2.1
Let . Under assumption (A) the following inequalities hold, in the sense of signed measures on ,
(7) 
and
(8) 
The inequalities can be obtained by standard calculus (applying the monotonicity and Lipschitz property of function ).

[(A)]

are proper convex lower semicontinuous l.s.c. functions such that (consequently ).
Let us define
We recall now that the multivalued subdifferential operator is the maximal monotone operator
We define and and by we understand that and . We know that and .
If is a locally bounded variation function, is a real increasing function, is a continuous function and is like in (A), then notation means that for any continuous function , it holds
(9) 
Now we are able to introduce the rigorous definition of a solution for equation (1). First, using definitions of , and , respectively, we can rewrite (1) in the form
(10) 
We call a solution of (10) if has locally bounded variation and with for such that

[(iii)]

, a.s., for all ,

a.e.,

, as (where is given by (2.2)) and

, a.s., .
Let and the Moreau–Yosida regularization of given by , which is a convex function. We mention some properties (see Brézis [2], and Pardoux and Răşcanu [17] for the last one): for all
We introduce the compatibility conditions between (which have previously been used in Maticiuc and Răşcanu [11]):

[(A)]

For all , , ,
where and .
Let .

[B.]

Clearly, since and are increasing monotone, we see that, if and , , then compatibility assumptions (2.2) are satisfied.

If are the convexity indicator functions, that is,
where are such that (see assumption (A)), then and similar for .
Since (A)(i) is fulfilled, the compatibility assumptions become , for and , for , and, respectively, for and , for .
The last assumption is the following:

[(A)]

There exist the p.m.s.p. with and such that a.s. and, using notation
(11) we suppose
and the locally boundedness conditions:
We point out that the purpose of defining of the new process is due to the computations; see, e.g., inequalities (3) and (3) from the proof of the first main theorem, where it is necessary to have a new process such that and on .
3 Main result: The existence of the strong solution
We present first the definition of a solution in the strong case when the process is absolutely continuous with respect to (i.e., on ).
We call a strong solution of (10) if there exist two p.m.s.p. , and , such that is a solution of (10) with and
where is given by (2.2).
If there exists such that , a.s., then the condition (3)(iii) is equivalent to , as .
We can now formulate the first main result. In order to obtain the absolute continuity with respect to of the process (as in Definition 3) it is necessary to impose a supplementary assumption:

[(A)]

There exists such that, for all ,
(12)
We mention that without this assumption we are not able to prove, among other, that there exist two processes and such that (see step from the proof of the next theorem).
Theorem 3.1
Let assumptions (A)–(A) be satisfied. Then the backward stochastic variational inequality (10) has a unique solution such that for all ,
(13) 
and
(14) 
Moreover, for all , there exists a constant such that, for all , a.s.
and
(15) 