SABBAGH Wissal

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In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time tau. The main motivations are giving a probabilistic representation of the Sobolev's solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when tau is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided.

This paper presents existence and uniqueness results for reflected backward doubly stochastic differential equations (in short RBDSDEs) in a convex domain D without any regularity conditions on the boundary. Moreover, using a stochastic flow approach a probabilistic interpretation for a system of reflected SPDEs in a domain is given via such RBDSDEs. The solution is expressed as a pair (u, ν) where u is a predictable continuous process which takes values in a Sobolev space and ν is a random regular measure. The bounded variation process K, the component of the solution of the reflected BDSDE, controls the set when u reaches the boundary of D. This bounded variation process determines the measure ν from a particular relation by using the inverse of the flow associated to the the diffusion operator.

In this article, we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation (2BDSDE). We prove existence and uniqueness of the solution under a Lipschitz type assumption on the generator, and we investigate the links between our 2BDSDEs and a class of parabolic fully nonLinear Stochastic PDEs. Precisely, we show that the Markovian solution of 2BDSDEs provide a probabilistic interpretation of the classical and stochastic viscosity solution of fully nonlinear SPDEs.

In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time tau. The main motivations are giving a probabilistic representation of the Sobolev's solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when tau is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided.

The objective of this thesis is to study the probabilistic representation (Feynman-Kac for- mula) of different classes ofStochastic Nonlinear PDEs (semilinear, fully nonlinear, reflected in a domain) by means of backward doubly stochastic differential equations (BDSDEs). This thesis contains four different parts. We deal in the first part with the second order BDS- DEs (2BDSDEs). We show the existence and uniqueness of solutions of 2BDSDEs using quasi sure stochastic control technics. The main motivation of this study is the probabilistic representation for solution of fully nonlinear SPDEs. First, under regularity assumptions on the coefficients, we give a Feynman-Kac formula for classical solution of fully nonlinear SPDEs and we generalize the work of Soner, Touzi and Zhang (2010-2012) for deterministic fully nonlinear PDE. Then, under weaker assumptions on the coefficients, we prove the probabilistic representation for stochastic viscosity solution of fully nonlinear SPDEs. In the second part, we study the Sobolev solution of obstacle problem for partial integro-differentialequations (PIDEs). Specifically, we show the Feynman-Kac formula for PIDEs via reflected backward stochastic differentialequations with jumps (BSDEs). Specifically, we establish the existence and uniqueness of the solution of the obstacle problem, which is regarded as a pair consisting of the solution and the measure of reflection. The approach is based on stochastic flow technics developed in Bally and Matoussi (2001) but the proofs are more technical. In the third part, we discuss the existence and uniqueness for RBDSDEs in a convex domain D without any regularity condition on the boundary. In addition, using the approach based on the technics of stochastic flow we provide the probabilistic interpretation of Sobolev solution of a class of reflected SPDEs in a convex domain via RBDSDEs. Finally, we are interested in the numerical solution of BDSDEs with random terminal time. The main motivation is to give a probabilistic representation of Sobolev solution of semilinear SPDEs with Dirichlet null condition. In this part, we study the strong approximation of this class of BDSDEs when the random terminal time is the first exit time of an SDE from a cylindrical domain. Thus, we give bounds for the discrete-time approximation error. We conclude this part with numerical tests showing that this approach is effective.

The objective of this thesis is the study of the probabilistic representation of different classes of nonlinear EDPSs (semi-linear, completely nonlinear, reflected in a domain) using the double stochastic backward differential equations (EDDSRs). This thesis contains four different parts. In the first part, we treat the second order EDDSRs (2EDDSRs). We show the existence and uniqueness of solutions of EDDSRs using quasi-secure stochastic control techniques. The main motivation of this study is the probabilistic representation of completely nonlinear EDPSs. In the second part, we study weak Sobolev-type solutions of the obstacle problem for integro-differential partial differential equations (IDDEs). More precisely, we show the Feynman-Kac formula for the EDPIDs via the reflected backward stochastic differential equations with jumps (EDSRRs). More precisely, we establish the existence and uniqueness of the solution of the obstacle problem, which is considered as a pair consisting of the solution and the reflection measure. The approach used is based on the stochastic flow techniques developed in Bally and Matoussi (2001) but the proof is much more technical. In the third part, we treat the existence and uniqueness for EDDSRRs in a convex domain D without any regularity condition on the boundary. Moreover, using the stochastic flow techniques approach we prove the probabilistic interpretation of the weak Sobolev-type solution of a class of reflected EDPSs in a convex domain via EDDSRRs. Finally, we focus on the numerical resolution of random terminal time EDDSRs. The main motivation is to give a probabilistic representation of the Sobolev solutions of semi-linear EDPSs with zero Dirichlet condition at the edge. In this section, we study the strong approximation of this class of EDDSRs when the random terminal time is the first exit time of a cylindrical domain EDS. Thus, we give bounds for the approximation error in discrete time. This section concludes with numerical tests that demonstrate that this approach is effective.

We give a probabilistic interpretation for the weak Sobolev solution of obstacle problem for semilinear parabolic partial integro-differential equations (PIDE). The results of Léandre [29] about the homeomorphic property for the solution of SDE with jumps are used to construct random test functions for the variational equation for such PIDE. This yields to the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDE), namely the Feynman Kac's formula for the solution of the PIDE. MSC: 60H15. 60G46. 35R60 Keyword: Reflected backward stochastic differential equation, partial parabolic integro-differential equation, jump diffusion process, obstacle problem, stochastic flow, flow of diffeo-morphism.