Numerical Computation for Backward Doubly SDEs with random terminal time

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.Comment: 38, Monte Carlo Methods and Applications (MCMA) 201

The obstacle problem for semilinear parabolic partial integro-differential equations

This paper presents a probabilistic interpretation for the weak Sobolev solution of the obstacle problem for semilinear parabolic partial integro-differential equations (PIDEs). The results of Leandre (1985) concerning the homeomorphic property for the solution of SDEs with jumps are used to construct random test functions for the variational equation for such PIDEs. This results in the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDEs), namely Feynman Kac's formula for the solution of the PIDEs. Moreover it gives an application to the pricing and hedging of contingent claims with constraints in the wealth or portfolio processes in financial markets including jumps.Comment: 31 page

Quelques contributions dans la représentation probabiliste des solutions d'EDPs non linéaires

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.L'objectif de cette thèse est l'étude de la représentation probabiliste des différentes classes d'EDPSs non-linéaires(semi-linéaires, complètement non-linéaires, réfléchies dans un domaine) en utilisant les équations différentielles doublement stochastiques rétrogrades (EDDSRs). Cette thèse contient quatre parties différentes. Nous traitons dans la première partie les EDDSRs du second ordre (2EDDSRs). Nous montrons l'existence et l'unicité des solutions des EDDSRs en utilisant des techniques de contrôle stochastique quasi- sure. La motivation principale de cette étude est la représentation probabiliste des EDPSs complètement non-linéaires. Dans la deuxième partie, nous étudions les solutions faibles de type Sobolev du problème d'obstacle pour les équations à dérivées partielles inteégro-différentielles (EDPIDs). Plus précisément, nous montrons la formule de Feynman-Kac pour l'EDPIDs par l'intermédiaire des équations différentielles stochastiques rétrogrades réfléchies avec sauts (EDSRRs). Plus précisément, nous établissons l'existence et l'unicité de la solution du problème d'obstacle, qui est considérée comme un couple constitué de la solution et de la mesure de réflexion. L'approche utilisée est basée sur les techniques de flots stochastiques développées dans Bally et Matoussi (2001) mais les preuves sont beaucoup plus techniques. Dans la troisième partie, nous traitons l'existence et l'unicité pour les EDDSRRs dans un domaine convexe D sans aucune condition de régularité sur la frontière. De plus, en utilisant l'approche basée sur les techniques du flot stochastiques nous démontrons l'interprétation probabiliste de la solution faible de type Sobolev d'une classe d'EDPSs réfléchies dans un domaine convexe via les EDDSRRs. Enfin, nous nous intéressons à la résolution numérique des EDDSRs à temps terminal aléatoire. La motivation principale est de donner une représentation probabiliste des solutions de Sobolev d'EDPSs semi-linéaires avec condition de Dirichlet nul au bord. Dans cette partie, nous étudions l'approximation forte de cette classe d'EDDSRs quand le temps terminal aléatoire est le premier temps de sortie d'une EDS d'un domaine cylindrique. Ainsi, nous donnons les bornes pour l'erreur d'approximation en temps discret. Cette partie se conclut par des tests numériques qui démontrent que cette approche est effective

Numerical Computation for Backward Doubly SDEs with random terminal time

38, Monte Carlo Methods and Applications (MCMA) 2016In 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

Numerical Computation for Backward Doubly SDEs with random terminal time

38, Monte Carlo Methods and Applications (MCMA) 2016In 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

When Capital Is a Funding Source: The Anticipated Backward Stochastic Differential Equations of X-Value Adjustments

International audienceX-value adjustments (XVAs) refer to various financial derivative pricing adjustments accounting for counterparty risk and its funding (FVA) and capital (KVA) implications for a bank. In this paper we show that the XVA equations are well-posed, including in the realistic case where capital is deemed fungible as a source of funding for variation margin. This intertwining of capital at risk and the FVA, added to the fact that the KVA is part of capital at risk, leads to a system of backward SDEs (BSDEs) of the McKean type (anticipated BSDEs) for the FVA and the KVA, with coefficients entailing a conditional risk measure of the one-year-ahead increment of the martingale part of the FVA. This is first considered in the case of a hypothetical bank without debt. In the practical case of a defaultable bank, the resulting anticipated BSDEs, which are stopped before the default of the bank, are solved likewise after reduction to a reference market filtration