On some Properties of Forward and Backward Stochastic Differential Equations with Jumps

Madoui, Imene (2024) On some Properties of Forward and Backward Stochastic Differential Equations with Jumps. Doctoral thesis, Université Mohamed Khider (Biskra - Algérie).

Text Madoui_Imene_thesis.pdf Download (1MB)

Abstract

The aim of this Ph.D. thesis is to study the problem of existence and uniqueness by relaxing the Lipschitz condition on generators of backward stochastic differential equations with jumps. The first topic deals with multidimensional Markovian BSDEs driven by a Poisson random measure and independent Brownian motion. Existence results for such equations with continuous generators that satisfy the usual linear growth condition are proved. The second topic is concerned with a class of quadratic BSDEs with jumps where the generators show quadratic growth in the Brownian component and non-linear functional form with respect to the jump term. We establish the existence (and sometimes the uniqueness) of solutions as well as a comparison and strict comparison principles under no monotonicity condition in the third argument of the generator. Probabilistic representations of solutions to some classes of quadratic partial integral differential equations are given by means of solutions of these QBSDEJs. This thesis presents three chapters. The first chapter focuses on the existence and uniqueness of the solution to multidimensional Markovian BSDEJs under the global Lipschitz property of the generator and square-terminal value. We prove, under the Lipschitz condition, that the BSDEJ's adapted solution can be represented in terms of a given Markov process and some deterministic functions. The second chapter is concerned with multidimensional Markovian BSDEJs in two cases. In the first case, when the generator is continuous with respect to the first and second variables and satisfies the Lipschitz condition with respect to the third variable. In the second case, when the generator is continuous with respect to the three variables. The existence of a solution (not necessarily unique) to BSDEJs under study is proved by using the so-called L^2-domination technique and some regularization and approximations arguments. Furthermore, some special cases of linear and sub-linear growth conditions and the regularity of the generator are discussed. We conclude this chapter with several examples of the Markov process having the L^2-domination property. The third chapter is devoted to the existence and/or uniqueness of the solutions to a variety of types of QBSDEJs. More precisely, the solvability of some QBSDEJs via several examples dealing with different generators of other quadratic forms. Furthermore, two comparison theorems are established. Finally, this chapter deals with the relationship between quadratic BSDEJs and QPIDEs with measurable generators. This connection provides a probabilistic representation of viscosity solutions of some QPIDEs, which is proved by means of the Feynman-Kac formula.

Actions (login required)

View Item