Stochastic Differential Equations Driven by a Jump Markov Process and Their Applications

Abstract

In the present thesis we are interested in the well-posedness problem to a wide class of backward stochastic differential equations driven by Brownian motion and indepen- dent random measures related to pure jump Markov processes (BSDEJs for short). We first prove an existence and uniqueness result for this type of BSDEJs with globally Lips- chitz generators along with a comparison theorem for the solutions. Then, we propose to relax the Lipschitz framework in three directions as three different topics. The first topic is devoted to the study such BSDEJs with continuous generators (not necessarily Lipschitz) allowing a linear growth condition. We start by proving the existence of at least one (minimal) solution. Then, we extend this later result to the case when the generator is merely left continuous, increasing, and bounded. Finally, we prove that if the generator is assumed to be continuous and of linear growth in (y, z, k (·)) The BSDEJ has one or uncountable solutions. In the second topic we are concerned with locally Lipschitz setting. We establish an existence, uniqueness and stability theorems to such BSDEJs. We approximate the initial problem by a sequence of BSDEJs with globally Lipschitz generators, such that for each integer n the previous BSDEJ has a unique solution (Y n, Kn(·)). Then by passing to the limits, we show that the initial problem has a unique solution (Y, K (·)) as a limit of a Cauchy sequence (Y n, Kn(·)) in a Banach space to be determined later. Finally, we prove the existence of a unique solution to a Kolmogorov equation. In the third topic we give a result of existence and uniqueness to a class of BSDEJs driven by a jump Markov process with a generator allowing a logarithmic growth. Then we apply this result to prove the existence of a unique solution to one type of quadratic BSDEJs.