Backward Stochastic Differential Equations Driven by a Jump Markov Process with Continuous and Non-Necessary Continuous Generators

Abstract

We deal with backward stochastic differential equations driven by a pure jump Markov process and an independent Brownian motion (BSDEJs for short). We start by proving the existence and uniqueness of the solutions for this type of equation and present a comparison of the solutions in the case of Lipschitz conditions in the generator. With these tools in hand, we study the existence of a (minimal) solution for BSDE where the coefficient is continuous and satisfies the linear growth condition. An existence result for BSDE with a left-continuous, increasing and bounded generator is also discussed. Finally, the general result is applied to solve one kind of quadratic BSDEJ.

Keywords

• backward stochastic differential equations
• jump Markov process
• comparison principle
• quadratic BSDE with jumps