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

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.

Automated summary

This article deals with backward stochastic differential equations driven by a pure jump Markov process and an independent Brownian motion (BSDEJs). We first prove the existence and uniqueness of solutions for this type of equation and provide a comparison of solutions in the case of Lipschitz conditions in the generator. With these tools, 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.