Backward stochastic differential equations with Markov chains and associated PDEs

Abstract

In this paper we study the regularity of the solutions for backward stochastic differential equations (BSDEs) with finite state Markov chains and establish its link with associated partial differential equations (PDEs) in classical sense. Moreover, we study the existence and uniqueness of solutions for such BSDEs under Lipschitz conditions on f in the space Lρ2(Rd;Rk)⊗Lρ2(Rd;Rd×k)⊗Lρ2(Rd×I;Rk). In this way, we establish a new connection between Lρ2(Rd;Rk)⊗Lρ2(Rd;Rd×k)⊗Lρ2(Rd×I;Rk) valued solutions of BSDEs and the solutions of PDEs in a Sobolev space.