BSDEs with regime switching: Weak convergence and applications

• We prove weak convergence of BSDEs with regime-switching.
• The Markov chain has a two-time-scale-structure.
• Weak convergence is proved under the Meyer–Zheng topology.
• We show the convergence of the corresponding PDE system.

This paper is concerned with a system of backward stochastic differential equations (BSDEs) with regime switching. The BSDEs are coupled by a finite-state Markov chain. The underlying Markov chain is assumed to have a two-time scale (or weak and strong interactions) structure. Namely, the states of the Markov chain can be divided into a number of groups so that the chain jumps rapidly within a group and slowly between the groups. It is shown in this paper that the original BSDE system can be approximated by a limit system in which the states in each group are aggregated out and replaced by a single state. In particular, it is proved that the solution of the original BSDE system converges weakly under the Meyer–Zheng topology as the fast jump rate goes to infinity. The limit process is a solution of aggregated BSDEs which can be determined by the corresponding martingale problem. The results are applied to a set of partial differential equations and used to validate their convergence to the corresponding limit system. Finally, a numerical example is given to demonstrate the approximation results.