On Markovian solutions to Markov Chain BSDEs

TL;DR

This paper investigates Markovian solutions to backward stochastic differential equations driven by finite state Markov chains, establishing conditions for Markovianity and linking them to coupled ODE systems for efficient computation.

Contribution

It characterizes when solutions are Markovian and connects these equations to coupled ODEs, enabling faster numerical methods.

Findings

Solutions are Markovian only if the integrand has a specific form.

Markovian BSDEs can be linked to coupled systems of ODEs.

Provides a basis for efficient numerical evaluation of Markov-Chain BSDEs.

Abstract

We study (backward) stochastic differential equations with noise coming from a finite state Markov chain. We show that, for the solutions of these equations to be `Markovian', in the sense that they are deterministic functions of the state of the underlying chain, the integrand must be of a specific form. This allows us to connect these equations to coupled systems of ODEs, and hence to give fast numerical methods for the evaluation of Markov-Chain BSDEs.