Solutions of Backward Stochastic Differential Equations on Markov Chains

TL;DR

This paper establishes existence and uniqueness of solutions for backward stochastic differential equations on finite state Markov chains under Lipschitz conditions, expanding the theoretical framework beyond monotonic generators.

Contribution

It introduces a new approach to solving BSDEs on Markov chains without requiring monotonicity, relying solely on Lipschitz continuity for existence and uniqueness.

Findings

Solutions exist for arbitrary terminal conditions.

Solutions are unique up to measure zero sets.

The method relaxes previous monotonicity constraints.

Abstract

We consider backward stochastic differential equations (BSDEs) related to finite state, continuous time Markov chains. We show that appropriate solutions exist for arbitrary terminal conditions, and are unique up to sets of measure zero. We do not require the generating functions to be monotonic, instead using only an appropriate Lipschitz continuity condition.