Undiscounted Markov chain BSDEs to stopping times

TL;DR

This paper studies backward stochastic differential equations driven by Markov chain noise with terminal conditions at stopping times, establishing existence, uniqueness, and growth bounds without requiring driver monotonicity or discounting.

Contribution

It proves existence and uniqueness of solutions for BSDEs with Markov chain noise at stopping times without monotonicity assumptions, extending previous results.

Findings

Solutions exist and are unique under specified integrability and growth conditions.

The results apply to hitting times of states in the Markov chain.

New applications of BSDE theory to stopping times are demonstrated.

Abstract

We consider Backward Stochastic Differential Equations in a setting where noise is generated by a countable state, continuous time Markov chain, and the terminal value is prescribed at a stopping time. We show that, given sufficient integrability of the stopping time and a growth bound on the terminal value and BSDE driver, these equations admit unique solutions satisfying the same growth bound (up to multiplication by a constant). This holds without assuming that the driver is monotone in y, that is, our results do not require that the terminal value be discounted at some uniform rate. We show that the conditions are satisfied for hitting times of states of the chain, and hence present some novel applications of the theory of these BSDEs.