# On the Solution of Locally Lipschitz BSDE Associated to Jump Markov Process

## TL;DR

This paper establishes existence, uniqueness, and stability results for a class of backward stochastic differential equations driven by jump Markov processes with locally Lipschitz conditions, and applies these findings to constrained European option pricing.

## Contribution

It introduces a method to prove existence and uniqueness for locally Lipschitz BSDEs driven by jump Markov processes by approximation with globally Lipschitz equations, and demonstrates practical application in finance.

## Key findings

- Proved existence and uniqueness of solutions for locally Lipschitz BSDEs with jumps.
- Established a stability theorem in the local Lipschitz setting.
- Applied the theoretical results to European option pricing with constraints.

## Abstract

In this study, we consider a class of backward SDE driven by jump Markov process. An existence and uniqueness result to this kind of equations is obtained in a locally Lipschitz case. We essentially approximate the initial problem by constructing a convenient sequence of globally Lipschitz BSDEs having the existence and the uniqueness propriety. Then, we show, by passing to the limits, the existence, and uniqueness of a solution to the initial problem. After that, a stability theorem is also proved in the local Lipschitz setting. Applying the aforementioned result, we give an application to European option pricing with constraint.