Doubly reflected BSDEs driven by RCLL martingales under stochastic Lipschitz coefficient

Abstract

In this paper, we study a backward stochastic differential equation driven by a Right Continuous with Left Limits (RCLL) martingale with two completely separated RCLL barriers. When the coefficient is stochastically Lipschitz, we demonstrate the existence and uniqueness of a square-integrable adapted solution using the penalization method. Additionally, we provide a fair price for a game contingent claim between two traders with additional exogenous knowledge of the same stock price in a public financial market driven by an Azéma’s martingale. We also determine a saddle point for the game when the obstacles are left upper semi-continuous along stopping times.