[Paper Review] BSDEs with two RCLL Reflecting Obstacles driven by a Brownian Motion and Poisson Measure and related Mixed Zero-Sum Games

Saïd Hamadène, H.Wang|arXiv (Cornell University)|Mar 12, 2008

Stochastic processes and financial applications5 references4 citations

TL;DR

This paper establishes existence and uniqueness of solutions to backward stochastic differential equations with two right-continuous, left-limited reflecting barriers driven by independent Brownian motion and Poisson random measures, under uniform Lipschitz conditions and complete separation of barriers. The key contribution is proving the existence of a value in a related mixed zero-sum differential-integral game without requiring a difference of supermartingales between barriers.

ABSTRACT

In this paper we study Backward Stochastic Differential Equations with two reflecting right continuous with left limits obstacles (or barriers) when the noise is given by Brownian motion and a Poisson random measure mutually independent. The jumps of the obstacle processes could be either predictable or inaccessible. We show existence and uniqueness of the solution when the barriers are completely separated and the generator uniformly Lipschitz. We do not assume the existence of a difference of supermartingales between the obstacles. As an application, we show that the related mixed zero-sum differential-integral game problem has a value.

Motivation & Objective

• To study backward stochastic differential equations (BSDEs) with two reflecting right-continuous with left limits (RCLL) obstacles under general Lévy noise, specifically Brownian motion and Poisson random measures.
• To address the challenge of solving such BSDEs when the obstacles may have both predictable and inaccessible jumps.
• To remove the restrictive assumption that the difference between the two barriers must be expressible as a difference of supermartingales.
• To establish the existence of a value in a mixed zero-sum differential-integral game using the BSDE solution.

Proposed method

• Formulate a two-barrier reflected BSDE driven by independent Brownian motion and Poisson random measure.
• Use a fixed-point argument based on uniform Lipschitz continuity of the generator to prove existence and uniqueness of the solution.
• Model the two reflecting barriers as RCLL processes with arbitrary jump dynamics, including both predictable and inaccessible jumps.
• Ensure the barriers are completely separated, meaning the upper barrier always remains strictly above the lower barrier.
• Apply the solution to a mixed zero-sum game framework, where the value is derived from the BSDE's solution.
• Leverage the independence of Brownian motion and Poisson measure to decouple the stochastic dynamics in the analysis.

Experimental results

Research questions

• RQ1 Under what conditions does a two-barrier reflected BSDE driven by Brownian motion and Poisson measure admit a unique solution?
• RQ2 Can the solution be established without assuming the difference of the barriers is a difference of supermartingales?
• RQ3 How do predictable and inaccessible jumps in the reflecting barriers affect the solvability of the BSDE?
• RQ4 Does the solution to the reflected BSDE imply the existence of a value in a mixed zero-sum differential-integral game?
• RQ5 What structural assumptions on the generator and barriers are necessary for existence and uniqueness?

Key findings

• A unique solution exists for the two-barrier reflected BSDE when the barriers are completely separated and the generator is uniformly Lipschitz.
• The solution is established without requiring the difference of the barriers to be representable as a difference of supermartingales, a significant relaxation of prior assumptions.
• The existence of the solution implies that the associated mixed zero-sum differential-integral game has a value.
• The method applies to obstacles with both predictable and inaccessible jumps, broadening the class of admissible barriers.
• The independence of the Brownian motion and Poisson random measure is crucial for decoupling the dynamics and ensuring the fixed-point argument holds.