[Paper Review] Existence and uniqueness results for BSDEs with jumps: the whole nine yards

Antonis Papapantoleon, Dylan Possamaï+1 more|arXiv (Cornell University)|Jul 14, 2016

Stochastic processes and financial applications3 citations

TL;DR

This paper establishes existence and uniqueness for multidimensional backward stochastic differential equations (BSDEs) with jumps under general conditions: unbounded random time horizon, possibly stochastically discontinuous filtrations, and stochastic Lipschitz generators. The key contribution is a well-posedness result in weighted spaces, extending prior theory to include BSDEs driven by discrete-time approximations of general martingales.

ABSTRACT

This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete--time approximations of general martingales.

Motivation & Objective

• To establish a well-posedness result for multidimensional BSDEs with jumps under minimal assumptions on the filtration and generator.
• To extend existing theory beyond Brownian filtrations to general martingale-driven BSDEs, including stochastically discontinuous filtrations.
• To handle unbounded or infinite time horizons, which are common in financial and stochastic control applications.
• To provide a framework for BSDEs driven by discrete-time approximations of general martingales, enabling numerical and approximation schemes.
• To generalize the stochastic Lipschitz condition, allowing random Lipschitz constants, thus broadening applicability to non-uniformly Lipschitz generators.

Proposed method

• The authors use a fixed-point argument in appropriately weighted stochastic spaces to prove existence and uniqueness of solutions.
• They define weighted norms based on exponential local martingales and integrability conditions to control the growth of solutions over unbounded time horizons.
• The generator is assumed to satisfy a stochastic Lipschitz condition, where the Lipschitz constants are random processes adapted to the filtration.
• The analysis is conducted in a filtration satisfying only the usual hypotheses, allowing for stochastically discontinuous martingales and general semimartingale drivers.
• The solution space is constructed to ensure integrability and predictability of the process pair (Y, Z), even under unbounded time horizons.
• The proof relies on a priori estimates and contraction mapping arguments in weighted L²-type spaces, leveraging the structure of the jump component and the generator's stochastic Lipschitz property.

Experimental results

Research questions

• RQ1 Under what conditions does a multidimensional BSDE with jumps admit a unique solution when the time horizon is unbounded or random?
• RQ2 Can the classical Lipschitz condition be replaced by a stochastic Lipschitz condition with random Lipschitz constants, and still guarantee well-posedness?
• RQ3 Is it possible to extend the well-posedness theory of BSDEs to filtrations that are stochastically discontinuous, i.e., not necessarily continuous or quasi-left-continuous?
• RQ4 Can BSDEs driven by discrete-time approximations of general martingales be rigorously treated within the same framework as continuous-time BSDEs?
• RQ5 What are the minimal integrability and regularity conditions on the terminal condition and generator to ensure existence and uniqueness in weighted spaces?

Key findings

• The paper establishes existence and uniqueness of solutions to multidimensional BSDEs with jumps under a stochastic Lipschitz condition, even when the time horizon is unbounded or random.
• Solutions exist and are unique in appropriately weighted L²-type spaces, which accommodate the growth of the solution process over infinite time horizons.
• The result holds for general filtrations satisfying only the usual conditions, including those with stochastically discontinuous martingales.
• The framework allows for BSDEs driven by discrete-time approximations of general martingales, providing a rigorous foundation for numerical schemes.
• The stochastic Lipschitz condition is essential: it allows for random Lipschitz constants, which generalize the classical uniform Lipschitz condition and enhance applicability.
• The proof relies on a contraction argument in weighted spaces, with sufficient conditions for contraction derived explicitly in terms of the jump size, drift, and volatility parameters.