Locally Lipschitz Path Dependent FBSDEs with Unbounded Terminal Conditions in Brownian and L{é}vy Settings
Hannah Geiss, Céline Labart, Adrien Richou, Alexander Steinicke
TL;DR
The paper addresses forward-backward stochastic differential equations driven by a $\ Lévy$ process with path-dependent, locally Lipschitz coefficients and unbounded terminal conditions.The authors prove existence and uniqueness of a bounded solution $(Y,Z,U)$ and establish Malliavin differentiability by a truncation-then-limit approach, complemented by a stability analysis.A detailed Malliavin calculus framework on the Lévy-Itô space is developed, including criteria for Brownian and jump directions and derivative representations for the $Z$ and $U$ components, enabling precise regularity results.The results extend the Brownian and Lévy literature to the path-dependent, unbounded-terminal setting and provide structural insights and tools for numerical schemes and connections to integro-differential PDEs.
Abstract
This paper is dedicated to the analysis of forward backward stochastic differential equations driven by a L{é}vy process. We assume that the generator and the terminal condition are path-dependent and satisfy a local Lipschitz condition. We study solvability and Malliavin differentiability of such BSDEs. The proof of the existence and uniqueness is done in three steps. First of all, we truncate and localize the terminal condition and the generator. Then we use an iteration argument to get bounds for the solutions of the truncated BSDE (independent from the level of truncation). Finally, we let the level of truncation tend to infinity. A stability result ends the proof. The Malliavin differentiability result is based on a recent characterisation for the Malliavin Sobolev space D 1,2 by S. Geiss and Zhou.