Quadratic-exponential growth BSDEs with Jumps and their Malliavin's Differentiability

TL;DR

This paper studies a class of backward stochastic differential equations with jumps and quadratic-exponential growth, establishing existence, uniqueness, convergence, and conditions for Malliavin differentiability.

Contribution

It introduces new conditions ensuring existence, uniqueness, and differentiability of solutions for quadratic-exponential BSDEs with jumps, extending previous results.

Findings

Proved existence and uniqueness under general conditions.

Established strong convergence for non-monotone driver sequences.

Provided sufficient conditions for Malliavin differentiability.

Abstract

We investigate a class of quadratic-exponential growth BSDEs with jumps. The quadratic structure introduced by Barrieu & El Karoui (2013) yields the universal bounds on the possible solutions. With local Lipschitz continuity and the so-called A_gamma-condition for the comparison principle to hold, we prove the existence of a unique solution under the general quadratic-exponential structure. We have also shown that the strong convergence occurs under more general (not necessarily monotone) sequence of drivers, which is then applied to give the sufficient conditions for the Malliavin's differentiability.