Existence, Uniqueness and Malliavin Differentiability of L\'evy-driven BSDEs with locally Lipschitz Driver

TL;DR

This paper establishes conditions for existence, uniqueness, and Malliavin differentiability of Le9vy-driven BSDEs with locally Lipschitz generators, including cases with quadratic and exponential growth, extending previous results to broader settings.

Contribution

It extends solvability and differentiability results for Le9vy-driven BSDEs to generators with locally Lipschitz conditions and unbounded growth, including non-Lipschitz cases relevant for utility maximization.

Findings

Proved existence and uniqueness of solutions under boundedness conditions.

Established Malliavin differentiability for solutions with bounded terminal data.

Extended results to non-Lipschitz generators in U, applicable to utility maximization.

Abstract

We investigate conditions for solvability and Malliavin differentiability of backward stochastic differential equations driven by a L\'evy process. In particular, we are interested in generators which satisfy a locally Lipschitz condition in the $Z$ and $U$ variable. This includes settings of linear, quadratic and exponential growths in those variables. Extending an idea of Cheridito and Nam to the jump setting and applying comparison theorems for L\'evy-driven BSDEs, we show existence, uniqueness, boundedness and Malliavin differentiability of a solution. The pivotal assumption to obtain these results is a boundedness condition on the terminal value $\xi$ and its Malliavin derivative $D\xi$. Furthermore, we extend existence and uniqueness theorems to cases where the generator is not even locally Lipschitz in $U.$ BSDEs of the latter type find use in exponential utility maximization.