$L_2$-variation of L\'{e}vy driven BSDEs with non-smooth terminal conditions

TL;DR

This paper studies the $L_2$-regularity of solutions to Lévy-driven BSDEs with non-smooth terminal conditions, revealing how fractional smoothness affects solution regularity through chaos expansion analysis.

Contribution

It introduces a novel analysis of $L_2$-regularity for Lévy-driven BSDEs with non-Lipschitz terminal conditions based on chaos expansion structure.

Findings

Established $L_2$-regularity results for non-smooth terminal conditions.

Linked chaos expansion structure to solution regularity.

Extended understanding of Lévy-driven BSDEs with fractional smoothness.

Abstract

We consider the $L_2$-regularity of solutions to backward stochastic differential equations (BSDEs) with Lipschitz generators driven by a Brownian motion and a Poisson random measure associated with a L\'{e}vy process $(X_t)_{t\in[0,T]}$. The terminal condition may be a Borel function of finitely many increments of the L\'{e}vy process which is not necessarily Lipschitz but only satisfies a fractional smoothness condition. The results are obtained by investigating how the special structure appearing in the chaos expansion of the terminal condition is inherited by the solution to the BSDE.