On Malliavin's differentiability of BSDE with time delayed generators driven by Brownian motions and Poisson random measures

TL;DR

This paper studies backward stochastic differential equations with time-delayed generators driven by Levy processes, proving existence, uniqueness, and Malliavin differentiability of solutions, extending stochastic calculus to jump and delay settings.

Contribution

It introduces a framework for analyzing Malliavin differentiability of BSDEs with time delays driven by Levy processes, including jump components.

Findings

Existence and uniqueness of solutions under small time horizon or Lipschitz constant.

Derivation of equations for Malliavin gradient processes.

Extension of Malliavin calculus to jump components in Levy-driven BSDEs.

Abstract

We investigate solutions of backward stochastic differential equations (BSDE) with time delayed generators driven by Brownian motions and Poisson random measures, that constitute the two components of a Levy process. In this new type of equations, the generator can depend on the past values of a solution, by feeding them back into the dynamics with a time lag. For such time delayed BSDE, we prove existence and uniqueness of solutions provided we restrict on a sufficiently small time horizon or the generator possesses a sufficiently small Lipschitz constant. We study differentiability in the variational or Malliavin sense and derive equations that are satisfied by the Malliavin gradient processes. On the chosen stochastic basis this addresses smoothness both with respect to the continuous part of our Levy process in terms of the classical Malliavin derivative for Hilbert space valued random variables, as well as with respect to the pure jump component for which it takes the form of an increment quotient operator related to the Picard difference operator.