Anticipating Linear Stochastic Differential Equations Driven by a L\'{e}vy Process

TL;DR

This paper investigates the existence of unique solutions for linear stochastic differential equations driven by Lévy processes, extending Girsanov transformation methods to handle non-adapted initial conditions and coefficients.

Contribution

It introduces an extension of Girsanov transformation techniques from Wiener space to Lévy space for equations with non-adapted data.

Findings

Established conditions for existence and uniqueness of solutions.

Extended Girsanov method to Lévy processes.

Handled non-adapted initial conditions and coefficients.

Abstract

In this paper we study the existence of a unique solution for linear stochastic differential equations driven by a L\'evy process, where the initial condition and the coefficients are random and not necessarily adapted to the underlying filtration. Towards this end, we extend the method based on Girsanov transformation on Wiener space and developped by Buckdahn to the canonical L\'evy space.