Stochastic integration for fractional Levy process and stochastic differential equation driven by fractional Levy noise

TL;DR

This paper develops a framework for stochastic integration and differential equations driven by fractional Levy noises, extending white noise analysis to pure-jump Levy processes and establishing existence and uniqueness of solutions.

Contribution

It introduces fractional Levy noises via white noise analysis and defines Skorohod integrals, along with proving solution existence and uniqueness for related stochastic equations.

Findings

Defined fractional Levy noises as derivatives of fractional Levy processes.

Established existence and uniqueness of solutions for stochastic Volterra and differential equations driven by fractional Levy noises.

Extended white noise analysis to pure-jump Levy processes for stochastic calculus.

Abstract

In this paper, based on the white noise analysis of square integrable pure-jump Levy process given by [1], we define the formal derivative of fractional Levy process defined by the square integrable pure-jump Levy process as the fractional Levy noises by considering fractional Levy process as the generalized functional of Levy process, and then we define the Skorohod integral with respect to the fractional Levy process. Moreover, we propose a class of stochastic Volterra equations driven by fractional Levy noises and investigate the existence and uniqueness of their solutions; In addition, we propose a class of stochastic differential equations driven by fractional Levy noises and prove that under the Lipschtz and linear conditions there exists unique stochastic distribution-valued solution.