Time fractional stochastic differential equations driven by pure jump L\'evy noise

TL;DR

This paper introduces a novel variable order time fractional stochastic differential equation driven by pure jump Lévy noise, modeling particles with memory effects, and establishes well-posedness, moment estimates, and regularity of solutions.

Contribution

It is the first to analyze such equations driven by pure jump Lévy noise without integrability assumptions, providing foundational results on existence, estimates, and regularity.

Findings

Proved well-posedness without integrability conditions

Derived $L^p$ moment estimates for solutions

Established Hölder regularity of solutions

Abstract

In this paper we introduce a variable order time fractional differential equation driven by pure jump L\'evy noise, which models the motion of a particle exhibiting memory effect. We prove the well-posedness of this equation without assuming any integrability condition on the initial condition and the large jump coefficient, by using a truncation argument. Under some extra conditions, we also derive some $L^p$ moment estimates on the solutions. As an application of moment estimates, we prove the H\"older regularity of the solutions.