Impulsive stochastic fractional differential equations driven by fractional Brownian motion

Mahmoud Abouagwa,Ji Li,Feifei Cheng

doi:10.1186/s13662-020-2533-2

Abstract

In this research, we study the existence and uniqueness results for a new class of stochastic fractional differential equations with impulses driven by a standard Brownian motion and an independent fractional Brownian motion with Hurst index 1/2< H<1 under a non-Lipschitz condition with the Lipschitz one as a particular case. Our analysis depends on an approximation scheme of Carathéodory type. Some previous results are improved and extended.

Highlights

1 Introduction Traditionally, a dominant interest in practical applications is the existence of solutions to deterministic fractional differential equations and fractional stochastic differential equations (FSDEs) driven by Brownain motion due to their role for helping candidates explore the hidden properties of the dynamics of complex systems in viscoelasticity, diffusion, mechanics, electromagnetism, control, signal processing, and physics
The specific objective of this study is to prove the existence and uniqueness of solutions to the following impulsive stochastic fractional differential equations (ISFDEs) driven by a standard Brownian motion and an independent fractional Brownian motion (fBm) of the form:
3 Main results we present the existence and uniqueness of solutions to Eq (1)

Summary

Introduction

A dominant interest in practical applications is the existence of solutions to deterministic fractional differential equations and fractional stochastic differential equations (FSDEs) driven by Brownain motion due to their role for helping candidates explore the hidden properties of the dynamics of complex systems in viscoelasticity, diffusion, mechanics, electromagnetism, control, signal processing, and physics. Benchaabane and Sakthivel [8] used the fractional calculus, semigroup theory, and stochastic analysis techniques to obtain the unique mild solution for a class of nonlinear fractional Sobolev-type SDEs with non-Lipschitz coefficients in Hilbert spaces under a new set of sufficient conditions. The specific objective of this study is to prove the existence and uniqueness of solutions to the following impulsive stochastic fractional differential equations (ISFDEs) driven by a standard Brownian motion and an independent fBm of the form:. We give our main results on the existence and uniqueness theorem for ISFDEs driven by a standard Brownian motion and an independent fBm given by (1) followed by some remarks and corollaries in Sect.

Preliminaries

Ij Xn tj – n