Existence and uniqueness of solutions for stochastic differential equations of fractional-order q > 1 with finite delays

Xianmin Zhang,Yajun Zhu,Fang You,Praveen Agarwal,Zuohua Liu,Hui Peng

doi:10.1186/s13662-017-1169-3

Abstract

This paper is concerned with stochastic differential equations of fractional-order q in(m-1, m) (where m in mathbb{Z} and m geq 2) with finite delay at a space BC ( [ - tau, 0];R^{d} ). Some sufficient conditions are obtained for the existence and uniqueness of solutions for these stochastic fractional differential systems by applying the Picard iterations method and the generalized Gronwall inequality.

Highlights

Stochastic differential equations are valuable tools for description of some systems and processes with stochastic disturbances in many fields of science and engineering
Some results of the existence of solutions were obtained for some stochastic differential equations in [ – ], and the exponential stability was considered for a kind of impulsive neutral stochastic partial differential equations in [ ]
The existence of mild solutions was addressed for a class of fractional stochastic differential equations with impulses by the fixed point theorem in Hilbert spaces [ ]

Summary

Introduction

Stochastic differential equations are valuable tools for description of some systems and processes with stochastic disturbances in many fields of science and engineering. Some results of the existence of solutions were obtained for some stochastic differential equations in [ – ], and the exponential stability was considered for a kind of impulsive neutral stochastic partial differential equations in [ ]. The existence of solutions was considered for (impulsive) fractional differential equations in [ – ], and some progress was achieved in controls, stability, chaos synchronization, some other fractional derivatives and some new methods of numerical solutions etc. Motivated by the above mentioned works, we will first consider the existence of solution for a d-dimensional stochastic differential equation of fractional-order q ∈ ( , ) with finite delay, and consider broader stochastic differential equations of fractional order q ∈ (m – , m) (here m ∈ Z and m ≥ ) in the present paper.

Preliminaries

Main results