On a Neutral Itô and Arbitrary (Fractional) Orders Stochastic Differential Equation with Nonlocal Condition

Ahmed M A El-Sayed,Hoda A Fouad

doi:10.3390/fractalfract5040201

Abstract

In this paper, we are concerned with the combinations of the stochastic Itô-differential and the arbitrary (fractional) orders derivatives in a neutral differential equation with a stochastic, nonlinear, nonlocal integral condition. The existence of solutions will be proved. The sufficient conditions for the uniqueness of the solution will be given. The continuous dependence of the unique solution will be studied.

Highlights

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The existence and uniqueness of solutions to stochastic differential equations driven by the Winner Processes have been studied by many authors
The stochastic differential equations with nonlocal conditions and of fractional orders have been studied by some authors

Summary

Introduction

The existence and uniqueness of solutions to stochastic differential equations driven by the Winner Processes have been studied by many authors (see [1,2,3,4,5]). The stochastic differential equations with nonlocal conditions and of fractional orders have been studied by some authors (see, for example, [6,7,8] and references therein). We study the existence of solutions of an Itô and arbitrary (fractional) orders stochastic nonlinear differential equation with nonlocal integral conditions containing the involved Caputo fractional order derivative. In this paper we study the existence of solutions x ∈ C ( I, L2 (Ω)) of the Itô − arbitrary (fractional) orders stochastic nonlinear differential equation creativecommons.org/licenses/by/.

Integral Representations of the Solution

Existence Theorem

Uniqueness of the Solution

Continuous Dependence

Conclusions