Successive Approximation of Neutral Functional Stochastic Differential Equations in Hilbert Spaces

Brahim Boufoussi,Salah Hajji

doi:10.5802/ambp.282

Abstract

By using successive approximation, we prove existence and uniqueness result for a class of neutral functional stochastic differential equations in Hilbert spaces with non-Lipschitzian coefficients

Highlights

The purpose of this paper is to prove the existence and uniqueness of mild solutions for a class of neutral functional stochastic differential equations (FSDEs) described in the form d[x(t) + g(t, xt )] = [Ax(t) + f (t, xt )]dt + σ(t, xt )dW (t), 0 ≤ t ≤ T, x(t) = φ(t), −r ≤ t ≤ 0
Where A is the infinitesimal generator of an analytic semigroup of bounded linear operators, (T (t))t≥0, in a Hilbert space H; xt ∈ Cr = C([−r, 0], H)
In the infinite-dimensional Hilbert space, only a few results have been obtained in this field despite the importance and interest of the model (1.1)

Summary

Introduction

The purpose of this paper is to prove the existence and uniqueness of mild solutions for a class of neutral functional stochastic differential equations (FSDEs) described in the form d[x(t) + g(t, xt )] = [Ax(t) + f (t, xt )]dt + σ(t, xt )dW (t), 0 ≤ t ≤ T, x(t) = φ(t), −r ≤ t ≤ 0. In the infinite-dimensional Hilbert space, only a few results have been obtained in this field despite the importance and interest of the model (1.1) In this respect, it is worth mentioning that this kind of neutral equation arises from problems related to coupled oscillators in a noisy environment, or in problems of viscoeslastic materials under random or stochastic influences (see [15] for a description of these problems in the deterministic case).

Preliminaries

Findings

The main result