Abstract
In this paper we investigate the existence, uniqueness, asymptotic behavior of mild solutions to neutral stochastic differential equations with delays driven by a fractional Brownian motion in a Hilbert space. The cases of finite and infinite delays are analyzed.
Highlights
The theory of stochastic differential equations driven by a fractional Brownian motion has been studied intensively in recent years [1], [2], [3], [4], and [5]
In this paper we investigate the existence, uniqueness, asymptotic behavior of mild solutions to neutral stochastic differential equations with delays driven by a fractional Brownian motion in a Hilbert space
Maslowski and Nualart [12] have studied the existence and uniqueness of a mild solution for nonlinear stochastic evolution equations in a Hilbert space driven by a cylindrical fractional Brownian motion (fBm) under some regularity and boundedness conditions on the coefficients
Summary
The theory of stochastic differential equations driven by a fractional Brownian motion (fBm) has been studied intensively in recent years [1], [2], [3], [4], and [5]. It behaves completely in a different way than the standard Brownian motion, in particular neither is a semimartingale nor a Markov process It is a self-similar process with stationary increments and has a long-memory when. The existence and uniqueness of mild solutions for a class of stochastic differential equations in a Hilbert space with a standard, cylindrical fBm with the Hurst parameter in the interval has been studied [11]. Caraballo and et al [13] investigated the existence and uniqueness of mild solutions to stochastic delay equations driven by fBm with Hurst parameter. . An existence and uniqueness result of mild solutions for a class of neutral stochastic differential equation with finite delay, driven by an fBm in a Hilbert space has been investigated [14] in Boufoussi and Hajji. The main tool of this paper is the fixed point theory which was proposed by Burton [17]
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