Functional differential equations driven by a fractional Brownian motion

The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion

“…In Neuenkirch et al [16], using rough path theory, the authors prove existence and uniqueness of solutions to fractional equations with delays when H > 1/3. More recently, Ferrante and Rovira [5] established the existence and uniqueness of solutions to delayed SDEs with fBm for H > 1/2 and constant delay, by extending the results established in Nualart and Rȃşcanu [18], and the same sort of results have been shown recently for non-constant delay in Boufoussi and Hajji [2].…”

A note on the generation of random dynamical systems from fractional stochastic delay differential equations

“…Boufoussi and Hajji [4] proved the existence and uniqueness theorem for stochastic delay differential equations driven by fBm (fSDDE) in a finite dimensional space, and then extended the results for systems in a separable Hilbert space in Boufoussi et al [5,6]. They also proved that the solution of an fSDDE is continuous w.r.t.…”

“…In this paper, we follow the technique developed by Boufoussi et al [4] to study a class of fSDDE in which the coefficient functions are time independent. An important remark here is that, unlike for the SDDE case in which the usual phase space A note on the generation of random dynamical systems M = R × L 2 ([−r, 0], R d ) is a separable Banach space, in the context of fSDDE, the phase space is often a Hölder space of the form C 1−α ([−r, 0], R d ) and is therefore not separable.…”

Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space

“…Ferrante and Rovira (2010) studied the existence and convergence when the delay goes to zero by using the Riemann-Stieltjes integral. Using also the Riemann-Stieltjes integral, Boufoussi and Hajji (2011) and Boufoussi, Hajji, and Lakhel (2012) proved the existence and uniqueness of a mild solution and studied the dependence of the solution on the initial condition in finite and infinite dimensional space. studied the existence, uniqueness and asymptotic behavior of mild solutions for the neutral stochastic differential equation with finite delay.…”