Abstract
We focus on a class of neutral stochastic delay partial differential equations perturbed by a standard Brownian motion and a fractional Brownian motion. Under some suitable assumptions, the existence, uniqueness, and controllability results for these equations are investigated by means of the Banach fixed point method. Moreover, an example is presented to illustrate our main results.
Highlights
Fractional Brownian motion with Hurst parameter H ∈ (0, 1) is a centered Gaussian process {βH(t), t ≥ 0} which is often used to model many complex phenomena in applications when the systems contain rough external forcing
We focus on a class of neutral stochastic delay partial differential equations perturbed by a standard Brownian motion and a fractional Brownian motion
When H = 1/2, the fBm is the standard Brownian motion which is a Markov process and martingale, and we can use the classical Itotheory to construct a stochastic integration with respect to the standard Brownian motion
Summary
Fractional Brownian motion (fBm for short) with Hurst parameter H ∈ (0, 1) is a centered Gaussian process {βH(t), t ≥ 0} which is often used to model many complex phenomena in applications when the systems contain rough external forcing. Boufoussi and Hajji in [8] considered the existence and uniqueness problems of a class of neutral stochastic delay differential equations; that is, B ≡ 0 and g ≡ 0 in (1) by means of the Banach fixed point theory. Liu and Luo in [13] studied the existence and uniqueness problems of a wide class of neutral stochastic delay partial differential equations, that is, B ≡ 0 in (1) by means of the Banach fixed point theory.
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