An Approach to Stochastic Integration for Fractional Brownian Motion in a Hilbert Space

On time-dependent neutral stochastic evolution equations with a fractional Brownian motion and infinite delays

In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory

Semilinear Stochastic Equations in a Hilbert Space with a Fractional Brownian Motion

The solutions of a family of semilinear stochastic equations in a Hilbert space with a fractional Brownian motion are investigated. The nonlinear term in these equations has primarily only a growth condition assumption. An arbitrary member of the family of fractional Brownian motions can be used in these equations. Existence and uniqueness for both weak and mild solutions are obtained for some ...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

Functional differential equations driven by a fractional Brownian motion

In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by a fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1), in a separable real Hilbert space. We prove an existence and uniqueness result of mild solution by means of the Banach fixed point principle. A practical example is provided to illustrate the viabil...

Small Deviations of Weighted Fractional Processes and Average Non–linear Approximation

We investigate the small deviation problem for weighted fractional Brownian motions in Lq–norm, 1 ≤ q ≤ ∞. Let BH be a fractional Brownian motion with Hurst index 0 < H < 1. If 1/r := H + 1/q, then our main result asserts lim ε→0 ε log P (∥∥∥ρBH∥∥∥ Lq(0,∞) < ε ) = −c(H, q) · ‖ρ‖ Lr(0,∞) , provided the weight function ρ satisfies a condition slightly stronger than the r– integrability. Thus we e...