Stochastic Volterra integro-differential equations driven by a fractional Brownian motion with delayed impulses

Xia Zhou, Xinzhi Liu, Shouming Zhong

In this paper, the problem of existence of mild solutions for a stochastic Volterra integro-differential equation with delayed impulses and driven by a fractional Brownian motion (Hurst parameter H ∈ (1/2 , 1)) is investigated. Here, we assume that the delayed impulses are linear and impulsive transients depend on not only their current but also historical states of the system. Utilizing the fixed point theorem combine with fractional power of operators and the semi-group theory, sufficient conditions that guarantee the existence and uniqueness of mild solutions for such equation are obtained. Finally, an example is presented to demonstrate the effectiveness of the proposed results.