Existence, uniqueness and stability of impulsive stochastic neutral functional differential equations driven by Rosenblatt process with varying-time delays

Abstract

Hermite processes are self-similar processes with stationary increments; the Hermite process of order 1 is fractional Brownian motion (fBm) and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of impulsive neutral stochastic functional differential equations with variable delays driven by Rosenblatt process with index {H\in(\frac{1}{2},1)} , which is a special case of a self-similar process with long-range dependence. More precisely, we prove an existence and uniqueness result, and we establish some conditions, ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point theory. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.

Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process

“…it allows to write, thanks to inequality (28), the boundedness of the function (60) and inequality ( 33)…”

Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process

“…For the last fifteen years the Rosenblatt process has received a significantly increasing interest in both theoretical and practical lines of research. Due to its self-similarity, its applications are numerous across a multitude of fields, including internet traffic [12] and turbulence [40,28]. From a statistical point of view, estimating the value of the Hurst index H is important for practical applications and various estimators exist, see [8,43].…”

Existence of solutions for non-autonomous second-order stochastic inclusions with Clarke’s subdifferential and non instantaneous impulses

“…Lakhel and Tlidi [36] employed the Banach fixed point theorem to discuss the existence, uniqueness, and established stability criteria for a neutral stochastic functional differential system with impulses involving variable delays governed by the Rosenblatt process.…”