On a Class of Measure-Dependent Stochastic Evolution Equations Driven by fBm

Abstract

We investigate a class of abstract stochastic evolution equations driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space. We establish the existence and uniqueness of a mild solution, a continuous dependence estimate, and various convergence and approximation results. Finally, the analysis of three examples is provided to illustrate the applicability of the general theory.

On a Class of Nonlocal Stochastic Functional Differential Equations with Infinite Delay

“…Hernandez et al [3] examined a class of stochastic evolution equations dependent a family of probability distributed measure of the solutions and established the existence and uniqueness of the solutions. [5,6] considered the stochastic differential equation of the form…”

“…These equations depend on all the sample and are named as the nonlocal stochastic differential equation. It is obvious that both Equation (1.3) and the equation in [3] contain Equation (1.2) as a special case. Although the so-called Efficient Market Hypothesis implies that all information available is already reflected in the present price of the stock and past stock performance gives no information, some statistical data of stock prices (see [1,9]) indicated the dependence on past behaviors.…”

“…It is clear that Equation (1.3) and the equation in [3] are more general than Equation (1.4), but it is difficult to test their conditions on the existence and uniqueness of solutions, for example, in [5,6], the conditions of the existence and uniqueness of solutions for Equation (1.3) contain the computation of the expectation which is difficult since the distributed function is generally unknown. In [3], the conditions of the existence and uniqueness are dependent on a metric function, which is also difficult to be computed. [3], this article will present some conditions under which the existence-and-uniqueness theorems are established.…”

Existence and stability results for neutral stochastic delay differential equations driven by a fractional Brownian motion

“…The fBm received much attention because of its huge range of potential applications in several fields like telecommunications, networks, finance markets, biology and so on [6], [7], [8], [9]. Moreover, one of the simplest stochastic processes that is Gaussian, self-similar, and has stationary increments is fBm [10].…”

“…Moreover, one of the simplest stochastic processes that is Gaussian, self-similar, and has stationary increments is fBm [10]. In particular, fBm is a generalization of the classical Brownian motion, which depends on a parameter called the Hurst index [9]. It should be mentioned that when , the stochastic process is a standard Brownian motion; when , it behaves completely in a different way than the standard Brownian motion, in particular neither is a semimartingale nor a Markov process.…”