Krein’s Spectral Theory and the Paley–wiener Expansion for Fractional Brownian Motion

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

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

Measure-preserving transformations of Volterra Gaussian processes and related bridges

We consider Volterra Gaussian processes on [0, T ], where T > 0 is a fixed time horizon. These are processes of type Xt = R t 0 zX(t, s)dWs, t ∈ [0, T ], where zX is a square-integrable kernel, and W is a standard Brownian motion. An example is fractional Brownian motion. By using classical techniques from operator theory, we derive measure-preserving transformations of X, and their inherently ...

A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4

We prove a change of variable formula for the 2D fractional Brownian motion of index H bigger or equal to 1/4. For H strictly bigger than 1/4, our formula coincides with that obtained by using the rough paths theory. For H = 1/4 (the more interesting case), there is an additional term that is a classical Wiener integral against an independent standard Brownian motion.

Series expansion of Wiener integrals via block pulse functions

In this paper, a suitable numerical method based on block pulse functions is introduced to approximate the Wiener integrals which the exact solution of them is not exist or it may be so hard to find their exact solutions. Furthermore, the error analysis of this method is given. Some numerical examples are provided which show that the approximation method has a good degree of accuracy. The main ...