Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications

Abstract

We consider a fractional Ornstein-Uhlenbeck process involving a stochastic forcing term in the drift, as a solution of a linear stochastic differential equation driven by a fractional Brownian motion. For such process we specify mean and covariance functions, concentrating on their asymptotic behavior. This gives us a sort of short-or long-range dependence, under specified hypotheses on the covariance of the forcing process. Applications of this process in neuronal modeling are discussed, providing an example of a stochastic forcing term as a linear combination of Heaviside functions with random center. Simulation algorithms for the sample path of this process are finally given.

Continuous-time multi-type Ehrenfest model and related Ornstein-Uhlenbeck diffusion on a star graph

“…In particular, the Ornstein-Uhlenbeck process is often used as it provides a fruitful compromise between the need to describe the dynamics of phenomena subject to fluctuations in the presence of an equilibrium point and the opportunity to have closed-form expressions of interest in applications, such as transition density and first-passage-time density through the equilibrium point. For instance, the recent papers by Ascione et al [3], Hongler and Filliger [24] and Ratanov [35] deal with suitable generalizations of the Ornstein-Uhlenbeck process. In various contexts, such as queueing and mathematical neurobiology, generalized Ornstein-Uhlenbeck processes arise trough a scaling of continuous-time processes on a discrete state space.…”

On the Construction of Some Fractional Stochastic Gompertz Models

“…Thus, we have that our equation is a perturbation of (17) with a fractional white noise. For ν = 1, we obtain the fractional Ornstein-Uhlenbeck process ( [39,40]).…”

Time-changed fractional Ornstein-Uhlenbeck process

“…Let us first study the uniform convergence of V 2,H . Let us recall that, as shown in [4], the function (H, t)…”

“…Such time-changed fOU process generally admits a probability density function, depending on both the Hurst index of the parent fOU process and the Bernstein function representing the Laplace exponent of the involved subordinator. Thus, as it is known that the fOU process converges towards the OU process as H → 1/2 (actually, the covariance of the fOU is a continuous function with respect to the Hurst index, as shown in [4]), a natural question that arises is linked to the behaviour of the density of the time-changed fOU process with respect to the Hurst index.…”