Ornstein-Uhlenbeck Processes of Bounded Variation

Abstract

Ornstein-Uhlenbeck process of bounded variation is introduced as a solution of an analogue of the Langevin equation with an integrated telegraph process replacing a Brownian motion. There is an interval I such that the process starting from the internal point of I always remains within I. Starting outside, this process a. s. reaches this interval in a finite time. The distribution of the time for which the process falls into this interval is obtained explicitly. The certain formulae for the mean and the variance of this process are obtained on the basis of the joint distribution of the telegraph process and its integrated copy. Under Kac's rescaling, the limit process is identified as the classical Ornstein-Uhlenbeck process.

First-passage probabilities and invariant distributions of Kac-Ornstein-Uhlenbeck processes

“…in a finite time. Moreover, once caught, the process remains there forever, see [26]. In this regard, we study the first passage through the threshold y, ρ 0 < y < ρ 1 .…”

“…The study of Markov-modulated Ornstein-Uhlenbeck processes has recently begun, first in [11,36], dealing with the transient behaviour of moments and some specific scaling of parameters, and then in [26][27][28] in terms of first passage distributions and with neural modelling applications.…”

“…Recently, some results have appeared on the Markov-modulated Ornstein-Uhlenbeck processes, see [11,[26][27][28]36]. This approach assumes that equation ( 1) is modified by Markov oscillation of all parameters.…”

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

“…Remark 4.4 If = 1, i.e. λ = µ, making use of (35) we can see that the quantities provided in Proposition 4.3 become respectively…”

“…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.…”