Stochastic Equivalence of Gaussian Process to the Wiener Process, Brownian Bridge, Ornstein–Uhlenbeck Process

Abstract

Background. We consider the Gaussian process with zero expectation and following covariance function: \[R(s,t)=u(s)v(t), s\leq t.\] It was found the representation of the equivalent Wiener process for such process (Doob’s Transformation Theorem). We consider the representation of the Gaussian process via Wiener process, Brownian bridge and Orn­­stein–Uhlenbeck process in the case of monotonous function

Objective. The purpose of this paper is to find the criteria of equivalence between Gaussian process and Wiener process and to formulate similar criteria for Brownian bridge and Ornstein–Uhlenbeck process.

Methods. We constructed the system of the functional equations based on properties of Gaussian processes.

Results. Representation of Gaussian process with covariance function R(s, t) to equivalent Wiener process, Brownian bridge, Ornstein–Uhlenbeck process is discovered. Results are formulated in the form of criterion. Cases of decreasing and strictly increasing function u(t)/v(t) are considered.

Conclusions. The received outcomes can be used for research of functionals of the Gaussian processes. For example, to find the probability that Gaussian process crossing certain level. Representation of restriction of the Chentsov random field on polygonal line to equivalent Wiener process allowed finding the exact distribution of the maximum of the Chentsov random field on polygonal lines.