Quasi Ornstein-Uhlenbeck processes

TL;DR

This paper investigates stationary solutions to Langevin equations driven by stationary increment noise, providing explicit kernel expressions for certain cases and analyzing autocorrelation behaviors, with applications to Gaussian and Lévy-driven fractional Ornstein-Uhlenbeck processes.

Contribution

It identifies cases where explicit kernels can be derived for quasi Ornstein-Uhlenbeck processes and analyzes their autocorrelation properties, extending understanding of such stochastic processes.

Findings

Explicit kernel expressions for certain quasi Ornstein-Uhlenbeck processes.

Asymptotic behavior of autocorrelation functions analyzed.

Applications to Gaussian and Lévy-driven fractional processes.

Abstract

The question of existence and properties of stationary solutions to Langevin equations driven by noise processes with stationary increments is discussed, with particular focus on noise processes of pseudo-moving-average type. On account of the Wold-Karhunen decomposition theorem, such solutions are, in principle, representable as a moving average (plus a drift-like term) but the kernel in the moving average is generally not available in explicit form. A class of cases is determined where an explicit expression of the kernel can be given, and this is used to obtain information on the asymptotic behavior of the associated autocorrelation functions, both for small and large lags. Applications to Gaussian- and L\'{e}vy-driven fractional Ornstein-Uhlenbeck processes are presented. A Fubini theorem for L\'{e}vy bases is established as an element in the derivations.