Generalized fractional Ornstein-Uhlenbeck processes

TL;DR

This paper introduces the generalized fractional Ornstein-Uhlenbeck (GFOU) process, extending the classical FOU process by incorporating an exponential of a Lévy process, and explores its properties including stationarity and long memory.

Contribution

It proposes a new GFOU process combining Lévy processes and fractional Brownian motion, and analyzes its existence, stationarity, long memory, and differential equation characteristics.

Findings

GFOU process is well-defined with an existing integral.

The process exhibits stationarity and long memory.

It satisfies a specific stochastic differential equation.

Abstract

We introduce an extended version of the fractional Ornstein-Uhlenbeck (FOU) process where the integrand is replaced by the exponential of an independent L\'evy process. We call the process the generalized fractional Ornstein-Uhlenbeck (GFOU) process. Alternatively, the process can be constructed from a generalized Ornstein-Uhlenbeck (GOU) process using an independent fractional Brownian motion (FBM) as integrator. We show that the GFOU process is well-defined by checking the existence of the integral included in the process, and investigate its properties. It is proved that the process has a stationary version and exhibits long memory. We also find that the process satisfies a certain stochastic differential equation. Our underlying intention is to introduce long memory into the GOU process which has short memory without losing the possibility of jumps. Note that both FOU and GOU processes have found application in a variety of fields as useful alternatives to the Ornstein-Uhlenbeck (OU) process.