Exponential Ergodicity and β-Mixing Property for Generalized Ornstein-Uhlenbeck Processes

Oesook Lee

doi:10.4236/tel.2012.21004

Abstract

The generalized Ornstein-Uhlenbeck process is derived from a bivariate Levy process and is suggested as a continuous time version of a stochastic recurrence equation [1]. In this paper we consider the generalized Ornstein-Uhlenbeck process and provide sufficient conditions under which the process is exponentially ergodic and hence holds the expo-nentially β-mixing property. Our results can cover a wide variety of areas by selecting suitable Levy processes and be used as fundamental tools for statistical analysis concerning the processes. Well known stochastic volatility models in finance such as Levy-driven Ornstein-Uhlenbeck process is examined as a special case.

Highlights

Many continuous time processes are suggested and studied as a natural continuous time generalization of a random recurrence equation, for example, diffusion model of Nelson [2], continuous time GARCH (COGARCH)(1,1) process of Klüppelberg et al [3] and Lévy-drivenOrnstein-Uhlenbeck (OU) process of Barndorff-Nielsen and Shephard [4] etc
The generalized Ornstein-Uhlenbeck process is derived from a bivariate Lévy process and is suggested as a continuous time version of a stochastic recurrence equation [1]
Well known stochastic volatility model in finance such as Lévy-driven Ornstein-Uhlenbeck process is examined as a special case

Summary

Introduction

Many continuous time processes are suggested and studied as a natural continuous time generalization of a random recurrence equation, for example, diffusion model of Nelson [2], continuous time GARCH (COGARCH). Lévy with processes are a class of continuous time independent and stationary increments and continuous in probability. Since Lévy processes t and t are semimartingales, stochastic integral in Equation (1) is well defined. The GOU process has recently attracted attention, especially in the financial modelling area such as option pricing, insurance and perpetuities, or risk theory. Stationarity, moment condition and autocovariance function of the GOU process are studied in Lindner and Maller [6]. Obtain the results for asymptotic behavior of extremes and sample autocovariance function of the GOU process. Mixing property of a stochastic process describes the temporal dependence in data and is used to prove consistency and asymptotic normality of estimators. Sato [13] for basic results and representations concerning Lévy processes

The Model

Drift Condition for Vnh n 0

Simultaneous -Irreducibility of Vt t 0

Exponential Ergodicity of Vt t 0

Examples

Conclusion