Model verification for Lévy-driven Ornstein-Uhlenbeck processes

Abstract

Lévy-Driven Ornstein-Uhlenbeck (or CAR(1)) processes were introduced by Barndorff-Nielsen and Shephard [1] as a model for stochastic volatility. Pham [17] developed a general formula to recover the unobserved driving process from the continuously observed CAR(1) process. When the CAR(1) process is observed at discrete times $0$, $h$, $2h$, $...$ $[T/h]h$ the driving process must be approximated. Approximated increments of the driving process are used to test the assumption that the CAR(1) process is Lévy-driven. Asymptotic behavior of the test statistic is investigated. Performance of the test is illustrated through simulation.