Statistical Inference for Lévy-Driven Ornstein-Uhlenbeck Processes

Abstract

When an Ornstein-Uhlenbeck (or CAR(1)) process is observed at discrete times 0, h, 2h,··· [T/h]h, the unobserved driving process can be approximated from the ob- served process. Approximated increments of the driving process are used to test the assumption that the process is L\'evy-driven. Asymptotic behavior of the test statis- tic at high sampling frequencies is developed assuming that the model parameters are known. The behavior of the test statistics using an estimated parameter is also studied. If it can be concluded that the driving process is L\'evy, the empirical process of the approximated increments can then be used to carry out more precise tests of goodness-of-fit. For example, one can test whether the driving process can be modeled as a Brownian motion or a gamma process. In each case, performance of the proposed test is illustrated through simulation.