Local Asymptotic Normality Property for Ornstein–Uhlenbeck Processes with Jumps Under Discrete Sampling

Abstract

We address the issue of the local asymptotic normality property and the Fisher information for three characterizing parameters of Ornstein–Uhlenbeck processes with jumps under low frequency and high frequency discrete sampling with expanding observation window. The martingale method with the Kolmogorov backward equation and the Malliavin calculus are employed to derive explicit formulas for derivatives of the likelihood ratio function in the form of conditional expectation, which serve as essential tools for justifying the passage to the limit by the dominated convergence theorem. This approach makes it possible to carry out the proof without specifying the law of the jump component and without knowing the tail behaviors of the transition probability density and, as a consequence, to keep various types of jump structure within the scope of this article. The Fisher information under high-frequency sampling is essentially identical to the one for purely Gaussian Ornstein–Uhlenbeck processes due to the dominance of the Gaussian component over the jump component in the short time framework.