LAN property for a linear model with jumps

TL;DR

This paper establishes the LAN property for a high-frequency observed linear jump model with unknown parameters, using advanced stochastic calculus techniques to derive asymptotic normality of the likelihood ratio.

Contribution

It introduces a novel application of Malliavin calculus and Girsanov's theorem to prove LAN for a jump-diffusion model with unknown parameters.

Findings

LAN property is proven for the model.

Likelihood ratio converges to a normal distribution.

Method can be applied to similar stochastic models.

Abstract

In this paper, we consider a linear model with jumps driven by a Brownian motion and a compensated Poisson process, whose drift and diffusion coefficients as well as its intensity are unknown parameters. Supposing that the process is observed discretely at high frequency we derive the local asymptotic normality (LAN) property. In order to obtain this result, Malliavin calculus and Girsanov's theorem are applied in order to write the log-likelihood ratio in terms of sums of conditional expectations, for which a central limit theorem for triangular arrays can be applied.