Quasi-likelihood ratio test for jump-diffusion processes based on adaptive maximum likelihood inference

TL;DR

This paper introduces an adaptive quasi-likelihood ratio test for multidimensional jump-diffusion processes, improving estimation stability and computational efficiency, with proven asymptotic properties and validated through simulations.

Contribution

It proposes a novel adaptive quasi-likelihood framework for jump-diffusion processes, enhancing parameter estimation and hypothesis testing with proven consistency and asymptotic normality.

Findings

Adaptive estimators are consistent and asymptotically normal.

The proposed test statistic converges to a chi-squared distribution.

Numerical simulations confirm the theoretical properties.

Abstract

In this paper, we consider parameter estimation and quasi-likelihood ratio tests for multidimensional jump-diffusion processes defined by stochastic differential equations. In general, simultaneous estimation faces challenges such as an increase of computational time for optimization and instability of estimation accuracy as the dimensionality of parameters grows. To address these issues, we propose an adaptive quasi-log likelihood function based on the joint quasi-log likelihood function introduced by Shimizu and Yoshida (2003, 2006) and Ogihara and Yoshida (2011). We then show that the resulting adaptive estimators possess consistency and asymptotic normality. Furthermore, we extend the joint quasi-log likelihood function proposed by Shimizu and Yoshida (2003, 2006) and Ogihara and Yoshida (2011) and construct a test statistic using the proposed adaptive estimators. We prove that the proposed test statistic converges in distribution to a $\chi^2$-distribution under the null hypothesis and that the associated test is consistent. Finally, we conduct numerical simulations using a specific jump-diffusion process model to examine the asymptotic behavior of the proposed adaptive estimators and test statistics.