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

Hiromasa Nishikawa, Tetsuya Kawai, Masayuki Uchida

TL;DR

The paper develops an adaptive quasi-maximum likelihood framework for multidimensional jump-diffusion processes observed at discrete times, addressing computational and numerical instability by decoupling diffusion from drift and jump estimation and introducing multi-threshold filters. It proves consistency and asymptotic normality of the adaptive estimators and constructs a quasi-likelihood ratio test with a chi-square limit under the null, proving consistency under alternatives. An independent-split extension and a concrete Lévy-OU example with Gaussian jumps illustrate the method, while simulations confirm the practical effectiveness and provide guidance on threshold choices. Overall, the approach yields computationally efficient, stable estimation and robust testing for complex jump-diffusion models with applications in finance and stochastic modeling.

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 $χ^2$-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.

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

TL;DR

The paper develops an adaptive quasi-maximum likelihood framework for multidimensional jump-diffusion processes observed at discrete times, addressing computational and numerical instability by decoupling diffusion from drift and jump estimation and introducing multi-threshold filters. It proves consistency and asymptotic normality of the adaptive estimators and constructs a quasi-likelihood ratio test with a chi-square limit under the null, proving consistency under alternatives. An independent-split extension and a concrete Lévy-OU example with Gaussian jumps illustrate the method, while simulations confirm the practical effectiveness and provide guidance on threshold choices. Overall, the approach yields computationally efficient, stable estimation and robust testing for complex jump-diffusion models with applications in finance and stochastic modeling.

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 -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.

Paper Structure

This paper contains 23 sections, 24 theorems, 274 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and assumptions
  3. Quasi-maximum likelihood estimation
  4. Joint and adaptive estimator
  5. The case in which the drift and jump parameter are split independently
  6. Quasi-likelihood ratio test
  7. Asymptotic distribution of test statistics
  8. Consistency of test
  9. The case in which the drift and jump parameter are split independently
  10. An example and simulation study
  11. Proofs
  12. Proofs of Chapter \ref{['sec4.3:m']}
  13. Preliminary results
  14. Proof of Theorem \ref{['thm1-2:jump']}
  15. Proof of Theorem \ref{['thm2-2:jump']}
  16. ...and 8 more sections

Key Result

Theorem 3.1

Assume [A1]-[A11],[B1], and either [$\textbf{C}_1 \textbf{1}$] or [$\textbf{C}_1 \textbf{2}$], which are the following: Then, $\check{\theta}_n \overset{P}{\longrightarrow} \theta_0$.

Figures (4)

  • Figure 1: QQ-plot of the simulated adaptive estimators. From left to right: the estimator for $\alpha,\beta,\lambda,\mu,\sigma^2$. From top to bottom: it is set that $\rho_1=\rho_2=\rho_3=0.255,0.26,0.265,0.27,0.275,0.28,0.285$. The solid line is $y=x$.
  • Figure 2: QQ-plot of the simulated adaptive test statistic with $\rho_1=0.285,\rho_2=0.26$, $\rho_3=0.255$, $\bar{\rho}_1$ and $\bar{\rho}_2$. From left to right: it is set that $\bar{\rho}_2=0.255,0.26,0.265,0.27,0.275$. From top to bottom: it is set that $\bar{\rho}_1=0.255,0.26,0.265,0.27,0.275,0.28,0.285$. The solid line is $y=x$.
  • Figure 3: QQ-plot of the simulated adaptive test statistic with $\rho_2=0.26,\rho_3=0.255,\rho_1=\bar{\rho}_1$ and $\bar{\rho}_2$. From left to right: it is set that $\bar{\rho}_2=0.255,0.26,0.265,0.27,0.275$. From top to bottom: it is set that $\rho_1=\bar{\rho}_1=0.255,0.26,0.265,0.27,0.275,0.28,0.285$. The solid line is $y=x$.
  • Figure 4: QQ-plot of the simulated adaptive test statistic with $\rho_1=0.285,\rho_2=\bar{\rho}_2$ and $\rho_3=\bar{\rho}_1$. From left to right: it is set that $\rho_2=\bar{\rho}_2=0.255,0.26,0.265,0.27,0.275$. From top to bottom: it is set that $\rho_3=\bar{\rho}_1=0.255,0.26,0.265,0.27,0.275,0.28,0.285$. The solid line is $y=x$.

Theorems & Definitions (54)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • Corollary 3.1
  • Remark 3.1
  • Theorem 3.2
  • Corollary 3.2
  • Remark 3.2
  • Remark 3.3
  • ...and 44 more