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Robust Inferential Methodology for Multidimensional Diffusion Processes

Sourojyoti Barick

Abstract

We investigate robust parameter estimation and testing procedure for multivariate diffusion processes observed at high frequency via the minimum density power divergence estimator (MDPDE). Within a general diffusion framework and under standard regularity conditions, we establish consistency and asymptotic normality for the estimators of both drift and diffusion parameters. The drift estimator converges at the $\sqrt{n h_n}$ rate, whereas the diffusion estimator attains the standard $\sqrt{n}$ rate, and the two estimators are shown to be asymptotically independent. The proposed methodology constitutes a robust alternative to quasi-likelihood and ordinary least squares based approaches, offering resilience against outliers, local contamination, and mild model misspecification, while remaining asymptotically equivalent to classical methods in the absence of contamination. Simulation studies demonstrate that the MDPDE achieves reliable finite-sample performance and enhanced numerical stability relative to likelihood-based estimators. These results underscore the practical relevance of divergence-based estimation for high-frequency diffusion models and point to natural extensions to more complex continuous-time settings.

Robust Inferential Methodology for Multidimensional Diffusion Processes

Abstract

We investigate robust parameter estimation and testing procedure for multivariate diffusion processes observed at high frequency via the minimum density power divergence estimator (MDPDE). Within a general diffusion framework and under standard regularity conditions, we establish consistency and asymptotic normality for the estimators of both drift and diffusion parameters. The drift estimator converges at the rate, whereas the diffusion estimator attains the standard rate, and the two estimators are shown to be asymptotically independent. The proposed methodology constitutes a robust alternative to quasi-likelihood and ordinary least squares based approaches, offering resilience against outliers, local contamination, and mild model misspecification, while remaining asymptotically equivalent to classical methods in the absence of contamination. Simulation studies demonstrate that the MDPDE achieves reliable finite-sample performance and enhanced numerical stability relative to likelihood-based estimators. These results underscore the practical relevance of divergence-based estimation for high-frequency diffusion models and point to natural extensions to more complex continuous-time settings.
Paper Structure (21 sections, 17 theorems, 228 equations, 5 tables)

This paper contains 21 sections, 17 theorems, 228 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Literature Survey
  3. MDPDE for Multidimensional Diffusion Process
  4. Minimum Density Power Divergence Estimator (MDPDE)
  5. Estimation and Main Results
  6. Simulation Study
  7. Conclusion
  8. Appendix
  9. Proof of Theorems
  10. Proof of Theorem \ref{['thm:mdpde-consistency']} (Consistency)
  11. Step 1:
  12. Step 2:
  13. Proof of Normality
  14. Proof of Theorem \ref{['thm:asymp-indep']} (Asymptotic Independence)
  15. Proof of Theorem \ref{['thm:joint-clt']} (Joint Normality)
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose that Assumptions assm:all_assumption(A1)--(A6) hold. Let $\alpha>0$ be fixed. If $h_n \to 0$, $n h_n \to \infty$ and $nh_n^q\to0$ for some $q>1$, then

Theorems & Definitions (35)

  • Remark 1
  • Example 1
  • Theorem 1: Consistency of parameters
  • Theorem 2: Asymptotic normality of the drift estimator
  • Corollary 1: Wald-type test for the drift parameter
  • Theorem 3: Asymptotic normality of the covariance estimator
  • Theorem 4: Asymptotic independence
  • Theorem 5: Joint asymptotic normality
  • Remark 2
  • proof
  • ...and 25 more