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On Finite Time Span Estimators of Parameters for Ornstein-Uhlenbeck Processes

Jun S. Han, Nino Kordzakhia

TL;DR

This work analyzes finite-time bias and mean-squared error for maximum likelihood estimators of the two-parameter mean-reverting model $(\lambda,\alpha)$, focusing on the OU process as a core example. It develops a likelihood-ratio framework and leverages change of measure and Ito calculus to derive explicit MLE representations and to compute bias and MSE, including through joint moment generating functions and Laplace transforms. The authors establish strong consistency and asymptotic efficiency for ergodic OU estimators, and they characterize distinct finite-time behavior in non-ergodic regimes, supported by numerical illustrations. The findings provide practical bias-correction insights and a pathway to robust finite-sample inference for mean-reverting diffusion models in finance and related fields.

Abstract

We study the bias and the mean-squared error of the maximum likelihood estimators (MLE) of parameters associated with a two-parameter mean-reverting process for a finite time $T$. Using the likelihood ratio process, we derive the expressions for MLEs, then compute the bias and the MSE via the change of measure and Ito's formula. We apply the derived expressions to the general Ornstein-Uhlenbeck process, where the bias and the MSE are numerically computed through a joint moment-generating function of key functionals of the O-U process. A numerical study is provided to illustrate the behaviour of bias and the MSE for the MLE of the mean-reverting speed parameter.

On Finite Time Span Estimators of Parameters for Ornstein-Uhlenbeck Processes

TL;DR

This work analyzes finite-time bias and mean-squared error for maximum likelihood estimators of the two-parameter mean-reverting model , focusing on the OU process as a core example. It develops a likelihood-ratio framework and leverages change of measure and Ito calculus to derive explicit MLE representations and to compute bias and MSE, including through joint moment generating functions and Laplace transforms. The authors establish strong consistency and asymptotic efficiency for ergodic OU estimators, and they characterize distinct finite-time behavior in non-ergodic regimes, supported by numerical illustrations. The findings provide practical bias-correction insights and a pathway to robust finite-sample inference for mean-reverting diffusion models in finance and related fields.

Abstract

We study the bias and the mean-squared error of the maximum likelihood estimators (MLE) of parameters associated with a two-parameter mean-reverting process for a finite time . Using the likelihood ratio process, we derive the expressions for MLEs, then compute the bias and the MSE via the change of measure and Ito's formula. We apply the derived expressions to the general Ornstein-Uhlenbeck process, where the bias and the MSE are numerically computed through a joint moment-generating function of key functionals of the O-U process. A numerical study is provided to illustrate the behaviour of bias and the MSE for the MLE of the mean-reverting speed parameter.

Paper Structure

This paper contains 15 sections, 82 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Maximum Likelihood Estimators
  3. Notations and Assumptions
  4. Maximum Likelihood Estimators
  5. Properties of MLEs
  6. The O-U Process and Results
  7. General Properties of O-U Process
  8. Properties of the MLE $\hat{\alpha}_{T}(\lambda)$ and $\bar{\alpha}_{T}$
  9. Properties of the MLE $\hat{\lambda}_{T}(\alpha)$
  10. Properties of the MLE $\bar{\lambda}_{T}=\hat{\lambda}_{T}(\bar{\alpha}_{T}).$
  11. Numerical Results
  12. Conclusion
  13. Derivation of Moment Generating Function $\psi$
  14. Derivation of Moment Generating Function $\Psi$
  15. Summary of Numerical Results as a Table

Figures (3)

  • Figure 1: Plots of bias of $\hat{\lambda}_{T}(\alpha)$ for cases when (i) $\alpha=-1$, (ii)$\alpha=0$; (iii)$\alpha=0.5$; (iv) $\alpha=1$ with different values of $\lambda$
  • Figure 2: Plots of mean-squared errors of $\hat{\lambda}_{T}(\alpha)$ for cases when (i) $\alpha=-1$, (ii)$\alpha=0$; (iii)$\alpha=0.5$; (iv) $\alpha=1$ with different values of $\lambda$
  • Figure 3: Plots of bias and mean-squared errors of $\hat{\lambda}_{T}(\alpha)$ for $\lambda=-1$ across different values of $\alpha$

Theorems & Definitions (2)

  • proof
  • proof