Maximum Likelihood Estimation of Diffusions by Continuous Time Markov Chain

TL;DR

This paper introduces a new maximum likelihood estimation method for diffusion processes using a continuous-time Markov chain approximation, avoiding discretization errors and improving estimation accuracy in econometric applications.

Contribution

The paper presents a novel CTMC-based MLE approach for diffusion processes that eliminates time-discretization errors and provides closed-form likelihood approximations.

Findings

CTMC-based MLE outperforms traditional discretization methods.

Method is effective with infrequently sampled data.

Validated with simulated and real financial data.

Abstract

In this paper we present a novel method for estimating the parameters of a parametric diffusion processes. Our approach is based on a closed-form Maximum Likelihood estimator for an approximating Continuous Time Markov Chain (CTMC) of the diffusion process. Unlike typical time discretization approaches, such as psuedo-likelihood approximations with Shoji-Ozaki or Kessler's method, the CTMC approximation introduces no time-discretization error during parameter estimation, and is thus well-suited for typical econometric situations with infrequently sampled data. Due to the structure of the CTMC, we are able to obtain closed-form approximations for the sample likelihood which hold for general univariate diffusions. Comparisons of the state-discretization approach with approximate MLE (time-discretization) and Exact MLE (when applicable) demonstrate favorable performance of the CMTC estimator. Simulated examples are provided in addition to real data experiments with FX rates and constant maturity interest rates.