Maximum likelihood estimation of diffusions by continuous time Markov chain

Abstract

A novel method is presented for estimating the parameters of a parametric diffusion process. The 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 pseudo-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, closed-form approximations are obtained 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 CTMC estimator. Simulated examples are provided in addition to real data experiments with FX rates and constant maturity interest rates.