Estimating parameters in diffusion processes using an approximate maximum likelihood approach

We present an approximate Maximum Likelihood estimator for univariate Itô stochastic differential equations driven by Brownian motion, based on numerical calculation of the likelihood function. The transition probability density of a stochastic differential equation is given by the Kolmogorov forward equation, known as the Fokker–Planck equation. This partial differential equation can only be solved analytically for a limited number of models, which is the reason for applying numerical methods based on higher order finite differences. The approximate likelihood converges to the true likelihood, both theoretically and in our simulations, implying that the estimator has many nice properties. The estimator is evaluated on simulated data from the Cox–Ingersoll–Ross model and a non-linear extension of the Chan–Karolyi–Longstaff–Sanders model. The estimates are similar to the Maximum Likelihood estimates when these can be calculated and converge to the true Maximum Likelihood estimates as the accuracy of the numerical scheme is increased. The estimator is also compared to two benchmarks; a simulation-based estimator and a Crank–Nicholson scheme applied to the Fokker–Planck equation, and the proposed estimator is still competitive.