Approximate discrete-time schemes for the estimation of diffusion processes from complete observations

TL;DR

This paper introduces a modified approximation method for estimating parameters of diffusion processes from discrete data, improving accuracy and bias reduction over traditional methods, especially with limited observations.

Contribution

It presents a convergent approximation to conditional moments for diffusion processes, enhancing quasi-maximum likelihood estimation accuracy from discrete observations.

Findings

New estimators outperform conventional ones in simulations

Estimators are asymptotically normal with decreasing bias

Performance improves as approximation error decreases

Abstract

In this paper, a modification of the conventional approximations to the quasi-maximum likelihood method is introduced for the parameter estimation of diffusion processes from discrete observations. This is based on a convergent approximation to the first two conditional moments of the diffusion process through discrete-time schemes. It is shown that, for finite samples, the resulting approximate estimators converge to the quasi-maximum likelihood one when the error between the discrete-time approximation and the diffusion process decreases. For an increasing number of observations, the approximate estimators are asymptotically normal distributed and their bias decreases when the mentioned error does it. A simulation study is provided to illustrate the performance of the new estimators. The results show that, with respect to the conventional approximate estimators, the new ones significantly enhance the parameter estimation of the test equations. The proposed estimators are intended for the recurrent practical situation where a nonlinear stochastic system should be identified from a reduced number of complete observations distant in time.