Closed-Form Likelihood Expansions for Multivariate Time-Inhomogeneous Diffusions

Abstract

The aim of this paper is to find approximate log-transition density functions for multivariate time-inhomogeneous diffusions in closed form. There are many empirical evidences that the underlying data generating processes for many economic variables might change over time. One possible way to explain the time-dependent behavior of state variables is to model the drift or volatility terms as functions of time t as well as state variables. Closed-form likelihood expansions for multivariate time-homogeneous diffusions have been obtained by Ait-Sahalia (2008). This research is built on his work and extends his results to time-inhomogeneous cases. Simulation study reveals that our method yields a very accurate approximate likelihood function that can be a good candidate when the true likelihood function is unavailable.likelihood function; multivariate time-inhomogeneous diffusion; reducible diffusions, irreducible diffusions