Parameter estimation in stochastic differential equations: An overview

Abstract

This paper presents an overview of the progress of research on parameter estimation methods for stochastic differential equations (mostly in the sense of Itô calculus) over the period 1981–1999. These are considered both without measurement noise and with measurement noise, where the discretely observed stochastic differential equations are embedded in a continuous-discrete time state space model. Every attempts has been made to include results from other scientific disciplines. Maximum likelihood estimation of parameters in nonlinear stochastic differential equations is in general not possible due to the unavailability of closed form expressions for the transition and stationary probability density functions of the states. However, major developments are classified according to their approximation to the “true” maximum likelihood solution as opposed to a historical order of presentation.