Maximum Likelihood Estimation of Generalized Itô Processes with Discretely Sampled Data

This paper considers the parametric estimation problem for continuous-time stochastic processes described by first-order nonlinear stochastic differential equations of the generalized Itô type (containing both jump and diffusion components). We derive a particular functional partial differential equation which characterizes the exact likelihood function of a discretely sampled Itô process. In addition, we show by a simple counterexample that the common approach of estimating parameters of an Itô process by applying maximum likelihood to a discretization of the stochastic differential equation does not yield consistent estimators.