This paper considers the parametric estimation problem for continuous-time stochastic processes described by first-order nonlinear stochastic differential equations of the generalized Ito 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 Ito process. In addition, we show by a simple of counterexample that the common approach of estimating parameters of an Ito process by applying maximum likelihood to a discretization of the stochastic differential equation does not yield consistent estimators.

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