Explicit correction formulae for parametric identification of stochastic differential systems

In this paper we consider the use of maximum likelihood estimators (MLEs) in the situation where the data come from a “smooth” dynamical system which is approximated by an Itô equation. It is well known that in such situation correction terms may appear already in the equation. Independently of this fact (in particular also when a correction term of a drift coefficient is vanishing) in most cases additional corrections of formulas for estimators of parameters of equation are necessary. The importance of the correction under consideration has been noticed and repeatedly indicated in several publications, but in most papers devoted to the estimation of parameters of Itô differential models it was not taken into account. The form of the correction terms seems to be known merely for linear systems. In this paper a general approach, based upon a known result of Mc Shane's, is employed to prove an appropriate convergence theorem concerning two types of integrals appearing in the formulas of MLEs and the explicit forms of correction terms are derived for nonlinear systems provided that the parameters enter the drift term linearly. The necessity of using corrections is underlined by an extremely simple example in which the neglection of correction terms leads to an infinite relative error of estimation. This theoretically established fact is also observed on the basis of numerical simulation. The second example, connected with a nonlinear stochastic oscillator, illustrates the effectiveness of estimation in presence of corrections in a practically important case (it also enlightens some earlier numerical experiences described in the literature).