No. 125 - On the estimation of stochastic differential equations: the continuous-time maximum-likelihood approach

Thanks to modern information technology, economic and in particular financial phenomena tend to be systematically observed at increased time frequency (weekly, daily, hourly etc.). This paper investigates the limit case of a continuous record of observations, presenting a general maximum-likelihood method for estimating the parameters of stochastic differential equations (SDEs) of observable point variables. This method, whose foundations can be found in the probabilistic literature, has the advantage of avoiding the main difficulties met by the discrete-time approach in the case of nonlinear and/or multivariate SDEs. In this paper, its properties are presented and some examples, covering a large part of the recent financial literature, are worked out. The case of discrete observations is considered, suggesting the importance of Monte Carlo experiments to evaluate the alternative approaches particularly in presence of nonstationary time series.