Estimating the parameters of stochastic differential equations

Two maximum Likelihood methods for estimating the parameters of stochastic differential equations (SDEs) from time-series data are proposed. The first is that of simulated maximum Likelihood in which a nonparametric kernel is used to construct the transitional density of an SDE from a series of simu lated trials. The second approach uses a spectral technique to solve the Ko lmogorov equation satisfied by the transitional probability density. The ex act likelihood function for a geometric random walk is used as a benchmark against which the performance of each method is measured. Both methods perf orm well with the spectral method returning results which are practically i dentical to those derived from the exact likelihood. The technique is illus trated by modelling interest rates in the UK gilts market using a fundament al one-factor term-structure equation for the instantaneous rate of interes t. (C) 1999 IMACS/Elsevier Science B.V. All rights reserved.