An approximate maximum likelihood method for direct estimation of embedded parameters in nonlinear, multivariate stochastic differential equations using discrete-time input-output data encumbered with additive measurement noise is proposed. The stochastic differential equations act as the system equation of a continuous-discrete time state space model which is introduced to describe nonlinear, multivariate and quasi-stationary systems. The likelihood is formulated as a function of the embedded parameters of the stochastic state space model, and an Iterated Extended Kalman filter is used in evaluating the likelihood function. A transformation is introduced to remove level effects (state-dependent diffusion terms) in some multivariate SDEs such that the filtering problem may be solved using the IEKF. Monte Carlo simulation of a nonlinear predator-prey system is used to study the statistical properties of the proposed method.
Stochastic modelling; continuous time systems; Brownian motion; random processes; Extended Kalman filters; Maximum likelihood estimation