Parameter estimation for a type of nonlinear stochastic models observed with error

摘要

An improved filtering method is provided to estimate the parameter for a type of nonlinear multivariate stochastic differential equations (SDEs) with multiplicative noise, when discrete observations contaminated with measurement error are given. First, a transformation is used to transform the diffusion terms of the SDEs into unit diffusion such that the improved filtering method can be used. After the transformation, the drift terms of the SDEs are local linearized by means of It? formula rather than Taylor expansion, and the predictions of the innovation estimators are approximated by a more rigorous theoretical form which guarantees that the improved method works well even when the Jacobian matrix of the drift terms is singular or ill-conditioned. The parameter is estimated from discrete observations by maximum likelihood technique. The improved method is compared to an existing software tool CTSM by estimating Van der Pol's random oscillation with unobserved state variables, the provided method proves to be robust particularly when observation noise is relatively large. Applying the improved method to modified stochastic Lotka–Volterra equations with multiplicative noise, where the performance of the linear approximation by It? formula and Taylor expansion is compared, in conclusion the provided method has better performance especially under long observation time interval.