Parameter Inference for Stochastic Differential Equations with Density Tracking by Quadrature

Abstract

We derive and experimentally test an algorithm for maximum likelihood estimation of parameters in stochastic differential equations (SDEs). Our innovation is to efficiently compute the transition densities that form the log likelihood and its gradient, and to then couple these computations with quasi-Newton optimization methods to obtain maximum likelihood estimates. We compute transition densities by applying quadrature to the Chapman–Kolmogorov equation associated with a time discretization of the original SDE. To study the properties of our algorithm, we run a series of tests involving both linear and nonlinear SDE. We show that our algorithm is capable of accurate inference, and that its performance depends in a logical way on problem and algorithm parameters.