A Particle Filter Method of Inference for Stochastic Differential Equations

Abstract

Stochastic Differential Equations (SDE) serve as an extremely useful modelling tool in areas including ecology, finance, population dynamics, and physics. Yet, parameter inference for SDEs is notoriously difficult due to the intractability of the likelihood function. A common approach is to approximate the likelihood by way of data augmentation, then integrate over the latent variables using particle filtering techniques. In the Bayesian setting, the particle filter is typically combined with various Markov chain Monte Carlo (MCMC) techniques to sample from the parameter posterior. However, MCMC can be excessive when this posterior is well-approximated by a normal distribution, in which case estimating the posterior mean and variance by stochastic optimization presents a much faster alternative. This thesis explores this latter approach. Specifically, we use a particle filter tailored to SDE models and consider various methods for approximating the gradient and hessian of the parameter log-posterior. Empirical results for several SDE models are presented.