Abstract

This talk studies the filtering of a partially-observed multidimensional diffusion process using the principle of least action, equivalently, maximum-likelihood estimation. We show how the most likely path of the unobserved part of the diffusion can be determined by solving a shooting ODE , and then we go on to study the (approximate) conditional distribution of the diffusion around the most likely path; this turns out to be a zero-mean Gaussian process which solves a linear SDE whose time-dependent coefficients can be identified by solving a first-order ODE with an initial condition. This calculation of the conditional distribution can be used as a way to guide SMC methods to search relevant parts of the state space, which may be valuable in high-dimensional problems, where SMC struggles; in contrast, ODE solution methods continue to work well even in moderately large dimension.