Approximations to the solution of the Kushner-Stratonovich equation for the stochastic chemostat

This paper concerns the inverse problem of characterising the state of a bioreactor from observations. In laboratory settings, the bioreactor is represented by a device called a chemostat. We consider a differential description of the evolution of the state of the chemostat under environmental fluctuations. First, we model the state evolution as a stochastic process driven by Brownian motion. Under this model, our best knowledge about the state of the chemostat is described by its probability distribution in time, given the distribution of the initial state. The corresponding probability density function solves a deterministic partial differential equation (PDE), the Kolmogorov forward equation. While this provides a probabilistic description, incorporating an observation process allows for a more refined characterisation of the state. More formally, we are interested in obtaining the distribution of the state conditional on an observation process as the solution to a filtering problem, with the corresponding conditional probability density function solving a non-linear stochastic PDE, the Kushner-Stratonovich equation. This paper focuses on the pathwise formulation of this filtering problem in which inferences about the state are obtained conditional on a fixed stream of observations. We establish the existence and uniqueness of solutions to the governing differential equations, ensuring well-posedness before presenting numerical approximations. We approximate the pathwise solution to the filtering problem by combining the finite difference and splitting methods for solving PDEs, and then compare the approximated solution with results from a linearisation method and a classical sequential Monte Carlo method.