Approximations to the Solution of the Kushner-Stratonovich Equation for the Stochastic Chemostat*

Abstract

AbstractIn order to characterise the dynamics of a biochemical system such as the chemostat, we consider a differential description of the evolution of its state under environmental fluctuations. We present solutions to the filtering problem for a chemostat subjected to geometric Brownian motion. Under this modelling assumption, our best knowledge about the state of the system is given by its distribution in time, given the distribution of the initial state. Such a function solves a deterministic partial differential equation, the Kolmogorov forward equation. In this paper, however, we refine our knowledge about the state of the chemostat when additional information about the system is available in the form of measurements. More formally, we are interested in obtaining the distribution of the state conditional on measurements as the solution to a non-linear stochastic partial integral differential equation, the Kushner-Stratonovich equation. For the chemostat, this solution is not available in closed form, and it must be approximated. We present approximations to the solution to the Kushner-Stratonovich equation based on methods for partial differential equations. We compare the solution with a linearisation method and with a classical sequential Monte Carlo method known as the bootstrap particle filter.