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

Abstract

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. We model the state evolution as a stochastic process driven by Brownian motion. Under this model, the state 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. This description is refined by incorporating observations. More formally, we seek the conditional distribution of the state given 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. In numerical studies of bioreactors, the well-posedness of these differential equations is frequently assumed rather than verified. Consequently, it remains unclear whether unique solutions exist and whether numerical approximations preserve key qualitative properties, such as positivity (e.g. non-negative chemical concentrations). This paper establishes the existence and uniqueness of solutions, ensuring well-posedness before presenting numerical approximations. Furthermore, we approximate the pathwise solution to the filtering problem by three approaches: a PDE-based approximation combining finite difference and splitting methods, a classical sequential Monte Carlo method, and a linearisation method. Our results show that the first two provide a richer probabilistic description of the state than the linearisation method, making them suitable for bioreactor control architectures.