On a class of constrained Bayesian filters and their numerical implementation in high-dimensional state-space Markov models

Bayesian filtering is a key tool in many problems that involve the online processing of data, including data assimilation, optimal control, nonlinear tracking and others. Unfortunately, the implementation of filters for nonlinear, possibly high-dimensional, dynamical systems is far from straightforward, as computational methods have to meet a delicate trade-off involving stability, accuracy and computational cost. In this paper we investigate the design, and theoretical features, of constrained Bayesian filters for state space models. The constraint on the filter is given by a sequence of compact subsets of the state space that determines the sources and targets of the Markov transition kernels in the dynamical model. Subject to such constraints, we provide sufficient conditions for filter stability and approximation error rates with respect to the original (unconstrained) Bayesian filter. Then, we look specifically into the implementation of constrained filters in a continuous-discrete setting where the state of the system is a continuous-time stochastic Itô process but data are collected sequentially over a time grid. We propose an implementation of the constraint that relies on a data-driven modification of the drift of the Itô process using barrier functions, and discuss the relation of this scheme with methods based on the Doob $h$-transform. Finally, we illustrate the theoretical results and the performance of the proposed methods in computer experiments for a partially-observed stochastic Lorenz 96 model.