Efficient projection filter algorithm for stochastic dynamical systems with correlated noises and state-dependent measurement covariance

This paper focuses on deriving the projection filter equation for a class of stochastic differential equations that incorporate correlated state and measurement noises, where the measurement process covariances depend on the state. To effectively implement the projection filter algorithm for exponential families, it is crucial to compute not only the expectation and variance of the natural statistics but also higher-dimensional statistics. However, computing these high-dimensional statistics can be computationally intensive and potentially compromise the numerical stability of the projection filter. To tackle this challenge, this study proposes a method for the careful selection of natural statistics. We shows that, subject to specific technical conditions, it is feasible to compute all the required statistics by utilizing only partial differentiation of an approximated cumulant-generating function. Notably, this approach eliminates the need to increase the parameter dimension, which was previously required in Emzir et al. (2023).