A theoretical analysis of one-dimensional discrete generation ensemble kalman particle filters

Despite the widespread usage of discrete generation ensemble Kalman particle filtering methodology to solve nonlinear and high-dimensional fil-tering and inverse problems, little is known about their mathematical foun-dations. As genetic-type particle filters (a.k.a. sequential Monte Carlo), this ensemble-type methodology can also be interpreted as mean-field particle ap-proximations of the Kalman-Bucy filtering equation. In contrast with conven-tional mean-field type interacting particle methods equipped with a globally Lipschitz interacting drift-type function, Ensemble Kalman filters depend on a nonlinear and quadratic-type interaction function defined in terms of the sample covariance of the particles. Most of the literature in applied mathematics and computer science on these sophisticated interacting particle methods amounts to designing dif-ferent classes of useable observer-type particle methods. These methods are based on a variety of inconsistent but judicious ensemble auxiliary transfor-mations or include additional inflation/localisation-type algorithmic innova-tions, in order to avoid the inherent time-degeneracy of an insufficient particle ensemble size when solving a filtering problem with an unstable signal. To the best of our knowledge, the first and the only rigorous mathemati-cal analysis of these sophisticated discrete generation particle filters is devel-oped in the pioneering articles by Le Gland-Monbet-Tran and by Mandel- Cobb-Beezley, which were published in the early 2010s. Nevertheless, be-sides the fact that these studies prove the asymptotic consistency of the en-semble Kalman filter, they provide exceedingly pessimistic mean-error esti-mates that grow exponentially fast with respect to the time horizon, even for linear Gaussian filtering problems with stable one-dimensional signals. In the present article we develop a novel self-contained and complete stochastic perturbation analysis of the fluctuations, the stability, and the long-time performance of these discrete generation ensemble Kalman particle fil-ters, including time-uniform and nonasymptotic mean-error estimates that ap-ply to possibly unstable signals. To the best of our knowledge, these are the first results of this type in the literature on discrete generation particle fil-ters, including the class of genetic-type particle filters and discrete generation ensemble Kalman filters. The stochastic Riccati difference equations consid-ered in this work are also of interest in their own right, as a prototype of a new class of stochastic rational difference equation.