A Lagged Particle Filter for Stable Filtering of Certain High-Dimensional State-Space Models

We consider the problem of high-dimensional filtering state-space models (SSMs) at discrete times. This is particularly challenging as analytical solutions are typically not available and many numerical approximation methods can have a cost that scales exponentially with dimension hidden state. Inspired by lag-approximation for smoothing [G. Kitagawa S. Sato, Monte Carlo self-organising model, in Sequential Methods Practice, Springer, New York, 2001, pp. 178–195; J. Olsson et al., Bernoulli, 14 (2008), 155–179], we introduce lagged distribution necessarily biased. For certain classes SSMs, those forget initial condition fast time, bias our shown to be uniformly controlled small time. develop sequential (SMC) method recursively estimate expectations respect biased distributions. Moreover, prove class SSMs contain dependencies amongst coordinates achieve stable mean square error estimation, expectations, per unit where number simulated samples SMC algorithm. Our methodology implemented on several examples including conservative shallow-water model.