Convergence Analysis of Splitting-Up Algorithm of the Zakai’s Equation with Correlated Noises

In the field of nonlinear filtering (NLF), it is well-known that the unnormalized conditional density of the states satisfies the Zakai’s equation. The splitting-up algorithm has been first studied in the independent noises case by Bensoussan, et al. (1990). In this paper, the authors extend this convergence analysis of the splitting-up algorithm to the correlated noises’ case. Given a time discretization, one splits the solution of the Zakai’s equation into two interlacing processes (with possibly computational advantage). These two processes correspond respectively to the prediction and updating. Under certain conditions, the authors show that both processes tend to the solution of the Zakai’s equation, as the time step goes to zero. The authors specify the conditions imposed on the way of splitting-up to guarantee the convergence. The major technical difficulty in the correlated noises’ case, compared with the independent case, is to control the gradient of the second process in some sense. To illustrate the potentially computational advantage of the schemes based on the splitting-up ways, the authors experiment on a toy NLF model using the feedback particle filter (FPF) developed based on the splitting-up method and the sampling importance and resampling (SIR) as comparison. The FPF outperforms in both accuracy and efficiency.