On the Convergence Analysis of Yau–Yau Nonlinear Filtering Algorithm: From a Probabilistic Perspective

Abstract. At the beginning of this century, a real time solution of the nonlinear filtering problem without memory was proposed in [S.-T. Yau and S. S.-T. Yau, Math. Res. Lett., 7 (2000), pp. 671–693], [S.-T. Yau and S. S.-T. Yau, SIAM J. Control Optim., 47 (2008), pp. 163–195] by the third author and his collaborator, and it was later referred to as the Yau–Yau algorithm. During the last two decades, a great many nonlinear filtering algorithms have been put forward and studied based on this framework. In this paper, we will generalize the results in the original works and conduct a novel convergence analysis of the Yau–Yau algorithm from a probabilistic perspective. Instead of considering a particular trajectory, we estimate the expectation of the approximation error and show that commonly used statistics of the conditional distribution (such as conditional mean and covariance matrix) can be accurately approximated with arbitrary precision by the Yau–Yau algorithm for general nonlinear filtering systems with very liberal assumptions. This novel probabilistic version of convergence analysis is more compatible with the development of modern stochastic control theory and will provide a more valuable theoretical guidance for practical implementations of the Yau–Yau algorithm.