Discretization and Simulation of the Zakai Equation

Abstract

This paper is concerned with numerical approximations for stochastic partial differential Zakai equation of nonlinear filtering problem. The approximation scheme is based on the representation of the solutions as weighted conditional distributions. We first accurately analyse the error caused by an Euler type scheme of time discretization. Sharp error bounds are calculated: we show that the rate of convergence is in general of order √ δ (δ is the time step), but in the case when there is no correlation between the signal and the observation for the Zakai equation, the order of convergence becomes δ. This result is obtained by carefully employing techniques of Malliavin calculus. In a second step, we propose a simulation of the time discretization Euler scheme by a quantization approach. This formally consists in an approximation of the weighted conditional distribution by a conditional discrete distribution on finite supports. We provide error bounds and rate of convergence in terms of the number N of the grids of this support. These errors are minimal at some optimal grids which are computed by a recursive method based on Monte Carlo simulations. Finally, we illustrate our results with some numerical experiments arising from correlated Kalman-Bucy filter.

A Multistage Wiener Chaos Expansion Method for Stochastic Advection-Diffusion-Reaction Equations

“…In [5,6] an SPDE is first discretized in time and then a finite element or finite difference method can be applied to this semidiscretization. Other numerical approaches include those making use of splitting techniques [2,17,12], quantization [9], or an approach based on the averaging-over-characteristic formula [26,27]. In [22,19] numerical algorithms based on the Wiener chaos expansion (WCE) were introduced for solving the nonlinear filtering problem for hidden Markov models.…”

Sparse Wiener Chaos approximations of nonlinear filtering with correlated noise

“…Another advantage compared with the previous works [10,11,12,13] is that the presents a unified treatment of both noise-correlated and noise-uncorrelated problems with possibly degenerate diffusion in the unobserved component.…”

“…Such applications require filtering algorithms with fast on line computations. Because of the large amount of calculations, many of the existing numerical schemes [10,11,12] cannot be implemented in real time when the dimension of the state process is more than three.…”

Introduction to vector quantization and its applications for numerics

“…It is also a challenging problem for the implementation of numerical schemes for Backward Stochastic Differential Equations (see [2,3]), Stochastic PDEs (see [32]), for non-linear filtering [57,68] or Stochastic Control Problems (see [13,14,58]). Further references are available in the survey paper [62] devoted to applications of optimal vector quantization to Numerical Probability.…”