Discretization and Simulation of the Zakai Equation

This paper is concerned with numerical approximations for a class of nonlinear stochastic partial differential equations: Zakai equation of nonlinear filtering problem and McKean-Vlasov type equations. The approximation scheme is based on the re\-pre\-sentation 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 $\sqrt{\delta}$ ( $\delta$ 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 $\delta$. 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 and Burgers equation.

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