Abstract. We consider the numerical solution of the stochastic partial differential equation ∂u/∂t = ∂ 2 u/∂x 2 + σ(u)Ẇ (x, t), whereẆ is space-time white noise, using finite differences. For this equation Gyöngy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals ofẆ over a rectangular grid. We investigate the extent to which this order of convergence can be improved, and find that better approximations are possible for the case of additive noise (σ(u) = 1) if we wish to estimate space averages of the solution rather than pointwise estimates, or if we are permitted to generate other functionals of the noise. But for multiplicative noise (σ(u) = u) we show that no such improvements are possible.
We construct a sequence of branching particle systems Un convergent in distribution to the solution of the Zakai equation. The algorithm based on this result can be used to solve numerically the filtering problem. The result is an improvement of the one presented in a recent paper [Crisan and T. Lyons, Prob. Theory Related Fields, 109 (1997), pp. 217-244], because it eliminates the extra degree of randomness introduced there.
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