In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman--Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.

翻译：本文提出了一种高效逼近高维Zakai方程解的数值方法。核心思想是将给定的Zakai随机偏微分方程（SPDE）转化为具有随机系数的偏微分方程（PDE）。我们证明，在Zakai方程系数满足适当正则性假设的条件下，相应的随机PDE存在一个解随机场，且对于随机系数的几乎所有实现，该解可表示为线性抛物型PDE的经典解。这使得能够应用Feynman-Kac公式，从而构建一种高效的蒙特卡罗方案来计算Zakai方程的近似解。该方法在高达25维的问题中取得了良好结果，且运行速度快。