  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.
