An efficient Monte Carlo scheme for Zakai equations

TL;DR

This paper introduces a Monte Carlo method for solving high-dimensional Zakai equations by transforming them into random PDEs, enabling efficient approximation with good scalability up to 25 dimensions.

Contribution

The paper presents a novel approach transforming Zakai SPDEs into random PDEs, allowing the use of Feynman–Kac formula for efficient Monte Carlo solutions in high dimensions.

Findings

Effective in up to 25 dimensions

Fast computational run times

Accurate approximation of Zakai equations

Abstract

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.