A high order time discretization of the solution of the non-linear filtering problem

TL;DR

This paper introduces high-order discretization schemes for the nonlinear filtering problem, enabling more accurate numerical approximations by generalizing classical first-order methods to arbitrary higher orders.

Contribution

It develops a class of discretization schemes with convergence rates proportional to the mesh size raised to any power, extending Picard's first-order approach to arbitrary order.

Findings

Discretization schemes with arbitrary order of convergence.

Generalization of classical first-order filtering discretizations.

Facilitates high-order numerical solutions for filtering problems.

Abstract

The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the $m$-power of the mesh of the partition for arbitrary $m\in\mathbb{N}$. The result paves the way for constructing high order numerical approximation for the solution of the filtering problem.