Asymptotic error distributions of the Euler method for continuous-time nonlinear filtering

TL;DR

This paper establishes the asymptotic error distribution of the Euler method in continuous-time nonlinear filtering, providing a rigorous probabilistic characterization of the approximation errors.

Contribution

It proves the stable convergence of conditional expectations of stochastic integrals, advancing the theoretical understanding of Euler method errors in filtering.

Findings

Error distribution characterized by stable convergence of conditional expectations

Application of martingale limit theorems to filtering errors

Provides a rigorous probabilistic framework for Euler method errors

Abstract

We deduce the asymptotic error distribution of the Euler method for the nonlinear filtering problem with continuous-time observations. Previous works by several authors have shown that the error structure of the method is characterized by conditional expectations of some functionals of multiple stochastic integrals. Our main result is a proof of the stable convergence of a sequence of such conditional expectations, using the technique of martingale limit theorems in the spirit of Jacod.