Stability of the optimal filter in continuous time: Beyond the bene{\v s} filter

TL;DR

This paper investigates the stability of the optimal filter in continuous time, extending beyond the Bene{ extendash}s filter, and introduces new techniques to establish the forgetting rate of initial conditions.

Contribution

It develops a novel approach to prove the stability of continuous-time optimal filters, including a new technique for handling the continuous setting.

Findings

The optimal filter's forgetting rate is at least a power of time t.

The method can be applied to analyze the stability of numerical approximations.

The approach extends stability results beyond the Bene{ extendash}s filter case.

Abstract

We are interested in the optimal filter in a continuous time setting. We want to show that the optimal filter is stable with respect to its initial condition. We reduce the problem to a discrete time setting and apply truncation techniques coming from [OR05]. Due to the continuous time setting, we need a new technique to solve the problem. In the end, we show that the forgetting rate is at least a power of the time t. The results can be re-used to prove the stability in time of a numerical approximation of the optimal filter.