Constrained non-linear estimation and links with stochastic filtering

TL;DR

This paper investigates the estimation of states in non-smooth constrained dynamics, linking it to stochastic filtering and Hamilton-Jacobi equations, and provides conditions for solution uniqueness and approximation methods.

Contribution

It establishes the viscosity solution framework for the estimation problem, analyzes boundary conditions, and connects the value function to stochastic filtering via large deviations.

Findings

Value function is a viscosity solution of a Hamilton-Jacobi-Bellman equation.

Conditions identified for the drift coefficient to ensure solution comparison and uniqueness.

Large deviation principles link the estimation problem to stochastic filtering.

Abstract

This article studies the problem of estimating the state variable of non-smooth subdifferential dynamics constrained in a bounded convex domain given some real-time observation. On the one hand, we show that the value function of the estimation problem is a viscosity solution of a Hamilton Jacobi Bellman equation whose sub and super solutions have different Neumann type boundary conditions. This intricacy arises from the non-reversibility in time of the non-smooth dynamics, and hinders the derivation of a comparison principle and the uniqueness of the solution in general. Nonetheless, we identify conditions on the drift (including zero drift) coefficient in the non-smooth dynamics that make such a derivation possible. On the other hand, we show in a general situation that the value function appears in the small noise limit of the corresponding stochastic filtering problem by establishing a large deviation result. We also give quantitative approximation results when replacing the non-smooth dynamics with a smooth penalised one.