Year: 2026
Type: article
Abstract: . This article studies the problem of estimating the state variable of nonsmooth 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 supersolutions have different Neumann type boundary conditions. This intricacy arises from the nonreversibility in time of the nonsmooth 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 nonsmooth 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 nonsmooth dynamics with a smooth penalized one. We eventually provide a numerical illustration of the estimation problem in a one-dimensional configuration with zero drift coefficient.
DOI: 10.1137/25m1730934
Institutions: Centre National de la Recherche Scientifique, Université Paris Sciences et Lettres, Département de mathématiques et applications, City University of Hong Kong, City University of Hong Kong, Shenzhen Research Institute, Institut national de recherche en sciences et technologies du numérique, École Polytechnique, Centre de Mathématiques Appliquées de l'École polytechnique, Institut Polytechnique de Paris, Institut National des Sciences Appliquées Rouen Normandie
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W4407131600