Existence and global Lipschitz estimates for unbounded classical solutions of a Hamilton–Jacobi equation

Louis-Pierre Chaintron

doi:10.5802/afst.1824

¹ DMA, École normale supérieure, Université PSL, CNRS, 75005 Paris, France
² CERMICS, École des ponts, 77420 Champs-sur-Marne, France. Inria, Team MΞDISIM, Inria Saclay, 91128 Palaiseau, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 34 (2025) no. 4, pp. 851-876

Abstract (Original language)
Résumé

The purpose of this article is to prove existence, uniqueness and uniform gradient estimates for unbounded classical solutions of a Hamilton–Jacobi–Bellman equation. Such an equation naturally arises in stochastic control problems. Contrary to the classical literature which handles the case of bounded regular coefficients, we only impose Lipschitz regularity conditions, allowing for a linear growth of coefficients. These Lipschitz assumptions are natural in a probabilistic setting. In principle, these assumptions are compatible with global Lipschitz regularity for the solution. However, to the best of our knowledge, this useful result had not been established before. Our proofs rely on the Ishii–Lions method [36]. We combine several elements from the viscosity solution theory to obtain estimates at the edges of what seems possible.

Cet article établit l’existence et l’unicité de la solution classique non-bornée d’une équation de Hamilton–Jacobi–Bellman, ainsi que des estimées uniformes sur son gradient. Cette équation apparait naturellement en contrôle stochastique. Contrairement aux résultats classiques qui traitent le cas de coefficients bornés et réguliers, nous ne demandons qu’une régularité Lipschitz qui inclut des coefficients non-bornés à croissance linéaire. Ces hypothèses Lipschitz sont naturelles dans un cadre probabiliste. En principe, ces hypothèses permettent d’obtenir une régularité Lipschitz pour la solution. Cependant ce résultat très utile n’avait jamais été établi à notre connaissance. Nos preuves reposent sur la méthode développée par Ishii–Lions [36] pour les solutions de viscosité. Plusieurs éléments classiques sont combinés afin d’obtenir ce résultat à la limite de ce qui semble possible.

Received: 2023-10-03
Accepted: 2024-07-18
Published online: 2025-10-02

DOI: 10.5802/afst.1824

Keywords: Hamilton–Jacobi equations, Lipschitz estimates, viscosity solutions, stochastic control
Mots-clés : Équations de Hamilton–Jacobi, estimées Lipschitz, solutions de viscosité, contrôle stochastique

Author's affiliations:

Louis-Pierre Chaintron ^{1, 2}

¹ DMA, École normale supérieure, Université PSL, CNRS, 75005 Paris, France
² CERMICS, École des ponts, 77420 Champs-sur-Marne, France. Inria, Team MΞDISIM, Inria Saclay, 91128 Palaiseau, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Louis-Pierre Chaintron},
     title = {Existence and global {Lipschitz} estimates for unbounded classical solutions of a {Hamilton{\textendash}Jacobi} equation},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {851--876},
     year = {2025},
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Louis-Pierre Chaintron. Existence and global Lipschitz estimates for unbounded classical solutions of a Hamilton–Jacobi equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 34 (2025) no. 4, pp. 851-876. doi: 10.5802/afst.1824

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