Viscosity Solutions of a class of Second Order Hamilton-Jacobi-Bellman Equations in the Wasserstein Space

Hang Cheung; Ho Man Tai; Jinniao Qiu

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arXiv · 2312.10322

Viscosity Solutions of a class of Second Order Hamilton-Jacobi-Bellman Equations in the Wasserstein Space

Hang Cheung·Ho Man Tai·Jinniao Qiu

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Abstract

This paper is devoted to solving a class of second order Hamilton-Jacobi-Bellman (HJB) equations in the Wasserstein space, associated with mean field control problems involving common noise. The well-posedness of viscosity solutions to the HJB equation under a new notion is established under general assumptions on the coefficients. Our approach adopts the smooth metric developed by Bayraktar, Ekren, and Zhang [Proc. Amer. Math. Soc. (2023)] as our gauge function for the purpose of smooth variational principle used in the proof of comparison theorem. Further estimates and regularity of the metric, including a novel second order derivative estimate with respect to the measure variable, are derived in order to ensure the uniqueness and existence.

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BibTeX RIS

Hang Cheung, Ho Man Tai, Jinniao Qiu. 2023-12-16. Viscosity Solutions of a class of Second Order Hamilton-Jacobi-Bellman Equations in the Wasserstein Space. https://arxiv.org/abs/2312.10322

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