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An optimal control problem in the space of probability measures, and the viscosity solu- tions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a novel smooth Fourier-Wasserstein metric. A comparison result between the Lipschitz viscosity sub and super solutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution.
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A comparison principle for semilinear Hamilton–Jacobi–Bellman equations in the Wasserstein space
Abstract The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton–Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the 1-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution.
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- PAR ID:
- 10523354
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Calculus of Variations and Partial Differential Equations
- Volume:
- 63
- Issue:
- 4
- ISSN:
- 0944-2669
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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