Path-dependent Hamilton-Jacobi-Bellman equation: Uniqueness of Crandall-Lions viscosity solutions

Andrea Cosso; Fausto Gozzi; Mauro Rosestolato; Francesco Russo

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arXiv · 2107.05959

Path-dependent Hamilton-Jacobi-Bellman equation: Uniqueness of Crandall-Lions viscosity solutions

Andrea Cosso·Fausto Gozzi·Mauro Rosestolato·Francesco Russo

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Abstract

We formulate a path-dependent stochastic optimal control problem under general conditions, for which weprove rigorously the dynamic programming principle and that the value function is the unique Crandall-Lions viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. Compared to the literature, the proof of our core result, that is the comparison theorem, is based on the fact that the valuefunction is bigger than any viscosity subsolution and smaller than any viscosity supersolution. It alsorelies on the approximation of the value function in terms of functions defined on finite-dimensionalspaces as well as on regularity results for parabolic partial differential equations.

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BibTeX RIS

Andrea Cosso, Fausto Gozzi, Mauro Rosestolato, Francesco Russo. 2021-07-13. Path-dependent Hamilton-Jacobi-Bellman equation: Uniqueness of Crandall-Lions viscosity solutions. https://arxiv.org/abs/2107.05959

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