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

Andrea Cosso, Fausto Gozzi, Mauro Rosestolato and 
Francesco Russo
submitted
Publication type:
Paper in peer-reviewed journals
Journal:
Preprint hal-03285204
arXiv:
assets/images/icons/icon_arxiv.png 2107.05959
Keywords :
Path-dependent SDEs; dynamic programming principle; pathwise derivatives; functional Itô calculus; path-dependent HJB equations; viscosity solutions.
Abstract:
We prove existence and uniqueness of Crandall-Lions viscosity solutions of Hamilton-Jacobi-Bellman equations in the space of continuous paths, associated to the optimal control of path-dependent SDEs. This seems the first uniqueness result in such a context. More precisely, similarly to the seminal paper of P.L. Lions, the proof of our core result, that is the comparison theorem, is based on the fact that the value function is bigger than any viscosity subsolution and smaller than any viscosity supersolution. Such a result, coupled with the proof that the value function is a viscosity solution (based on the dynamic programming principle, which we prove), implies that the value function is the unique viscosity solution to the Hamilton-Jacobi-Bellman equation. The proof of the comparison theorem in P.L. Lions' paper, relies on regularity results which are missing in the present infinite-dimensional context, as well as on the local compactness of the finite-dimensional underlying space. We overcome such non-trivial technical difficulties introducing a suitable approximating procedure and a smooth gauge-type function, which allows to generate maxima and minima through an appropriate version of the Borwein-Preiss generalization of Ekeland's variational principle on the space of continuous paths.
BibTeX:
@article{Cos-Goz-Ros-Rus-2200,
    author={Andrea Cosso and Fausto Gozzi and Mauro Rosestolato and 
           Francesco Russo },
    title={Path-dependent Hamilton-Jacobi-Bellman equation: Uniqueness of 
           Crandall-Lions viscosity solutions },
    journal={Preprint hal-03285204 },
    year={submitted },
}