Stochastic Perron's method for Hamilton-Jacobi-Bellman equations
Abstract
We show that the value function of a stochastic control problem is the unique solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, completely avoiding the proof of the so-called dynamic programming principle (DPP). Using Stochastic Perron's method we construct a super-solution lying below the value function and a sub-solution dominating it. A comparison argument easily closes the proof. The program has the precise meaning of verification for viscosity-solutions, obtaining the DPP as a conclusion. It also immediately follows that the weak and strong formulations of the stochastic control problem have the same value. Using this method we also capture the possible face-lifting phenomenon in a straightforward manner.
Cite
@article{arxiv.1212.2170,
title = {Stochastic Perron's method for Hamilton-Jacobi-Bellman equations},
author = {Erhan Bayraktar and Mihai Sirbu},
journal= {arXiv preprint arXiv:1212.2170},
year = {2013}
}
Comments
Final version. To appear in the SIAM Journal on Control and Optimization. Keywords: Perron's method, viscosity solutions, non-smooth verification, comparison principle