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 equation, completely avoiding the proof of the so-called dynamic programming principle (DPP). Using the stochastic Perron's method we construct a supersolution lying below the value function and a subsolution 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.