Stochastic Perron's method for Hamilton-Jacobi-Bellman equations

TL;DR

This paper introduces Stochastic Perron's method to establish the uniqueness of the value function as the solution to the HJB equation without relying on the dynamic programming principle, simplifying verification and linking weak and strong formulations.

Contribution

It develops a novel approach using Stochastic Perron's method to prove the uniqueness of viscosity solutions for HJB equations in stochastic control, bypassing traditional DPP proofs.

Findings

Proves the value function is the unique viscosity solution of the HJB equation.

Shows the equivalence of weak and strong formulations of stochastic control.

Captures face-lifting phenomena straightforwardly.

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.