Stochastic Perron's Method for Hamilton-Jacobi-Bellman Equations

SIAM Journal on Control and Optimization, 51(6), 4274-4294, 2013

21 Pages Posted: 27 Jan 2014  

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

Mihai Sirbu

University of Texas at Austin

Date Written: September 23, 2013

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.

Keywords: Perron's method, viscosity solutions, non-smooth verification, comparison principle

Suggested Citation

Bayraktar, Erhan and Sirbu, Mihai, Stochastic Perron's Method for Hamilton-Jacobi-Bellman Equations (September 23, 2013). SIAM Journal on Control and Optimization, 51(6), 4274-4294, 2013. Available at SSRN: https://ssrn.com/abstract=2384840

Erhan Bayraktar (Contact Author)

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Mihai Sirbu

University of Texas at Austin ( email )

Austin, TX 78712
United States

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