Abstract
We apply the stochastic Perron method of Bayraktar and Sîrbu to a general infinite horizon optimal control problem, where the state $X$ is a controlled diffusion process, and the state constraint is described by a closed set. We prove that the value function $v$ is bounded from below (resp., from above) by a viscosity supersolution (resp., subsolution) of the related state constrained problem for the Hamilton-Jacobi-Bellman equation. In the case of a smooth domain, under some additional assumptions, these estimates allow to identify $v$ with a unique continuous constrained viscosity solution of this equation.
Highlights
Introduction and the main resultThe aim of the paper is to extend the scope of applications of the stochastic Perron method, developed by Bayraktar and Sîrbu
We apply the stochastic Perron method of Bayraktar and Sîrbu to a general infinite horizon optimal control problem, where the state X is a controlled diffusion process, and the state constraint is described by a closed set
We prove that the value function v is bounded from below by a viscosity supersolution of the related state constrained problem for the Hamilton-Jacobi-Bellman equation
Summary
The aim of the paper is to extend the scope of applications of the stochastic Perron method, developed by Bayraktar and Sîrbu This method allows to characterise the value function of a controlled diffusion problem as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation, bypassing the dynamic programming principle. If a comparison result, providing the inequality u− ≥ w+, holds true, it follows that u− = v = w+ is a unique (continuous) viscosity solution This construction differs from Perron’s method of [17], which is not linked to the value function. Denote by A (x), x ∈ G the set of F-progressively measurable control processes α with values in A and such that Xtx,α ∈ G, t ≥ 0 a.s. Elements of A (x) are called admissible controls for the initial condition x.
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