Stochastic Optimal Control with Measurable Coefficients and Applications

TL;DR

This paper develops a novel theoretical framework for solving fully non-linear stochastic optimal control problems with measurable coefficients, establishing existence, uniqueness, and verification theorems for the associated HJB equations.

Contribution

It introduces the first comprehensive results for such control problems, utilizing $L^p$-viscosity solutions to characterize the value function and optimal controls.

Findings

Proved existence of $L^p$-viscosity solutions for the HJB equation.

Established verification theorems for optimality conditions.

Applied the theory to an economics problem in optimal advertising.

Abstract

Stochastic optimal control control problems with merely measurable coefficients are not well understood. In this manuscript, we consider fully non-linear stochastic optimal control problems in infinite horizon with measurable coefficients and (local) uniformly elliptic diffusion. Using the theory of $L^p$-viscosity solutions, we show existence of an $L^p$-viscosity solution $v\in W_{\rm loc}^{2,p}$ of the Hamilton-Jacobi-Bellman (HJB) equation, which, in turn, is also a strong solution (i.e. it satisfies the HJB equation pointwise a.e.). We are then led to prove verification theorems, providing necessary and sufficient conditions for optimality. These results allow us to construct optimal feedback controls and to characterize the value function as the unique $L^p$-viscosity solution of the HJB equation. To the best of our knowledge, these are the first results for fully non-linear stochastic optimal control problems with measurable coefficients. We use the theory developed to solve a stochastic optimal control problem arising in economics within the context of optimal advertising.