Stochastic optimal control problems with measurable coefficients via $L^p$-viscosity solutions and applications to optimal advertising models
Filippo de Feo
TL;DR
The paper develops a framework for fully nonlinear stochastic control on an infinite horizon with merely measurable coefficients by leveraging $L^p$-viscosity solutions to the Hamilton-Jacobi-Bellman equation. It proves local $W^{2,p}$-regularity and strong (a.e.) solvability of $v$ when $p\ge n$, and it connects $L^p$-viscosity solutions with standard viscosity solutions under continuity assumptions; Perron-type methods yield existence results. Verification theorems are established via Dynkin's formula for $W^{2,n}_{loc}$-functions, yielding necessary and sufficient optimality conditions that apply to $L^p$-viscosity and viscosity solutions, and enabling the construction of optimal feedback controls. The theory is then applied to a stochastic advertising model with discontinuous drift, demonstrating how to obtain $L^p$-viscosity solutions and optimal feedbacks, and proving uniqueness of the resulting value function. Overall, the work advances analysis and control design for irregular, fully nonlinear stochastic systems with practical economic applications.
Abstract
We consider fully non-linear stochastic optimal control problems in infinite horizon with measurable coefficients and 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. We use the theory developed to solve a stochastic optimal control problem arising in economics within the context of optimal advertising.