Stochastic optimal control of delay equations arising in advertising models

TL;DR

This paper studies stochastic delay differential equations in optimal advertising models, reformulating the problem as an infinite-dimensional stochastic control problem and deriving explicit solutions in special cases.

Contribution

It extends classical advertising control models to include delays in both state and control, using infinite-dimensional stochastic control and HJB equations.

Findings

Reformulation as an infinite-dimensional stochastic control problem.

Verification theorem for the stochastic control problem.

Explicit solutions for certain cases enabling optimal feedback controls.

Abstract

We consider a class of optimal control problems of stochastic delay differential equations (SDDE) that arise in connection with optimal advertising under uncertainty for the introduction of a new product to the market, generalizing classical work of Nerlove and Arrow (1962). In particular, we deal with controlled SDDE where the delay enters both the state and the control. Following ideas of Vinter and Kwong (1981) (which however hold only in the deterministic case), we reformulate the problem as an infinite dimensional stochastic control problem to which we associate, through the dynamic programming principle, a second order Hamilton-Jacobi-Bellman equation. We show a verification theorem and we exhibit some simple cases where such equation admits an explicit smooth solution, allowing us to construct optimal feedback controls.