Stochastic Maximum Principle for optimal advertising models with delay and non-convex control space

TL;DR

This paper develops a stochastic maximum principle for optimal advertising models with delay and non-convex control spaces, incorporating memory effects and past dependencies in a complex, realistic setting.

Contribution

It introduces a novel stochastic Pontryagin maximum principle applicable to models with delays, non-convex controls, and memory effects, using explicit BSDE-based adjoint equations.

Findings

Explicit formulation of the maximum principle for delayed stochastic control

Inclusion of non-convex control spaces in the optimal advertising framework

Derivation of first- and second-order adjoint BSDEs for the problem

Abstract

In this paper we study optimal advertising problems that models the introduction of a new product into the market in the presence of carryover effects of the advertisement and with memory effects in the level of goodwill. In particular, we let the dynamics of the product goodwill to depend on the past, and also on past advertising efforts. We treat the problem by means of the stochastic Pontryagin maximum principle, that here is considered for a class of problems where in the state equation either the state or the control depend on the past. Moreover the control acts on the martingale term and the space of controls can be chosen to be non-convex. The maximum principle is thus formulated using a first-order adjoint Backward Stochastic Differential Equations (BSDEs), which can be explicitly computed due to the specific characteristics of the model, and a second-order adjoint relation.