Stochastic Maximum Principle for Optimal Advertising Models with Delay and Non-Convex Control Spaces  

作　　者：Giuseppina Guatteri Federica Masiero Giuseppina Guatteri;Federica Masiero(Dipartimento di Matematica, Politecnico di Milano via Bonardi, Milano, Italia;Dipartimento di Matematica e Applicazioni, Universit di Milano-Bicocca via Cozzi, Milano, Italia)

机构地区：[1]Dipartimento di Matematica, Politecnico di Milano via Bonardi, Milano, Italia [2]Dipartimento di Matematica e Applicazioni, Universit di Milano-Bicocca via Cozzi, Milano, Italia

出　　处：《Advances in Pure Mathematics》2024年第6期442-450,共9页理论数学进展（英文）

摘　　要：In this paper we study optimal advertising problems that model 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 U can be chosen to be non-convex but now the space of controls U 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.In this paper we study optimal advertising problems that model 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 U can be chosen to be non-convex but now the space of controls U 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.

关 键 词：Stochastic Optimal Control Delay Equations Advertisement Models Stochastic Maximum Principle 

分 类 号：O17[理学—数学]