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 goodwi...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.展开更多
Many dynamic optimal control models for advertising make efforts to solve the problem of determining optimal advertising expenditures and other variables of interest over time for a firm or several competing firms,How...Many dynamic optimal control models for advertising make efforts to solve the problem of determining optimal advertising expenditures and other variables of interest over time for a firm or several competing firms,However,after analyzing the extant literature,one can find that few dynamic optimal advertising models available consider the problem within the product diffusion framework.Furthermore,the established models involving product diffusion are inspired by the Bass model,which has been out of date.This paper poses a dynamic optimal advertising model for new products,which considers the product diffusion based on the relative newly developed generalized version of the Bass model.In this paper,the optimal control model is used to derive the optimal advertising expenditure policy,which gives some implications to advertising practice.展开更多
In this paper, stability and Hopf bifurcation of a nonlinear advertising ca- pital model with time delayed are studied. By analyzing the change of delay, we obtain several sufficient conditions on stable and unstable ...In this paper, stability and Hopf bifurcation of a nonlinear advertising ca- pital model with time delayed are studied. By analyzing the change of delay, we obtain several sufficient conditions on stable and unstable properties. When delay passes a critical value, Hopf bifurcation may appear. Furthermore, the direction and stability of bifurcating periodic solutions are investigated by normal form and center manifold theory. Additionally, we also have some discussion about the model with continuous time delay.展开更多
文摘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.
基金This research is supported by the National Natural Science Foundation of China.
文摘Many dynamic optimal control models for advertising make efforts to solve the problem of determining optimal advertising expenditures and other variables of interest over time for a firm or several competing firms,However,after analyzing the extant literature,one can find that few dynamic optimal advertising models available consider the problem within the product diffusion framework.Furthermore,the established models involving product diffusion are inspired by the Bass model,which has been out of date.This paper poses a dynamic optimal advertising model for new products,which considers the product diffusion based on the relative newly developed generalized version of the Bass model.In this paper,the optimal control model is used to derive the optimal advertising expenditure policy,which gives some implications to advertising practice.
文摘In this paper, stability and Hopf bifurcation of a nonlinear advertising ca- pital model with time delayed are studied. By analyzing the change of delay, we obtain several sufficient conditions on stable and unstable properties. When delay passes a critical value, Hopf bifurcation may appear. Furthermore, the direction and stability of bifurcating periodic solutions are investigated by normal form and center manifold theory. Additionally, we also have some discussion about the model with continuous time delay.
