Mean field games incorporating carryover effects: optimizing advertising models

We consider a class of optimal control problems that arise in connection with optimal advertising under uncertainty. Two main features appear in the model: a delay in the control variable driving the state dynamics; a mean-field term both in the state dynamics and in the utility functional, taking into account for other agents. We interpret the model in a competitive environment, hence we set it in the framework of Mean Field Games. We rephrase the problem in an infinite dimensional setting, in order to obtain the associated Mean Field Game system. Finally, we specify the problem to a simple case, and solve it providing an explicit solution.