Stochastic optimal control of delay equations arising in advertising models

By Fausto Gozzi, Carlo Marinelli. 2004

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AbstractWe consider a class of optimal control problems of stochastic delay differential equations(SDDE) that arise in connection with optimal advertising under uncertainty for the intro-duction of a new product to the market, generalizing classical work of Nerlove and Arrow[30]. In particular, we deal with controlled SDDE where the delay enters both the stateand the control. Following ideas of Vinter and Kwong [34] (which however hold only in thedeterministic case), we reformulate the problem as an infinite dimensional stochastic controlproblem to which we associate, through the dynamic programming principle, a second orderHamilton-Jacobi-Bellman equation. We show a verification theorem and we exhibit somesimple cases where such equation admits an explicit smooth solution, allowing us to constructoptimal feedback controls. 1 Introduction In this paper we consider a class of stochastic optimal control problems where the state equationis a stochastic delay differential equations (SDDE). Such problems arise for instance in modelingoptimal advertising under uncertainty for the introduction of a new product to the market,generalizing classical work of Nerlove and Arrow [30]. The main novelty in our model is thatwe deal also with delays in the control: this is interesting from the applied point of view andintroduces new mathematical difficulties in the problem.Dynamic models in marketing have a long history, that begins at least with the seminalpapers of Vidale and Wolfe [33] and Nerlove and Arrow [30]. Since then a considerable amountof work has been devoted to the extension of these models and to their application to problemsof optimal advertising, both in the monopolistic and the competitive settings, mainly underdeterministic assumptions. Models with uncertainty have received instead relatively less attention(see Feichtinger, Hartl, and Sethi [9] for a review of the existing work until 1994, Prasad and Sethi[31] for a Vidale and Wolf-like model in the competitive setting, and e.g. Grosset and Viscolani[21], Marinelli [29] for recent work on the case of a monopolistic firm). Moreover, it has beenadvocated in the literature (see e.g. Hartl [22], Feichtinger et al. [9] and references therein), asit reasonable to assume, that more realistic dynamic models of goodwill should allow for lags inthe effect of advertisement. Namely, it is natural to assume that there will be a time lag betweenadvertising expenditure and the corresponding effect on the goodwill level. More generally, afurther lag structure has been considered, allowing a distribution of the forgetting time.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/26c113ef8495d61fa9220b154804dc472e9bea8f

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