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On Controlled Linear Diffusions with Delay in a Model of Optimal Advertising under Uncertainty with Memory Effects

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Abstract

We consider a class of dynamic advertising problems under uncertainty in the presence of carryover and distributed forgetting effects, generalizing the classical model of Nerlove and Arrow (Economica 29:129–142, 1962). In particular, we allow the dynamics of the product goodwill to depend on its past values, as well as previous advertising levels. Building on previous work (Gozzi and Marinelli in Lect. Notes Pure Appl. Math., vol. 245, pp. 133–148, 2006), the optimal advertising model is formulated as an infinite-dimensional stochastic control problem. We obtain (partial) regularity as well as approximation results for the corresponding value function. Under specific structural assumptions, we study the effects of delays on the value function and optimal strategy. In the absence of carryover effects, since the value function and the optimal advertising policy can be characterized in terms of the solution of the associated HJB equation, we obtain sharper characterizations of the optimal policy.

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Authors and Affiliations

  1. Facoltà di Economia, Libera Università degli Studi Sociali “Guido Carli”, 00198, Roma, Italy

    F. Gozzi

  2. Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, 53115, Bonn, Germany

    C. Marinelli

  3. Graduate School of Business, Columbia University, New York, NY, 10027, USA

    S. Savin

Authors
  1. F. Gozzi
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  2. C. Marinelli
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  3. S. Savin
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Correspondence to F. Gozzi.

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Communicated by C. Deissenberg.

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Gozzi, F., Marinelli, C. & Savin, S. On Controlled Linear Diffusions with Delay in a Model of Optimal Advertising under Uncertainty with Memory Effects. J Optim Theory Appl 142, 291–321 (2009). https://doi.org/10.1007/s10957-009-9524-5

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