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Risks 2015, 3(4), 553-572;

Stochastic Optimal Control for Online Seller under Reputational Mechanisms

Mathematics of Networks and Systems, Bell Labs, 600 Mountain Avenue, Murray Hill, NJ 07974, USA
Department of Mathematics, Kettering University, Flint, MI 48504, USA
Department of Statistics and Probability and Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Author to whom correspondence should be addressed.
Academic Editors: Emiliano A. Valdez and Nick Costanzino
Received: 28 August 2015 / Accepted: 25 November 2015 / Published: 4 December 2015
(This article belongs to the Special Issue Recent Advances in Mathematical Modeling of the Financial Markets)
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In this work we propose and analyze a model which addresses the pulsing behavior of sellers in an online auction (store). This pulsing behavior is observed when sellers switch between advertising and processing states. We assert that a seller switches her state in order to maximize her profit, and further that this switch can be identified through the seller’s reputation. We show that for each seller there is an optimal reputation, i.e., the reputation at which the seller should switch her state in order to maximize her total profit. We design a stochastic behavioral model for an online seller, which incorporates the dynamics of resource allocation and reputation. The design of the model is optimized by using a stochastic advertising model from [1] and used effectively in the Stochastic Optimal Control of Advertising [2]. This model of reputation is combined with the effect of online reputation on sales price empirically verified in [3]. We derive the Hamilton-Jacobi-Bellman (HJB) differential equation, whose solution relates optimal wealth level to a seller’s reputation. We formulate both a full model, as well as a reduced model with fewer parameters, both of which have the same qualitative description of the optimal seller behavior. Coincidentally, the reduced model has a closed form analytical solution that we construct. View Full-Text
Keywords: Stochastic optimal control models; online stores; Hamilton-Jacobi-Bellman equation Stochastic optimal control models; online stores; Hamilton-Jacobi-Bellman equation

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Bradonjić, M.; Causley, M.; Cohen, A. Stochastic Optimal Control for Online Seller under Reputational Mechanisms. Risks 2015, 3, 553-572.

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