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**Competitive advertising under uncertainty: a stochastic differential game approach.**
*(English)*
Zbl 1114.90050

Summary: We analyze optimal advertising spending in a duopolistic market where each firm’s market share depends on its own and its competitor”s advertising decisions, and is also subject to stochastic disturbances. We develop a differential game model of advertising in which the dynamic behavior is based on the Sethi stochastic advertising model and the Lanchester model of combat. Particularly important to note is the morphing of the sales decay term in the Sethi model into decay caused by competitive advertising and noncompetitive churn that acts to equalize market shares in the absence of advertising. We derive closed-loop Nash equilibria for symmetric as well as asymmetric competitors. For all cases, explicit solutions and comparative statics are presented.

### MSC:

90B60 | Marketing, advertising |

91A15 | Stochastic games, stochastic differential games |

91A23 | Differential games (aspects of game theory) |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H20 | Stochastic integral equations |

### Keywords:

Advertising; dynamic duopoly; competitive strategy; differential games; stochastic differential equations
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\textit{A. Prasad} and \textit{S. P. Sethi}, J. Optim. Theory Appl. 123, No. 1, 163--185 (2004; Zbl 1114.90050)

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### References:

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