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Stochastic optimal control problems with measurable coefficients via $L^p$-viscosity solutions and applications to optimal advertising models

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  • Filippo de Feo

Abstract

We consider fully non-linear stochastic optimal control problems in infinite horizon with measurable coefficients and uniformly elliptic diffusion. Using the theory of $L^p$-viscosity solutions, we show existence of an $L^p$-viscosity solution $v\in W_{\rm loc}^{2,p}$ of the Hamilton-Jacobi-Bellman (HJB) equation, which, in turn, is also a strong solution (i.e. it satisfies the HJB equation pointwise a.e.). We are then led to prove verification theorems, providing necessary and sufficient conditions for optimality. These results allow us to construct optimal feedback controls. We use the theory developed to solve a stochastic optimal control problem arising in economics within the context of optimal advertising.

Suggested Citation

  • Filippo de Feo, 2025. "Stochastic optimal control problems with measurable coefficients via $L^p$-viscosity solutions and applications to optimal advertising models," Papers 2502.02352, arXiv.org.
    • Handle: RePEc:arx:papers:2502.02352
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    References listed on IDEAS

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    1. Luca Grosset & Bruno Viscolani, 2004. "Advertising for a new product introduction: A stochastic approach," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 12(1), pages 149-167, June.
    2. Gozzi, Fausto & Russo, Francesco, 2006. "Verification theorems for stochastic optimal control problems via a time dependent Fukushima-Dirichlet decomposition," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1530-1562, November.
    3. Gozzi, Fausto & Russo, Francesco, 2006. "Weak Dirichlet processes with a stochastic control perspective," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1563-1583, November.
    4. Olivier Menoukeu-Pamen & Ludovic Tangpi, 2023. "Maximum Principle for Stochastic Control of SDEs with Measurable Drifts," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1195-1228, June.
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