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Verification theorems for stochastic optimal control problems via a time dependent Fukushima-Dirichlet decomposition

Author

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  • Gozzi, Fausto
  • Russo, Francesco

Abstract

This paper is devoted to presenting a method of proving verification theorems for stochastic optimal control of finite dimensional diffusion processes without control in the diffusion term. The value function is assumed to be continuous in time and once differentiable in the space variable (C0,1) instead of once differentiable in time and twice in space (C1,2), like in the classical results. The results are obtained using a time dependent Fukushima-Dirichlet decomposition proved in a companion paper by the same authors using stochastic calculus via regularization. Applications, examples and a comparison with other similar results are also given.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:11:p:1530-1562
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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. Lejay, Antoine, 2004. "A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE," Stochastic Processes and their Applications, Elsevier, vol. 110(1), pages 145-176, March.
    3. Takayama,Akira, 1985. "Mathematical Economics," Cambridge Books, Cambridge University Press, number 9780521314985, January.
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    Cited by:

    1. Anna Battauz & Marzia Donno & Alessandro Sbuelz, 2017. "Reaching nirvana with a defaultable asset?," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 40(1), pages 31-52, November.
    2. Giorgio Fabbri & Francesco Russo, 2017. "HJB Equations in Infinite Dimension and Optimal Control of Stochastic Evolution Equations via Generalized Fukushima Decomposition," AMSE Working Papers 1704, Aix-Marseille School of Economics, France.
    3. Fabbri, Giorgio & Russo, Francesco, 2017. "Infinite dimensional weak Dirichlet processes and convolution type processes," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 325-357.
    4. Filippo de Feo, 2025. "Stochastic optimal control with measurable coefficients via $L^p$-viscosity solutions and applications to optimal advertising," Papers 2502.02352, arXiv.org, revised Jan 2026.
    5. Cristina Girolami & Giorgio Fabbri & Francesco Russo, 2014. "The covariation for Banach space valued processes and applications," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 51-104, January.
    6. Bandini, Elena & Russo, Francesco, 2017. "Weak Dirichlet processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4139-4189.
    7. Huberts, Nick F.D. & Rossi Silveira, Rafael, 2023. "How economic depreciation shapes the relationship of uncertainty with investments’ size & timing," International Journal of Production Economics, Elsevier, vol. 260(C).
    8. Huberts, N.F.D. & Dawid, H. & Huisman, K.J.M. & Kort, P.M., 2019. "Entry deterrence by timing rather than overinvestment in a strategic real options framework," European Journal of Operational Research, Elsevier, vol. 274(1), pages 165-185.
    9. Giorgio Fabbri & Francesco Russo, 2016. "Infinite Dimensional Weak Dirichlet Processes and Convolution Type Processes," Working Papers halshs-01309384, HAL.
    10. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.

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