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Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations

Author

Listed:
  • Giorgio Fabbri

    () (GREQAM - Groupement de Recherche en Économie Quantitative d'Aix-Marseille - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche Scientifique - AMU - Aix Marseille Université - EHESS - École des hautes études en sciences sociales)

  • Fausto Gozzi

    (Dipartimento di Economia e Finanza [Roma] - LUISS - Libera Università Internazionale degli Studi Sociali Guido Carli [Roma])

  • Andrzej Swiech

    (School of Mathematics - Georgia Institute of Technology - Georgia Institute of Technology [Atlanta])

Abstract

Providing an introduction to stochastic optimal control in infinite dimensions, this book gives a complete account of the theory of second-order HJB equations in infinite-dimensional Hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. It features a general introduction to optimal stochastic control, including basic results (e.g. the dynamic programming principle) with proofs, and provides examples of applications. A complete and up-to-date exposition of the existing theory of viscosity solutions and regular solutions of second-order HJB equations in Hilbert spaces is given, together with an extensive survey of other methods, with a full bibliography. In particular, Chapter 6, written by M. Fuhrman and G. Tessitore, surveys the theory of regular solutions of HJB equations arising in infinite-dimensional stochastic control, via BSDEs. The book will be of interest to both pure and applied researchers working in the control theory of stochastic PDEs, and in PDEs in infinite dimensions. Readers from other fields who want to learn the basic theory will also find it useful. The prerequisites are: standard functional analysis, the theory of semigroups of operators and its use in the study of PDEs, some knowledge of the dynamic programming approach to stochastic optimal control problems in finite dimensions, and the basics of stochastic analysis and stochastic equations in infinite-dimensional spaces.

Suggested Citation

  • Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
  • Handle: RePEc:hal:journl:hal-01505767
    Note: View the original document on HAL open archive server: https://hal-amu.archives-ouvertes.fr/hal-01505767
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    References listed on IDEAS

    as
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    Citations

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    Cited by:

    1. Emmanuelle Augeraud-Véron & Raouf Boucekkine & Vladimir Veliov, 2019. "Distributed Optimal Control Models in Environmental Economics: A Review," AMSE Working Papers 1902, Aix-Marseille School of Economics, France.
    2. Silvia Faggian & Fausto Gozzo & Peter M. Kort, 2019. "Optimal investment with vintage capital: equilibrium distributions," Working Papers 2019: 12, Department of Economics, University of Venice "Ca' Foscari".
    3. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2019. "Growth and agglomeration in the heterogeneous space: a generalized AK approach," Journal of Economic Geography, Oxford University Press, vol. 19(6), pages 1287-1318.
    4. Giorgio Fabbri & Francesco Russo, 2017. "HJB equations in infinite dimension and optimal control of stochastic evolution equations via generalized Fukushima decomposition," Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) 2017003, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    5. Ulrich Horst & Xiaonyu Xia, 2018. "Continuous viscosity solutions to linear-quadratic stochastic control problems with singular terminal state constraint," Papers 1809.01972, arXiv.org, revised Jun 2019.
    6. René Carmona & Jean-Pierre Fouque & Seyyed Mostafa Mousavi & Li-Hsien Sun, 0. "Systemic Risk and Stochastic Games with Delay," Journal of Optimization Theory and Applications, Springer, vol. 0, pages 1-34.
    7. René Carmona & Jean-Pierre Fouque & Seyyed Mostafa Mousavi & Li-Hsien Sun, 2018. "Systemic Risk and Stochastic Games with Delay," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 366-399, November.
    8. Graewe, Paulwin & Horst, Ulrich & Séré, Eric, 2018. "Smooth solutions to portfolio liquidation problems under price-sensitive market impact," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 979-1006.
    9. Paulwin Graewe & Ulrich Horst & Eric S'er'e, 2013. "Smooth solutions to portfolio liquidation problems under price-sensitive market impact," Papers 1309.0474, arXiv.org, revised Jun 2017.

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