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HJB Equations and Stochastic Control on Half-Spaces of Hilbert Spaces

Author

Listed:
  • Alessandro Calvia

    (Università di Parma)

  • Gianluca Cappa

    (LUISS University)

  • Fausto Gozzi

    (LUISS University)

  • Enrico Priola

    (University of Pavia)

Abstract

In this paper, we study a first extension of the theory of mild solutions for Hamilton–Jacobi–Bellman (HJB) equations in Hilbert spaces to the case where the domain is not the whole space. More precisely, we consider a half-space as domain, and a semilinear HJB equation. Our main goal is to establish the existence and the uniqueness of solutions to such HJB equations, which are continuously differentiable in the space variable. We also provide an application of our results to an exit-time optimal control problem, and we show that the corresponding value function is the unique solution to a semilinear HJB equation, possessing sufficient regularity to express the optimal control in feedback form. Finally, we give an illustrative example.

Suggested Citation

  • Alessandro Calvia & Gianluca Cappa & Fausto Gozzi & Enrico Priola, 2023. "HJB Equations and Stochastic Control on Half-Spaces of Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 198(2), pages 710-744, August.
    • Handle: RePEc:spr:joptap:v:198:y:2023:i:2:d:10.1007_s10957-023-02208-1
    DOI: 10.1007/s10957-023-02208-1
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    References listed on IDEAS

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    1. Boucekkine, R. & Camacho, C. & Fabbri, G., 2013. "Spatial dynamics and convergence: The spatial AK model," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2719-2736.
    2. 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.
    3. Djehiche, Boualem & Gozzi, Fausto & Zanco, Giovanni & Zanella, Margherita, 2022. "Optimal portfolio choice with path dependent benchmarked labor income: A mean field model," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 48-85.
    4. Enrico Biffis & Fausto Gozzi & Cecilia Prosdocimi, 2020. "Optimal portfolio choice with path dependent labor income: the infinite horizon case," Papers 2002.00201, arXiv.org.
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