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Gambling for resurrection and the heat equation on a triangle

Author

Listed:
  • Stefan Ankirchner

    (Friedrich-Schiller-Universität = Friedrich Schiller University Jena [Jena, Germany])

  • Christophette Blanchet-Scalliet

    (ICJ - Institut Camille Jordan - ECL - École Centrale de Lyon - Université de Lyon - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - INSA Lyon - Institut National des Sciences Appliquées de Lyon - Université de Lyon - INSA - Institut National des Sciences Appliquées - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique)

  • Nabil Kazi-Tani

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

  • Chao Zhou

    (NUS - National University of Singapore)

Abstract

We consider the problem of controlling the diffusion coefficient of a diffusion with constant negative drift rate such that the probability of hitting a given lower barrier up to some finite time horizon is minimized. We assume that the diffusion rate can be chosen in a progressively measurable way with values in the interval [0, 1]. We prove that the value function is regular, concave in the space variable, and that it solves the associated HJB equation. To do so, we show that the heat equation on a right triangle, with a boundary condition that is discontinuous in the corner, possesses a smooth solution.

Suggested Citation

  • Stefan Ankirchner & Christophette Blanchet-Scalliet & Nabil Kazi-Tani & Chao Zhou, 2021. "Gambling for resurrection and the heat equation on a triangle," Post-Print hal-02405853, HAL.
    • Handle: RePEc:hal:journl:hal-02405853
    DOI: 10.1007/s00245-020-09741-9
    Note: View the original document on HAL open archive server: https://hal.science/hal-02405853
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    References listed on IDEAS

    as
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    Keywords

    Hitting probability; Stochastic control; Heat equation;
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