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Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries

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  • Francesco C. De Vecchi

    (Institute for Applied Mathematics & Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
    These authors contributed equally to this work.)

  • Elisa Mastrogiacomo

    (Dipartimento di Economia, Università degli Studi dell’Insubria, Via Montegeneroso 71, 21100 Varese, Italy
    These authors contributed equally to this work.)

  • Mattia Turra

    (Institute for Applied Mathematics & Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
    These authors contributed equally to this work.)

  • Stefania Ugolini

    (Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20113 Milano, Italy
    These authors contributed equally to this work.)

Abstract

We establish a generalization of the Noether theorem for stochastic optimal control problems. Exploiting the tools of jet bundles and contact geometry, we prove that from any (contact) symmetry of the Hamilton–Jacobi–Bellman equation associated with an optimal control problem it is possible to build a related local martingale. Moreover, we provide an application of the theoretical results to Merton’s optimal portfolio problem, showing that this model admits infinitely many conserved quantities in the form of local martingales.

Suggested Citation

  • Francesco C. De Vecchi & Elisa Mastrogiacomo & Mattia Turra & Stefania Ugolini, 2021. "Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries," Mathematics, MDPI, vol. 9(9), pages 1-34, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:953-:d:542540
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    References listed on IDEAS

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    5. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
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