Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries

Francesco C. De Vecchi, Elisa Mastrogiacomo, Mattia Turra and Stefania Ugolini
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Francesco C. De Vecchi: Institute for Applied Mathematics & Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Elisa Mastrogiacomo: Dipartimento di Economia, Università degli Studi dell’Insubria, Via Montegeneroso 71, 21100 Varese, Italy
Mattia Turra: Institute for Applied Mathematics & Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Stefania Ugolini: Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20113 Milano, Italy

Mathematics, 2021, vol. 9, issue 9, 1-34

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.

Keywords: Noether theorem; stochastic optimal control; contact symmetries; Merton’s optimal portfolio problem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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