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Optimal position targeting via decoupling fields

Author

Listed:
  • Stefan Ankirchner

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

  • Alexander Fromm

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

  • Thomas Kruse

    (Universität Duisburg-Essen = University of Duisburg-Essen [Essen])

  • Alexandre Popier

    (LMM - Laboratoire Manceau de Mathématiques - UM - Le Mans Université)

Abstract

We consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation of the terminal state from a given target position. Using the Pontryagin maximum principle we characterize a solution of the unconstrained control problem in terms of a fully coupled forward-backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution.

Suggested Citation

  • Stefan Ankirchner & Alexander Fromm & Thomas Kruse & Alexandre Popier, 2018. "Optimal position targeting via decoupling fields," Working Papers hal-01500311, HAL.
    • Handle: RePEc:hal:wpaper:hal-01500311
    Note: View the original document on HAL open archive server: https://hal.science/hal-01500311v2
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    References listed on IDEAS

    as
    1. Alfonsi Aurélien & Alexander Schied & Alla Slynko, 2012. "Order Book Resilience, Price Manipulation, and the Positive Portfolio Problem," Post-Print hal-00941333, HAL.
    2. Kruse, T. & Popier, A., 2016. "Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2554-2592.
    3. Delarue, François, 2002. "On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 209-286, June.
    4. Popier, A., 2006. "Backward stochastic differential equations with singular terminal condition," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 2014-2056, December.
    5. Ma, Jin & Yin, Hong & Zhang, Jianfeng, 2012. "On non-Markovian forward–backward SDEs and backward stochastic PDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 3980-4004.
    6. Ulrich Horst & Jinniao Qiu & Qi Zhang, 2014. "A Constrained Control Problem with Degenerate Coefficients and Degenerate Backward SPDEs with Singular Terminal Condition," Papers 1407.0108, arXiv.org, revised Jul 2015.
    7. Paulwin Graewe & Ulrich Horst & Jinniao Qiu, 2013. "A Non-Markovian Liquidation Problem and Backward SPDEs with Singular Terminal Conditions," Papers 1309.0461, arXiv.org, revised Jan 2015.
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