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Malliavin Calculus and Optimal Control of Stochastic Volterra Equations

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  • Nacira Agram

    (University Med Khider)

  • Bernt Øksendal

    (University of Oslo)

Abstract

Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore, classical methods, such as dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that using Malliavin calculus, it is possible to formulate modified functional types of maximum principle suitable for such systems. This principle also applies to situations where the controller has only partial information available to base her decisions upon. We present both a Mangasarian sufficient condition and a Pontryagin-type maximum principle of this type, and then, we use the results to study some specific examples. In particular, we solve an optimal portfolio problem in a financial market model with memory.

Suggested Citation

  • Nacira Agram & Bernt Øksendal, 2015. "Malliavin Calculus and Optimal Control of Stochastic Volterra Equations," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 1070-1094, December.
    • Handle: RePEc:spr:joptap:v:167:y:2015:i:3:d:10.1007_s10957-015-0753-5
    DOI: 10.1007/s10957-015-0753-5
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    References listed on IDEAS

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    1. Jan Ubøe & Bernt Øksendal & Knut Aase & Nicolas Privault, 2000. "White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance," Finance and Stochastics, Springer, vol. 4(4), pages 465-496.
    2. Yong, Jiongmin, 2006. "Backward stochastic Volterra integral equations and some related problems," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 779-795, May.
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    Cited by:

    1. Farkhondeh Rouz, O. & Shahmorad, S. & Ahmadian, D., 2024. "Double weakly singular kernels in stochastic Volterra integral equations with application to the rough Heston model," Applied Mathematics and Computation, Elsevier, vol. 475(C).
    2. Hu, Yaozhong & Øksendal, Bernt, 2019. "Linear Volterra backward stochastic integral equations," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 626-633.
    3. Michele Giordano & Anton Yurchenko-Tytarenko, 2024. "Optimal control in linear-quadratic stochastic advertising models with memory," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 47(1), pages 275-298, June.
    4. Andrés Cárdenas & Sergio Pulido & Rafael Serrano, 2022. "Existence of optimal controls for stochastic Volterra equations," Working Papers hal-03720342, HAL.

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