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Diffusive Limit Approximation of Pure-Jump Optimal Stochastic Control Problems

Author

Listed:
  • Marc Abeille

    (Criteo AI Lab.)

  • Bruno Bouchard

    (Université Paris-Dauphine, PSL, CNRS)

  • Lorenzo Croissant

    (Criteo AI Lab.
    Université Paris-Dauphine, PSL, CNRS)

Abstract

We consider the diffusive limit of a typical pure-jump Markovian control problem as the intensity of the driving Poisson process tends to infinity. We show that the convergence speed is provided by the Hölder exponent of the Hessian of the limit problem, and explain how correction terms can be constructed. This provides an alternative efficient method for the numerical approximation of the optimal control of a pure-jump problem in situations with very high intensity of jumps. We illustrate this approach in the context of a display advertising auction problem.

Suggested Citation

  • Marc Abeille & Bruno Bouchard & Lorenzo Croissant, 2023. "Diffusive Limit Approximation of Pure-Jump Optimal Stochastic Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 147-176, January.
    • Handle: RePEc:spr:joptap:v:196:y:2023:i:1:d:10.1007_s10957-022-02135-7
    DOI: 10.1007/s10957-022-02135-7
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    References listed on IDEAS

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    1. Bouchard, Bruno, 2002. "Stochastic targets with mixed diffusion processes and viscosity solutions," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 273-302, October.
    2. Bulow, Jeremy & Klemperer, Paul, 1996. "Auctions versus Negotiations," American Economic Review, American Economic Association, vol. 86(1), pages 180-194, March.
    3. Joaquin Fernandez-Tapia & Olivier Gu'eant & Jean-Michel Lasry, 2015. "Optimal Real-Time Bidding Strategies," Papers 1511.08409, arXiv.org, revised Jun 2016.
    4. William Vickrey, 1961. "Counterspeculation, Auctions, And Competitive Sealed Tenders," Journal of Finance, American Finance Association, vol. 16(1), pages 8-37, March.
    5. Cohen, Asaf & Young, Virginia R., 2020. "Rate of convergence of the probability of ruin in the Cramér–Lundberg model to its diffusion approximation," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 333-340.
    6. Roger B. Myerson, 1981. "Optimal Auction Design," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 58-73, February.
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    Cited by:

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    2. Yoshioka, Hidekazu & Yoshioka, Yumi, 2024. "Assessing fluctuations of long-memory environmental variables based on the robustified dynamic Orlicz risk," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).

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