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Stochastic target problems with controlled loss in jump diffusion models

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  • Ludovic Moreau

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

Abstract

In this paper, we consider a mixed diffusion version of the stochastic target problem introduced by Bouchard et al. (2009). This consists in finding the minimum initial value of a controlled process which guarantees to reach a controlled stochastic target with a given lovel of expected loss. As in Bouchard et al. (2009), it can be converted into a standard stochastic target problem, as already studied by Soner and Touzi (2002) or Bouchard (2002) for the mixed diffusion case, by increasing both the state space and the dimension of the control. In our mixed-diffusion setting, the main difficulty comes from the presence of jumps, which leads to the introduction of a new kind of controls that take values in an unbounded set of measurable maps. This has non trivial impacts on the formulation and derivation of the associated partial differential equations.

Suggested Citation

  • Ludovic Moreau, 2011. "Stochastic target problems with controlled loss in jump diffusion models," Post-Print hal-00515522, HAL.
    • Handle: RePEc:hal:journl:hal-00515522
    DOI: 10.1137/100802268
    Note: View the original document on HAL open archive server: https://hal.science/hal-00515522v3
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    References listed on IDEAS

    as
    1. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(1), pages 193-194, February.
    2. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(2), pages 541-545, April.
    3. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(4), pages 1007-1017, August.
    4. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(6), pages 1461-1465, December.
    5. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(5), pages 1273-1289, October.
    6. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(3), pages 819-821, June.
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    Citations

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    Cited by:

    1. Erhan Bayraktar & Jiaqi Li, 2016. "Stochastic Perron for Stochastic Target Problems," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 1026-1054, September.
    2. Bruno Bouchard & Ludovic Moreau & Mete H. Soner, 2016. "Hedging under an expected loss constraint with small transaction costs," Post-Print hal-00863562, HAL.
    3. Adrien Nguyen Huu & Nadia Oudjane, 2014. "Hedging Expected Losses on Derivatives in Electricity Futures Markets," Papers 1401.8271, arXiv.org.
    4. Cyril B'en'ezet & Jean-Franc{c}ois Chassagneux & Mohan Yang, 2023. "An optimal transport approach for the multiple quantile hedging problem," Papers 2308.01121, arXiv.org.
    5. Bruno Bouchard & Ludovic Moreau & Mete H. Soner, 2013. "Hedging under an expected loss constraint with small transaction costs," Papers 1309.4916, arXiv.org, revised Sep 2014.
    6. Cyril B'en'ezet & Jean-Franc{c}ois Chassagneux & Christoph Reisinger, 2019. "A numerical scheme for the quantile hedging problem," Papers 1902.11228, arXiv.org.

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    Keywords

    stochastic target problem; mixed diffusion process; discontinuous viscosity solutions; quantile hedging;
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