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Stochastic Perron for Stochastic Target Problems

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  • Erhan Bayraktar
  • Jiaqi Li

Abstract

In this paper, we adapt stochastic Perron's method to analyze a stochastic target problem with unbounded controls in a jump diffusion set-up. With this method, we construct a viscosity sub-solution and super-solution to the associated Hamiltonian-Jacobi-Bellman (HJB) equations. Under comparison principles, uniqueness of the viscosity solutions holds and the value function coincides with the unique solution in the parabolic interior. Since classical control problems can be analyzed under the framework of stochastic target problems (with unbounded controls), we use our results to generalize the results in ArXiv:1212.2170 to problems with controlled jumps.

Suggested Citation

  • Erhan Bayraktar & Jiaqi Li, 2016. "Stochastic Perron for Stochastic Target Problems," Papers 1604.03906, arXiv.org, revised May 2016.
    • Handle: RePEc:arx:papers:1604.03906
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    References listed on IDEAS

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    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. 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.
    4. 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.
    5. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(4), pages 1007-1017, August.
    6. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(6), pages 1461-1465, December.
    7. Erhan Bayraktar & Jiaqi Li, 2014. "Stochastic Perron for stochastic target games," Papers 1408.6799, arXiv.org, revised Apr 2016.
    8. Ludovic Moreau, 2011. "Stochastic target problems with controlled loss in jump diffusion models," Post-Print hal-00515522, HAL.
    9. Bruno Bouchard & Marcel Nutz, 2016. "Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 109-124, February.
    10. Bruno Bouchard & Marcel Nutz, 2015. "Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions," Post-Print hal-00846830, HAL.
    11. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(5), pages 1273-1289, October.
    12. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(3), pages 819-821, June.
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    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.
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