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BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations

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  • El-Karoui, N.
  • Hamadène, S.

Abstract

We deal with the risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Using backward stochastic differential equations we show the existence of an optimal control and, a saddle-point and an equilibrium point for respectively the zero-sum and nonzero-sum games.

Suggested Citation

  • El-Karoui, N. & Hamadène, S., 2003. "BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 145-169, September.
    • Handle: RePEc:eee:spapps:v:107:y:2003:i:1:p:145-169
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    References listed on IDEAS

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    1. Pauline Barrieu & Nicole El Karoui, 2002. "Optimal design of derivatives in illiquid markets," Quantitative Finance, Taylor & Francis Journals, vol. 2(3), pages 181-188.
    2. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
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    Cited by:

    1. Jun Moon, 2022. "State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations," Mathematics, MDPI, vol. 10(10), pages 1-32, May.
    2. Robertson, Scott & Xing, Hao, 2015. "Large time behavior of solutions to semi-linear equations with quadratic growth in the gradient," LSE Research Online Documents on Economics 60578, London School of Economics and Political Science, LSE Library.
    3. Dmitry Kramkov & Sergio Pulido, 2016. "Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model," Post-Print hal-01181147, HAL.
    4. Hamadène, Said & Mu, Rui, 2020. "Discontinuous Nash equilibrium points for nonzero-sum stochastic differential games," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6901-6926.
    5. Arnab Basu & Mrinal K. Ghosh, 2012. "Zero-Sum Risk-Sensitive Stochastic Differential Games," Mathematics of Operations Research, INFORMS, vol. 37(3), pages 437-449, August.
    6. Bäuerle, Nicole & Rieder, Ulrich, 2017. "Zero-sum risk-sensitive stochastic games," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 622-642.
    7. Giovanni Mottola, 2014. "Generalized Dynkin game of switching type representation for defaultable claims in presence of contingent CSA," Papers 1410.0594, arXiv.org, revised Jan 2015.
    8. Xing, Hao & Žitković, Gordan, 2018. "A class of globally solvable Markovian quadratic BSDE systems and applications," LSE Research Online Documents on Economics 73440, London School of Economics and Political Science, LSE Library.
    9. Arnab Basu & Mrinal K. Ghosh, 2018. "Nonzero-Sum Risk-Sensitive Stochastic Games on a Countable State Space," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 516-532, May.
    10. Hamadène, S. & Wang, H., 2009. "BSDEs with two RCLL reflecting obstacles driven by Brownian motion and Poisson measure and a related mixed zero-sum game," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2881-2912, September.
    11. Said Hamadène & Rui Mu, 2021. "Risk-Sensitive Nonzero-Sum Stochastic Differential Game with Unbounded Coefficients," Dynamic Games and Applications, Springer, vol. 11(1), pages 84-108, March.
    12. Basu, Arnab & Ghosh, Mrinal Kanti, 2014. "Zero-sum risk-sensitive stochastic games on a countable state space," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 961-983.
    13. E. Savku & G.-W Weber, 2022. "Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market," Annals of Operations Research, Springer, vol. 312(2), pages 1171-1196, May.
    14. Chen, Fang & Guo, Xianping, 2023. "Two-person zero-sum risk-sensitive stochastic games with incomplete reward information on one side," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 218-245.
    15. Zhou, Qing & Ren, Yong, 2012. "Reflected backward stochastic differential equations with time delayed generators," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 979-990.
    16. Djordjević, Jasmina & Janković, Svetlana, 2015. "Backward stochastic Volterra integral equations with additive perturbations," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 903-910.
    17. Boualem Djehiche & Hamidou Tembine, 2014. "Risk-Sensitive Mean-Field Type Control under Partial Observation," Papers 1411.7231, arXiv.org.
    18. Bahlali, Khaled & Hamadène, SaI¨d & Mezerdi, Brahim, 2005. "Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient," Stochastic Processes and their Applications, Elsevier, vol. 115(7), pages 1107-1129, July.
    19. Kihun Nam, 2019. "Global Well-posedness of Non-Markovian Multidimensional Superquadratic BSDE," Papers 1912.03692, arXiv.org, revised Jan 2022.

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