Skip to main content
SpringerLink
Log in

Stochastic differential games with reflection and related obstacle problems for Isaacs equations

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997), which turns out to be very useful because it allows us to estimate the L p-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barles, G., Buckdahn, R., Pardoux, E. Backward stochastic differential equations and integral-partial differential equations. Stochastics and Stochastics Reports, 60: 57–83 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Buckdahn, R., Cardaliaguet, P., Rainer, C. Nash equilibrium payoffs for nonzero-sum stochastic differential games. SIAM J. Cont. Opt., 43(2): 624–642 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buckdahn, R., Li, J. Stochastic Differential Games and Viscosity Solutions of Hamilton-Jacobi-Bellman-Isaacs Equations. Siam. J. Control. Optim., 47(1): 444–475 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buckdahn, R., Li, J. Probabilistic Interpretation for Systems of Isaacs Equations with Two Reflecting Barriers. Nonlinear Differential Equations and Applications, 16: 381–420 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Crandall, M.G., Ishii, H., Lions, P.L. User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 27: 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cvitanic, J., Karatzas, I. Backward SDE’s with reflection and Dynkin Games. Annals of Probability, 24: 2024–2056 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. El Karoui, N., Peng, S., Quenez, M.C. Backward stochastic differential equations in finance. Math. Finance, 7(1): 1–71 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M.C. Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Annals. Probability, 25(2): 702–737 (1997)

    Article  MATH  Google Scholar 

  9. El Karoui, N., Pardoux, E., Quenez, M.C. Reflected Backward SDEs and American options. In: Numerical methods in finance, ed. by L. Robers and D. Talay, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1997, 215–231

    Google Scholar 

  10. Fleming, W.H., Souganidis, P.E. On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J., 38(2): 293–314 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hamadène, S., Lepeltier, J.P. Zero-sum stochastic differential games and backward equations. Systems and Control Letters, 24: 259–263 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hamadène, S., Lepeltier, J.P. Reflected BSDEs and mixed game problems. Stochastic processes and their applications, 85: 177–188 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hamadène, S., Lepeltier, J.P., Peng, S. BSDEs with continuous coefficients and stochastic differential games. In: Backward stochastic differential equations, ed. by N. El Karoui and L. Mazliak, Harlow: Longman. Pitman Res. Notes Math. Ser. 364, 1997, 115–128

    Google Scholar 

  14. Hamadène, S., Rotenstein, E., Zalinescu, Z. A Generalized Mixed Zero-sum Stochastic Differential Game and Double Barrier Reflected BSDEs with Quadratic Growth Coefficient. arXiv, posted on 9 Jul 2008, last revision on 6 Sep 2008; accepted for publication

  15. Hou, S., Tang, S. Switching games of stochastic differential systems. SIAM J. Cont. Opt., 46(3): 900–929 (2006)

    MathSciNet  Google Scholar 

  16. Ikeda, N., Watanabe, S. Stochastic differential equations and diffusion processes. North Holland-Kodansha, Amsterdam-Tokyo, 1989

    MATH  Google Scholar 

  17. Karatzas, I., Shreve, S.E. Brownian motion and stochastic calculus. Springer-Verlag, New York, 1987

    Google Scholar 

  18. Pardoux, E., Peng, S. Adapted solution of a backward stochastic differential equation. Systems Control Lett., 14(2): 61–74 (1990)

    MathSciNet  Google Scholar 

  19. Peng, S. BSDE and stochastic optimizations; Topics in stochastic analysis. YAN, J., PENG, S., FANG, S. and WU, L. Ch.2, Science Press. Beijing, 1997 (in Chinese)

    Google Scholar 

  20. Peng, S. A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics and Stochastics Reports, 38: 119–134 (1992)

    MathSciNet  MATH  Google Scholar 

  21. Wu, Z., Yu, Z. Dynamic programming principle for one kind of stochastic recursive optimal control problem and Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim., 47(5): 2616–2641 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Bretagne Occidentale, 6, avenue Victor-le-Gorgeu, B.P. 809, 29285, Brest cedex, France

    Rainer Buckdahn

  2. School of Mathematics and Statistics, Shandong University at Weihai, Weihai, 264209, China

    Juan Li

  3. Institute for Advanced Study, Shangdong University, Jinan, 250100, China

    Rainer Buckdahn & Juan Li

Authors
  1. Rainer Buckdahn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Li.

Additional information

The first author was supported by the Agence Nationale de la Recherche (France), reference ANR-10-BLAN 0112, and by the Marie Curie ITN “Controlled Systems”, call: FP7-PEOPLE-2007-1-1-ITN, no. 213841-2. The second author was supported by the National Natural Science Foundation of China (No. 10701050, 11071144), National Basic Research Program of China (973 Program) (No. 2007CB814904), Shandong Province (No. Q2007A04), Independent Innovation Foundation of Shandong University and the Project-sponsored by SRF for ROCS, SEM.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Buckdahn, R., Li, J. Stochastic differential games with reflection and related obstacle problems for Isaacs equations. Acta Math. Appl. Sin. Engl. Ser. 27, 647–678 (2011). https://doi.org/10.1007/s10255-011-0068-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-011-0068-8

Keywords

2000 MR Subject Classification

Search

Navigation