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On weak solutions of forward–backward SDEs
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  • Published: 25 June 2010

On weak solutions of forward–backward SDEs

  • Jin Ma1 &
  • Jianfeng Zhang1 

Probability Theory and Related Fields volume 151, pages 475–507 (2011)Cite this article

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Abstract

In this paper we continue exploring the notion of weak solution of forward–backward stochastic differential equations (FBSDEs) and associated forward–backward martingale problems (FBMPs). The main purpose of this work is to remove the constraints on the martingale integrands in the uniqueness proofs in our previous work (Ma et al. in Ann Probab 36(6):2092–2125, 2008). We consider a general class of non-degenerate FBSDEs in which all the coefficients are assumed to be essentially only bounded and uniformly continuous, and the uniqueness is proved in the space of all the square integrable adapted solutions, the standard solution space in the FBSDE literature. A new notion of semi-strong solution is introduced to clarify the relations among different definitions of weak solution in the literature, and it is in fact instrumental in our uniqueness proof. As a by-product, we also establish some a priori estimates of the second derivatives of the solution to the decoupling quasilinear PDE.

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References

  1. Antonelli F.: Backward-forward stochastic differential equations. Ann. Appl. Probab. 3(3), 777–793 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antonelli F., Ma J.: Weak solutions of forward-backward SDE’s. Stoch. Anal. Appl. 21(3), 493–514 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barles G., Biton S., Bourgoing M., Ley O.: Uniqueness results for quasilinear parabolic equations through viscosity solutions’ methods. Calc. Var. 18, 159–179 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouchard, B., Touzi, N.: Weak dynamic programming principle for viscosity solutions. (2009, preprint)

  5. Buckdahn, R., Engelbert, H.-J.: A backward stochastic differential equation without strong solution. Teor. Veroyatn. Primen. 50(2), 390–396 (2005); translation in Theory Probab. Appl. 50(2), 284–289 (2006)

  6. Buckdahn, R., Engelbert, H.-J.: On the continuity of weak solutions of backward stochastic differential equations. Teor. Veroyatn. Primen. 52(1), 190–199 (2007); translation in Theory Probab. Appl. 52(1), 152–160 (2008)

  7. Buckdahn R., Engelbert H.-J., Rascanu A.: On weak solutions of backward stochastic differential equations. Teor. Veroyatn. Primen. 49(1), 70–108 (2004)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Delarue F.: On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stoch. Process. Appl. 99, 209–286 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Delarue, F.: Estimates of the Solutions of a System of Quasi-linear PDEs. A Probabilistic Scheme. Séminaire de Probabilités XXXVII. Lecture Notes in Mathematics, vol. 1832, pp. 290–332. Springer, Berlin (2003)

  11. Delarue F., Guatteri G.: Weak existence and uniqueness for forward-backward SDEs. Stoch. Process. Appl. 116(12), 1712–1742 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fabes E., Kenig C.: Examples of singular parabolic measures and singular transition probability densities. Duke Math. J. 48, 845–856 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hu Y., Peng S.: Solution of forward-backward stochastic differential equations. Probab. Theory Relat. Fields 103(2), 273–283 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ikeda N., Watanabe S.: Stochastic Differential Equations and Diffusion Processes. North Holland, Amstrerdam (1981)

    MATH  Google Scholar 

  15. Ishii H.: Degenerate parabolic PDEs with discontinuities and generalized evolutions of surfaces. Adv. Differ. Equ. 1(1), 51–72 (1996)

    MATH  Google Scholar 

  16. Ishii H., Lions P.-L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83, 26–78 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kobylanski M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28(2), 558–602 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kurtz T.: The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities. Electron. J. Probab. 12, 951–965 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Lepeltier M., San Martin J.: Backward stochastic differential equations with nonLipschitz coefficients. Stat. Prob. Lett. 32(4), 425–430 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lieberman G.: Second Order Parabolic Differential Equations. World Scientific Publishing Co., Inc., River Edge (1996)

    MATH  Google Scholar 

  21. Ma J., Protter P., Yong J.: Solving forward-backward stochastic differential equations explicitly— a four step scheme. Probab. Theory Relat. Fields 98, 339–359 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ma J., Yong J.: Solvability of forward-backward SDEs and the nodal set of Hamilton-Jaccobi-Bellman equations. Chin. Ann. Math. 16B, 279–298 (1995)

    MathSciNet  Google Scholar 

  23. Ma, J., Yong, J.: Forward-backward stochastic differential equations and their applications. Lecture Notes in Mathematics, vol. 1702. Springer, New York (1999)

  24. Ma J., Zhang J., Zheng Z.: Weak solutions for forward-backward SDEs—a martingale problem approach. Ann. Probab. 36(6), 2092–2125 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80(4), 931–954 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pardoux E., Peng S.: Adapted solutions of backward stochastic equations. Syst. Control Lett. 14, 55–61 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. Lecture Notes in CIS, vol. 176, pp. 200–217. Springer, Berlin (1992)

  28. Pardoux E., Tang S.: Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Relat. Fields 114(2), 123–150 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Peng S., Wu Z.: Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim. 37(3), 825–843 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Protter P.: Integration and Stochastic Differential Equations. Springer, Berlin (1990)

    MATH  Google Scholar 

  31. Stroock D.W., Varadhan S.R.S.: Multidimensional Diffusion Processes. Springer-Verlag, Berlin (1979)

    MATH  Google Scholar 

  32. Yong J.: Finding adapted solutions of forward-backward stochastic differential equations: method of continuation. Probab. Theory Relat. Fields 107(4), 537–572 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang J.: The wellposedness of FBSDEs. Discrete Contin. Dyn. Syst. Ser. B 6(4), 927–940 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Jin Ma & Jianfeng Zhang

Authors
  1. Jin Ma
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  2. Jianfeng Zhang
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Corresponding author

Correspondence to Jin Ma.

Additional information

J. Ma is supported in part by NSF grants #0835051 and #0806017. J. Zhang is supported in part by NSF grant #0631366.

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Cite this article

Ma, J., Zhang, J. On weak solutions of forward–backward SDEs. Probab. Theory Relat. Fields 151, 475–507 (2011). https://doi.org/10.1007/s00440-010-0305-8

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  • Received: 01 June 2009

  • Revised: 19 February 2010

  • Published: 25 June 2010

  • Issue Date: December 2011

  • DOI: https://doi.org/10.1007/s00440-010-0305-8

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Keywords

  • Forward–backward stochastic differential equations
  • Weak solution
  • Forward–backward martingale problems
  • Viscosity solutions

Mathematics Subject Classification (2000)

  • Primary 60H10
  • Secondary 34F05
  • 90A12
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