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Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

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Abstract

In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson (C-N) scheme for solving decoupled FBSDEs. Based on the Taylor and Itô-Taylor expansions, the Malliavin calculus theory (e.g., the multiple Malliavin integration-by-parts formula), and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bender C, Denk R. A forward scheme for backward SDE. Stochastic Process Appl, 2007, 117: 1793–1812

    Article  MathSciNet  MATH  Google Scholar 

  3. Bender C, Zhang J. Time discretization and Markovian iteration for coupled FBSDEs. Ann Appl Probab, 2008, 18: 143–177

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouchard B, Touzi N. Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process Appl, 2004, 111: 175–206

    Article  MathSciNet  MATH  Google Scholar 

  5. Crisan D, Manolarakis K. Second order discretization of backward SDE and simulation with the cubature method. Ann Appl Probab, 2014, 24: 652–678

    Article  MathSciNet  MATH  Google Scholar 

  6. Cvitanic J, Zhang J. The steepest descent method for forward-backward SDE. Electron J Probab, 2005, 10: 1468–1495

    Article  MathSciNet  MATH  Google Scholar 

  7. Delarue F, Menozzi S. A forward-backward stochastic algorithm for quasi-linear PDEs. Ann Appl Probab, 2006, 16: 140–184

    Article  MathSciNet  MATH  Google Scholar 

  8. Douglas J, Ma J, Protter P. Numerical methods for forward-backward stochastic differential equations. Ann Appl Probab, 1996, 6: 940–968

    Article  MathSciNet  MATH  Google Scholar 

  9. Evans L C. Partial Differential Equations. Providence: Amer Math Soc, 1998

    MATH  Google Scholar 

  10. Gobet E, Labart C. Error expansion for the discretization of backward stochastic differential equations. Stochastic Process Appl, 2007, 117: 803–829

    Article  MathSciNet  MATH  Google Scholar 

  11. Gobet E, Lemmor J P, Warin X. A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann Appl Probab, 2005, 15: 2172–2202

    Article  MathSciNet  MATH  Google Scholar 

  12. Karoui N E, Peng S G, Quenez M C. Backward stochastic differential equations in finance. Math Finance, 1997, 7: 1–71

    Article  MathSciNet  MATH  Google Scholar 

  13. Kloeden P E, Platen E. Numerical Solution of Stochastic Differential Equations. Berlin: Springer-Verlag, 1992

    Book  MATH  Google Scholar 

  14. Ladyzenskaja O, Solonnikov V, Uralceva N. Linear and Quasilinear Equations of Parabolic Type. Providence: Amer Math Soc, 1968

    Google Scholar 

  15. Ma J, Protter P, San J M, et al. Numerical methods for backward stochastic differential equations. Ann Appl Probab, 2002, 12: 302–316

    Article  MathSciNet  MATH  Google Scholar 

  16. Ma J, Yong J. Forward-Backward Stochastic Differential Equations and Their Applications. Berlin: Springer-Verlag, 1999

    MATH  Google Scholar 

  17. Nualart D. The Malliavin Calculus and Related Topics. Berlin: Springer Verlag, 1995

    Book  MATH  Google Scholar 

  18. Pardoux E, Peng S G. Adapted solution of a backward stochastic differntial equation. Systems Control Lett, 1990, 14: 55–61

    Article  MathSciNet  MATH  Google Scholar 

  19. Pardoux E, Rascanu A. Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Berlin: Springer, 2014

    Book  MATH  Google Scholar 

  20. Peng S G. A general stochastic maximum principle for optimal control problems. SIAM J Control Optim, 1990, 28: 966–979

    Article  MathSciNet  MATH  Google Scholar 

  21. Peng S G. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch Stoch Rep, 1991, 37: 61–74

    Article  MathSciNet  MATH  Google Scholar 

  22. Peng S G. Backward SDE and related g-expectation, in backward stochastic differential equations In: Pitman Res Notes Math Ser, vol. 364. Harlow: Longman, 1997, 141–159

    Google Scholar 

  23. Peng S G, Wang F L. BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci China Math, 2016, 59: 19–36

    Article  MathSciNet  MATH  Google Scholar 

  24. Peng S G, Zhang H L. Stochastic calculus with respect to G-Brownian motion viewed through rough paths. Sci China Math, 2017, 60: 1–20

    Article  MathSciNet  Google Scholar 

  25. Yang J, Zhao W D. Convergence of recent multistep schemes for a forward-backward stochastic differential equation. East Asian J Appl Math, 2015, 5: 387–404

    Article  MathSciNet  Google Scholar 

  26. Zhang J F. A numerical scheme for BSDE. Ann Appl Probab, 2004, 14: 459–488

    Article  MathSciNet  Google Scholar 

  27. Zhang W, Zhao W D. Euler-type schemes for weakly coupled forward-backward stochastic differential equations and optimal convergence analysis. Front Math China, 2015, 10: 415–434

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhao W D, Chen L F, Peng S G. A new kind of accurate numerical method for backward stochastic differential equations. SIAM J Sci Comput, 2006, 28: 1563–1581

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhao W D, Fu Y, Zhou T. New kinds of high-order multi-step schemes for coupled forward backward stochastic differential equations. SIAM J Sci Comput, 2014, 36: 1731–1751

    Article  MathSciNet  Google Scholar 

  30. Zhao W D, Li Y, Fu Y. Second-order schemes for solving decoupled forward backward stochastic differential equations. Sci China Math, 2014, 57: 665–686

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhao W D, Li Y, Ju L L. Error estimates of the Crank-Nicolson scheme for solving backward stochastic differential equations. Int J Numer Anal Model, 2013, 10: 876–898

    MathSciNet  MATH  Google Scholar 

  32. Zhao W D, Li Y, Zhang G. A generalized θ-scheme for solving backward stochastic differential equations. Discrete Contin Dyn Syst Ser B, 2012, 17: 1585–1603

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhao W D, Wang J L, Peng S G. Error estimates of the θ-scheme for backward stochastic differential equations. Discrete Contin Dyn Syst Ser B, 2009, 12: 905–924

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhao W D, Zhang G N, Ju L L. A stable multistep scheme for solving backward stochastic differential equations. SIAM J Numer Anal, 2010, 48: 1369–1394

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhao W D, Zhang W, Ju L L. A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations. Commun Comput Phys, 2014, 15: 618–646

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by Shanghai University Young Teacher Training Program (Grant No. slg14032) and National Natural Science Foundations of China (Grant Nos. 11501366 and 11571206). The authors thank the referees very much for their valuable comments, which help them to improve our paper a lot.

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Yang Li

  2. School of Mathematics, Shandong University, Jinan, 250100, China

    Jie Yang & WeiDong Zhao

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  1. Yang Li
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  2. Jie Yang
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  3. WeiDong Zhao
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Correspondence to WeiDong Zhao.

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Li, Y., Yang, J. & Zhao, W. Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs. Sci. China Math. 60, 923–948 (2017). https://doi.org/10.1007/s11425-016-0178-8

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  • DOI: https://doi.org/10.1007/s11425-016-0178-8

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