Skip to main content
SpringerLink
Log in

A regression-based numerical scheme for backward stochastic differential equations

  • Original Paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

Based on Fourier cosine expansion, two approximations of conditional expectations are studied, and the local errors for these approximations are analyzed. Using these approximations and the theta-time discretization, a new and efficient numerical scheme, which is based on least-squares regression, for forward–backward stochastic differential equations is proposed. Numerical experiments are done to test the availability and stability of this new scheme for Black–Scholes call and calls combination under an empirical expression about volatility. Some conclusions are given.

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

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

    Article  MATH  MathSciNet  Google Scholar 

  • Bally V (1997) Approximation scheme for solutions of BSDE. Pitman research notes in mathematics series. Longman, Harlow

    MATH  Google Scholar 

  • Bender C, Steiner J (2012) Least-squares Monte Carlo for backward SDEs. In: Carmona RA et al (eds) Numerical methods in finance. Springer, Berlin

  • Bender CM, Orszag SA (1978) Advanced mathematical methods for scientists and engineers. International series in pure and applied mathematics. McGraw-Hill Book Co, New York

    MATH  Google Scholar 

  • Bismut JM (1973) Conjugate convex functions in optimal stochastic control. J Math Anal Appl 44(2):384–404

    Article  MATH  MathSciNet  Google Scholar 

  • Bouchaud B, Touzi N (2004) Discrete time approximation and Monte Carlo simulation of backward stochastic differential equations. Stoch Process Appl 111(2):175–206

    Article  MATH  MathSciNet  Google Scholar 

  • Boyd JP (2001) Chebyshev and Fourier spectral methods, 2nd edn. Dover Publications Inc, Mineola

    MATH  Google Scholar 

  • Fang F, Oosterlee CW (2008) A novel pricing method for European options based on Fourier-cosine series expansions. SIAM J Sci Comput 31(2):826–848

    Article  MATH  MathSciNet  Google Scholar 

  • Gobet E, Lemor JP, Warin X (2005) A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann Appl Probab 15(3):2172–2202

    Article  MATH  MathSciNet  Google Scholar 

  • Huijskens TP, Ruijter MJ, Oosterlee CW (2016) Efficient numerical Fourier methods for coupled forward–backward SDEs. J Comput Appl Math 296:593–612

    Article  MATH  MathSciNet  Google Scholar 

  • Lemor JP, Gobet E, Warin X (2006) Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12(5):889–916

    Article  MATH  MathSciNet  Google Scholar 

  • Ma J, Protter P, Yong JM (1994) Solving forward–backward stochastic differential equations explicitly: a four step scheme. Probab Theory Related Fields 98(3):339–359

    Article  MATH  MathSciNet  Google Scholar 

  • Pardoux E, Peng SG (1990) Adapted solution of a backward stochastic differential equation. Syst Control Lett 14(1):55–61

    Article  MATH  MathSciNet  Google Scholar 

  • Pardoux E, Peng SG (1992) Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic partial differential equations and their applications (Charlotte, NC, 1991). Lecture notes control and information sciences, vol 176. Springer, Berlin, pp 200–217

  • Pham H (2015) Feynman–Kac representation of fully nonlinear PDEs and applications. Acta Math Vietnam 40(2):255–269

    Article  MATH  MathSciNet  Google Scholar 

  • Ruijter MJ, Oosterlee CW (2015) A Fourier cosine method for an efficient computation of solutions to BSDEs. SIAM J Sci Comput 37(2):859–889

    Article  MATH  MathSciNet  Google Scholar 

  • Ruijter MJ, Oosterlee CW (2016) Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance. Appl Numer Math 103:1–26

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang J (2004) A numerical scheme for BSDEs. Ann Appl Probab 14(1):459–488

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao W, Chen L, Peng SG (2006) A new kind of accurate numerical method for backward stochastic differential equations. SIAM J Sci Comput 28(4):1563–1581

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao W, Wang J, Peng SG (2009) Error estimates of the theta-scheme for backward stochastic differential equations. Discrete Contin Dyn Syst Ser B 12(4):905–924

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao W, Zhang W, Ju L (2014) A numerical method and its error estimates for the decoupled forward–backward stochastic differential equations. Commun Comput Phys 15(3):618–646

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees for their patience and significant suggestions which greatly improve the quality of this paper. We would also like to thank the editors for handling this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Macau, Macao, China

    Deng Ding, Xiaofei Li & Yiqi Liu

  2. Kuang-Chi Institute of Advanced Technology, Shenzhen, China

    Yiqi Liu

Authors
  1. Deng Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaofei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yiqi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yiqi Liu.

Additional information

This work was supported by the research Grants MYRG068(Y2-L2)-FST13-DD from University of Macau and 081/2016/A2 from FDCT of Macao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ding, D., Li, X. & Liu, Y. A regression-based numerical scheme for backward stochastic differential equations. Comput Stat 32, 1357–1373 (2017). https://doi.org/10.1007/s00180-017-0763-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-017-0763-x

Keywords

Search

Navigation