Skip to main content

Optimal Error Estimates for a Fully Discrete Euler Scheme for Decoupled Forward Backward Stochastic Differential Equations

  • Bo Gong (a1) and Weidong Zhao (a2)

In error estimates of various numerical approaches for solving decoupled forward backward stochastic differential equations (FBSDEs), the rate of convergence for one variable is usually less than for the other. Under slightly strengthened smoothness assumptions, we show that the fully discrete Euler scheme admits a first-order rate of convergence for both variables.

Corresponding author
*Corresponding author. Email addresses: (B. Gong), (W. Zhao)
Hide All
[1] Bally V., Approximation scheme for solutions of BSDE, in “Backward Stochastic Differential Equations” (Paris, 1995–1996), pp. 177191, Pitman Res. Notes Math. Ser. 364, Longman, Harlow (1997).
[2] Bender C. and Denk R., A forward scheme for backward SDEs, Stochastic Process. Appl. 117, 17931812 (2007).
[3] Fu Y., Zhao W. and Zhou T., Multistep schemes for forward backward stochastic differential equations with jumps, J. Sci. Comput. 69, 651672 (2016).
[4] Gianin E.R., Risk measures via g-expectations, Insurance Math. Econom. 39, 1934 (2006).
[5] El Karoui N., Peng S. and Quenez M.C., Backward stochastic differential equations in finance, Math. Finance 7, 171 (1997).
[6] Li Y., Yang J. and Zhao W., Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs, Sci. China Math. 60, doi: 10.1007/s11425-016-0178-8 (2017).
[7] Kong T., Zhao W. and Zhou T., Probabilistic high order numerical schemes for fully nonlinear parabolic PDEs, Commun. Comput. Phys. 18, 14821503 (2015).
[8] Mastroianni G. and Monegato G., Error estimates for Gauss-Laguerre and Gauss-Hermite quadrature formulas, in Approximation and Computation: A Festschrift in Honor of Walter Gautschi. Eds. Zahar R. V. M., pp. 421434, Birkhäuser, Boston (1994).
[9] Milstein G.N. and Tretyakov M.V., Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput. 28, 561582 (2006).
[10] Pardoux E. and Peng S., Adapted solution of a backward stochastic differential equation, Syst. Control Lett. 14, 5561 (1990).
[11] Peng S., A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim. 28, 966979 (1990).
[12] Peng S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Rep. 37, 6174 (1991).
[13] Peng S., Nonlinear expectations, nonlinear evaluations and risk measures in Stochastic Methods in Finance: Lectures given at the C.I.M.E.-E.M.S. Summer School held in Bressanone/Brixen, Italy, July 6-12, 2003, Springer Berlin Heidelberg, Berlin, Heidelberg, 165253 (2004).
[14] Zhao W., Chen L. and Peng S., A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput. 28, 15631581 (2006).
[15] Zhao W., Li Y. and Zhang G., A generalized θ-scheme for solving backward stochastic differential equations, Discrete and Continuous Dynamic Systems Series B 17, 15851603 (2012).
[16] Zhao W., Wang J. and Peng S., Error Estimates of the θ-scheme for backward stochastic differential equations, Dis. Cont. Dyn. Sys. B 12, 905924 (2009).
[17] Zhao W., Zhang G. and Ju L., A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal. 48, 13691394 (2010).
[18] Zhang G., Gunzburger M. and Zhao W., A sparse-grid method for multi-dimensional backward stochastic differential equations, J. Comput. Math. 31, 221248 (2013).
[19] Zhao W., Fu Y. and Zhou T., New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput. 36, A17311751 (2014).
[20] Zhao W., Li Y. and Fu Y., Second-order schemes for solving decoupled forward backward stochastic differential equations, Science China Math. 57, 665686 (2014).
[21] Zhao W., Zhang W. and Ju L., A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations, Commun. Comput. Phys. 15, 618646 (2014).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 2
Total number of PDF views: 23 *
Loading metrics...

Abstract views

Total abstract views: 130 *
Loading metrics...

* Views captured on Cambridge Core between 7th September 2017 - 19th February 2018. This data will be updated every 24 hours.