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

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic analysis

Published online by Cambridge University Press:  07 September 2017

Bo Gong  and
Weidong Zhao
Bo Gong*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Hong Kong, China
Weidong Zhao*
Affiliation:
School of Mathematics & Finance Institute, Shandong University, Jinan, Shandong 250100, China
*
*Corresponding author. Email addresses:[email protected] (B. Gong), [email protected] (W. Zhao)
*Corresponding author. Email addresses:[email protected] (B. Gong), [email protected] (W. Zhao)
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Abstract

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.

Keywords

Forward backward stochastic differential equationsfully discrete schemeerror estimate

MSC classification

Secondary: 60H35: Computational methods for stochastic equations 65C30: Stochastic differential and integral equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 548 - 565
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Bally, V., Approximation scheme for solutions of BSDE, in “Backward Stochastic Differential Equations” (Paris, 1995–1996), pp. 177–191, Pitman Res. Notes Math. Ser. 364, Longman, Harlow (1997).Google Scholar
[2] Bender, C. and Denk, R., A forward scheme for backward SDEs, Stochastic Process. Appl. 117, 1793–1812 (2007).CrossRefGoogle Scholar
[3] Fu, Y., Zhao, W. and Zhou, T., Multistep schemes for forward backward stochastic differential equations with jumps, J. Sci. Comput. 69, 651–672 (2016).CrossRefGoogle Scholar
[4] Gianin, E.R., Risk measures via g-expectations, Insurance Math. Econom. 39, 19–34 (2006).Google Scholar
[5] El Karoui, N., Peng, S. and Quenez, M.C., Backward stochastic differential equations in finance, Math. Finance 7, 1–71 (1997).Google Scholar
[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).Google Scholar
[7] Kong, T., Zhao, W. and Zhou, T., Probabilistic high order numerical schemes for fully nonlinear parabolic PDEs, Commun. Comput. Phys. 18, 1482–1503 (2015).Google Scholar
[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. 421–434, Birkhäuser, Boston (1994).Google Scholar
[9] Milstein, G.N. and Tretyakov, M.V., Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput. 28, 561–582 (2006).CrossRefGoogle Scholar
[10] Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Syst. Control Lett. 14, 55–61 (1990).CrossRefGoogle Scholar
[11] Peng, S., A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim. 28, 966–979 (1990).Google Scholar
[12] Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Rep. 37, 61–74 (1991).Google Scholar
[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, 165–253 (2004).Google Scholar
[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, 1563–1581 (2006).CrossRefGoogle Scholar
[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, 1585–1603 (2012).Google Scholar
[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, 905–924 (2009).Google Scholar
[17] Zhao, W., Zhang, G. and Ju, L., A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal. 48, 1369–1394 (2010).CrossRefGoogle Scholar
[18] Zhang, G., Gunzburger, M. and Zhao, W., A sparse-grid method for multi-dimensional backward stochastic differential equations, J. Comput. Math. 31, 221–248 (2013).CrossRefGoogle Scholar
[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, A1731–1751 (2014).Google Scholar
[20] Zhao, W., Li, Y. and Fu, Y., Second-order schemes for solving decoupled forward backward stochastic differential equations, Science China Math. 57, 665–686 (2014).Google Scholar
[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, 618–646 (2014).Google Scholar