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A Multistep Scheme for Decoupled Forward-Backward Stochastic Differential Equations

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

Published online by Cambridge University Press:  24 May 2016

Weidong Zhao ,
Wei Zhang  and
Lili Ju
Weidong Zhao*
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, P. R. China
Wei Zhang*
Affiliation:
College of Applied Sciences, Beijing University of Technology, Beijing, 100022, P. R. China
Lili Ju*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
*
*Corresponding author. Email addresses:[email protected](W. Zhao), [email protected](W. Zhang), [email protected](L. Ju)
*Corresponding author. Email addresses:[email protected](W. Zhao), [email protected](W. Zhang), [email protected](L. Ju)
*Corresponding author. Email addresses:[email protected](W. Zhao), [email protected](W. Zhang), [email protected](L. Ju)
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Abstract

Upon a set of backward orthogonal polynomials, we propose a novel multi-step numerical scheme for solving the decoupled forward-backward stochastic differential equations (FBSDEs). Under Lipschtiz conditions on the coefficients of the FBSDEs, we first get a general error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are presented to demonstrate the accuracy of the proposed multi-step scheme and to numerically verify the theoretical results.

Keywords

Decoupled forward-backward stochastic differential equationsbackward orthogonal polynomialsmulti-step numerical schemeerror estimatenumerical analysis

MSC classification

Secondary: 60H35: Computational methods for stochastic equations 65C20: Models, numerical methods 60H10: Stochastic ordinary differential equations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 2 , May 2016 , pp. 262 - 288
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Bally, V., An approximation scheme for BSDEs and applications to control and nonlinear PDEs, Pitman Research Notes in Mathematics Series, Longman, 364 (1997).Google Scholar
[2]Bender, C. and Denk, R., A forward scheme for backward SDEs, Stochastic Processes and their Applications, 117 (2007), pp. 1793–1812.Google Scholar
[3]Bouchard, B. and Elie, R., Discrete-time approximation of decoupled Forward-Backward SDE with jumps, Stochastic Processes and their Applications, 118 (2008), pp. 53–75.Google Scholar
[4]Bouchard, B. and Touzi, N., Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications, 111 (2004), pp. 175–206.Google Scholar
[5]Chassagneux, J.F., Linear multi-step schemes for BSDEs, arxiv:1306.5548, preprint.Google Scholar
[6]Chassagneux, J.F. and Crisen, D., Runge-Kutta schemes for BSDEs, forthcoming in Annals of Applied Probability, 2012.Google Scholar
[7]Chevance, D., Numerical Methods for Backward Stochastic Differential Equations, in Numerical Methods in Finance, Edited by Rogers, L. C. G. and Talay, D., Cambridge University Press, 1997, pp. 232–244.Google Scholar
[8]Cvitanic, J. and Zhang, J., The Steepest Descent Method for FBSDEs, Electronic J. Probab., 10 (2005), pp. 1468–1495.Google Scholar
[9]Douglas, J. Jr., Ma, J. and Protter, P., Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), pp. 940–968.Google Scholar
[10]Delarue, F. and Menozzi, S., A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl, Probab., 16 (2006), pp. 140–184.Google Scholar
[11]Gianin, E., Risk measures via g-expectations, Insurance: Mathematics and Economics, 39 (2006), pp. 19–34.Google Scholar
[12]Gobet, E. and Labart, C., Error expansion for the discretization of Backward Stochastic Differential Equations, Stochastic Process anf their Applications, 117 (2007) pp. 803–829.CrossRefGoogle Scholar
[13]Karoui, N., Peng, S., and Quenez, M. C., Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), pp. 1–71.Google Scholar
[14]Kloeden, P. and Platen, E., Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, Third Printing, 1999.Google Scholar
[15]Ladyzenskaja, O., Solonnikov, O., Uralceva, N., Linear and Quasi-Linear Equations of Parabolic Type, Translations of Math. Monographs, 23, AMS, Providence Rhode Island, 1968.Google Scholar
[16]Li, Y. and Zhao, W., Lp-error estimates for numerical schemes for solving certain kinds of backward stochastic differential equations, Statistics and Probability Letters, 80 (2010), pp. 1612–1617.Google Scholar
[17]Milstein, G. and Tretyakov, M. V., Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), pp. 561–582.CrossRefGoogle Scholar
[18]Milstein, G. and Tretyakov, M. V., Discretization of forward-backward stochastic differential equations and related quasi-linear parabolic equations, IMA J. Numer. Anal., 27 (2007), pp. 24–44.CrossRefGoogle Scholar
[19]Ma, J., Protter, P., San Martin, J., and Torres, S., Numerical methods for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), pp. 302–316.Google Scholar
[20]Ma, J., Protter, P., and Yong, J., Solving forward-backward stochastic differential equations explicitly-a four step scheme, Probability Theory and Related Fields, 98 (1994), pp. 339–359.Google Scholar
[21]Ma, J., Shen, J., and Zhao, Y., On Numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal., 46 (2008), pp. 2636–2661.Google Scholar
[22]Ma, J. and Yong, J., Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702, Springer, 1999.Google Scholar
[23]Ma, J. and Zhang, J., Representation theorems for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), pp. 1390–1418.Google Scholar
[24]Mémin, J., Peng, S., and Xu, M., Convergence of solutions of discrete reflected backward SDE's and simulations, preprint of Institut de Recherche Mathmatique de Rennes, Rennes, 2003.Google Scholar
[25]Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differntial equation, Systems and Control Letters, 14 (1990), pp. 55–61.Google Scholar
[26]Peng, S., A general stochastic maximum principle for optimal control problems, SIAM J. Control Optimization, 28 (1990), pp. 966–979.Google Scholar
[27]Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports, 37 (1991), pp. 61–74.Google Scholar
[28]Peng, S., Backward SDE and Related g-Expectation, Backward Stochastic Differential Equations, Pitman Research Notes in Math. Series, 364 (1997), pp. 141–159.Google Scholar
[29]Peng, S., A linear approximation algorithm using BSDE, Pacific Economic Review, 4 (1999), pp. 285–291.Google Scholar
[30]Prato, G. and Zabczyk, J., Second Order Partial Differential Equations in Hilbert Spaces, Cambridge University Press, 2002.Google Scholar
[31]Süli, E. and Mayers, D., An Introduction to Numerical Analysis, Cambridge University Press, 2003.Google Scholar
[32]Wang, J., Luo, C. and Zhao, W., Crank-Nicolson scheme and its error estimates for Backward Stochastic Differential Equations, Acta Mathematicae Applicatae Sinica (English Series), 2010. DOI: 10.1007/s10255-009-9051-z.Google Scholar
[33]Zhang, J., A numerical scheme for BSDEs, Ann. Appl. Prob., 14 (2004), pp. 459–488.Google Scholar
[34]Zhang, Y. and Zheng, W., Discretizing a backward stochastic differential equation, Inter. J. Math. Mathematical., 32 (2002), pp. 103–116.Google Scholar
[35]Zhang, G., Gunzburger, M. and Zhao, W., A sparse-grid method for multi-dimensional backward stochastic differential equations, J. Comput. Math., 31 (2013), pp. 221–248.Google Scholar
[36]Zhao, W., Wang, J., and Peng, S., Error estimates of the θ-scheme for backward Stochastic differential equations, Dis. Con. Dyn. Sys. B, 12 (2009), pp. 905–924.Google Scholar
[37]Zhao, W., Chen, L., and Peng, S., A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), pp. 1563–1581.Google Scholar
[38]Zhao, W., Li, Y. and Ju, L., Error Estimates of the Crank-Nicolson Scheme for Solving Backward Stochastic Differential Equations, Inter. J. Numer. Anal. Model., 10 (2013), pp. 876–898.Google Scholar
[39]Zhao, W., Li, Y. and Zhang, G., A generalized θ-scheme for solving backward stochastic differential equations, Dis. Con. Dyn. Sys. B, 5 (2012), pp. 1585–1603.Google Scholar
[40]Zhao, W., Zhang, G., Ju, L., A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), pp. 1369–1394.Google Scholar