June  2022, 7(2): 133-150. doi: 10.3934/puqr.2022009

On the speed of convergence of Picard iterations of backward stochastic differential equations

1. 

Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, Essen, Nordrhein-Westfalen 45127, Germany

2. 

Institute of Mathematics, University of Giessen, Arndtstraße 2, Giessen 35392, Germany

thomas.kruse@math.uni-giessen.de

Received  July 07, 2021 Accepted  June 23, 2022 Published  June 2022 Early access  June 2022

Fund Project: This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, Grant No. HU1889/7-1).

It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearities converge at least exponentially fast to the solution. In this paper we prove that this convergence is in fact at least square-root factorially fast. We show for one example that no higher convergence speed is possible in general. Moreover, if the nonlinearity is z-independent, then the convergence is even factorially fast. Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations.

Citation: Martin Hutzenthaler, Thomas Kruse, Tuan Anh Nguyen. On the speed of convergence of Picard iterations of backward stochastic differential equations. Probability, Uncertainty and Quantitative Risk, 2022, 7 (2) : 133-150. doi: 10.3934/puqr.2022009
References:
[1]

Beck, C., Hornung, F., Hutzenthaler, M., Jentzen, A. and Kruse, T., Overcoming the curse of dimensionality in the numerical approximation of Allen−Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations, Journal of Numerical Mathematics, 2020, 28(4): 197−222. doi: 10.1515/jnma-2019-0074.

[2]

Bender, C. and Denk, R., A forward scheme for backward SDEs, Stochastic Process. Appl., 2007, 117(12): 1793−1812.

[3]

Bender, C. and Zhang, J., Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab., 2008, 18(1): 143−177.

[4]

Briand, P. and Labart, C., Simulation of BSDEs by Wiener chaos expansion, Ann. Appl. Probab., 2014, 24(3): 1129−1171.

[5]

Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1992.

[6]

E, W., Hutzenthaler, M., Jentzen, A. and Kruse, T., On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations, Journal of Scientific Computing, 2019, 79(3): 1534−1571. doi: 10.1007/s10915-018-00903-0.

[7]

E, W., Hutzenthaler, M., Jentzen, A. and Kruse, T., Multilevel Picard iterations for solving smooth semilinear parabolic heat equations, Partial Differential Equations and Applications, 2021, 2(6): 80.

[8]

El Karoui, N., Peng, S. and Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 1997, 7(1): 1−71. doi: 10.1111/1467-9965.00022.

[9]

Geiss, C. and Labart, C., Simulation of BSDEs with jumps by Wiener chaos expansion, Stochastic Process. Appl., 2016, 126(7): 2123−2162. doi: 10.1016/j.spa.2016.01.006.

[10]

Gobet, E. and Labart, C., Solving BSDE with adaptive control variate, SIAM J. Numer. Anal., 2010, 48(1): 257−277. doi: 10.1137/090755060.

[11]

Hutzenthaler, M., Jentzen, A. and Kruse, T., Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities, Foundations of Computational Mathematics, 2021.

[12]

Hutzenthaler, M., Jentzen, A., Kruse, T. and Nguyen, T. A., Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities, arXiv: 2009.02484, 2020.

[13]

Hutzenthaler, M., Jentzen, A., Kruse, T. and Nguyen, T. A., Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations, Accepted in Journal of Numerical Mathematics, 2022.

[14]

Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T. A. and von Wurstem-berger, P., Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2020, 476(2244): 20190630. doi: 10.1098/rspa.2019.0630.

[15]

Hutzenthaler, M. and Kruse, T., Multilevel Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities, SIAM Journal on Numerical Analysis, 2020, 58(2): 929−961. doi: 10.1137/17M1157015.

[16]

Labart, C. and Lelong, J., A parallel algorithm for solving BSDEs, Monte Carlo Methods Appl., 2013, 19(1): 11−39.

[17]

Øksendal, B., Stochastic Differential Equations: An Introduction with Applications, Universitext, Springer, Berlin, Heidelberg, 1985.

[18]

Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 1990, 14(1): 55−61. doi: 10.1016/0167-6911(90)90082-6.

[19]

Pardoux, E. and Răşcanu, A. Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied Probability, Springer, 2016.

[20]

Pham, H., Continuous-time Stochastic Control and Optimization with Financial Applications, Stochastic Modelling and Applied Probability, Springer, Berlin, Heidelberg, 2009.

[21]

Ren, Y.-F., On the Burkholder-Davis-Gundy inequalities for continuous martingales, Statistics & Probability Letters, 2008, 78(17): 3034−3039.

[22]

Zhang, J., Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory, Probability Theory and Stochastic Modelling, Springer, 2017.

show all references

References:
[1]

Beck, C., Hornung, F., Hutzenthaler, M., Jentzen, A. and Kruse, T., Overcoming the curse of dimensionality in the numerical approximation of Allen−Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations, Journal of Numerical Mathematics, 2020, 28(4): 197−222. doi: 10.1515/jnma-2019-0074.

[2]

Bender, C. and Denk, R., A forward scheme for backward SDEs, Stochastic Process. Appl., 2007, 117(12): 1793−1812.

[3]

Bender, C. and Zhang, J., Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab., 2008, 18(1): 143−177.

[4]

Briand, P. and Labart, C., Simulation of BSDEs by Wiener chaos expansion, Ann. Appl. Probab., 2014, 24(3): 1129−1171.

[5]

Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1992.

[6]

E, W., Hutzenthaler, M., Jentzen, A. and Kruse, T., On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations, Journal of Scientific Computing, 2019, 79(3): 1534−1571. doi: 10.1007/s10915-018-00903-0.

[7]

E, W., Hutzenthaler, M., Jentzen, A. and Kruse, T., Multilevel Picard iterations for solving smooth semilinear parabolic heat equations, Partial Differential Equations and Applications, 2021, 2(6): 80.

[8]

El Karoui, N., Peng, S. and Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 1997, 7(1): 1−71. doi: 10.1111/1467-9965.00022.

[9]

Geiss, C. and Labart, C., Simulation of BSDEs with jumps by Wiener chaos expansion, Stochastic Process. Appl., 2016, 126(7): 2123−2162. doi: 10.1016/j.spa.2016.01.006.

[10]

Gobet, E. and Labart, C., Solving BSDE with adaptive control variate, SIAM J. Numer. Anal., 2010, 48(1): 257−277. doi: 10.1137/090755060.

[11]

Hutzenthaler, M., Jentzen, A. and Kruse, T., Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities, Foundations of Computational Mathematics, 2021.

[12]

Hutzenthaler, M., Jentzen, A., Kruse, T. and Nguyen, T. A., Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities, arXiv: 2009.02484, 2020.

[13]

Hutzenthaler, M., Jentzen, A., Kruse, T. and Nguyen, T. A., Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations, Accepted in Journal of Numerical Mathematics, 2022.

[14]

Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T. A. and von Wurstem-berger, P., Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2020, 476(2244): 20190630. doi: 10.1098/rspa.2019.0630.

[15]

Hutzenthaler, M. and Kruse, T., Multilevel Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities, SIAM Journal on Numerical Analysis, 2020, 58(2): 929−961. doi: 10.1137/17M1157015.

[16]

Labart, C. and Lelong, J., A parallel algorithm for solving BSDEs, Monte Carlo Methods Appl., 2013, 19(1): 11−39.

[17]

Øksendal, B., Stochastic Differential Equations: An Introduction with Applications, Universitext, Springer, Berlin, Heidelberg, 1985.

[18]

Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 1990, 14(1): 55−61. doi: 10.1016/0167-6911(90)90082-6.

[19]

Pardoux, E. and Răşcanu, A. Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied Probability, Springer, 2016.

[20]

Pham, H., Continuous-time Stochastic Control and Optimization with Financial Applications, Stochastic Modelling and Applied Probability, Springer, Berlin, Heidelberg, 2009.

[21]

Ren, Y.-F., On the Burkholder-Davis-Gundy inequalities for continuous martingales, Statistics & Probability Letters, 2008, 78(17): 3034−3039.

[22]

Zhang, J., Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory, Probability Theory and Stochastic Modelling, Springer, 2017.

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