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Volume 21, Issue 2
Richardson Extrapolation of the Crank-Nicolson Scheme for Backward Stochastic Differential Equations

Yafei Xu & Weidong Zhao

DOI: 10.4208/ijnam2024-1011

Int. J. Numer. Anal. Mod., 21 (2024), pp. 268-294.

Published online: 2024-04

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  • Abstract

In this work, we consider Richardson extrapolation of the Crank-Nicolson (CN) scheme for backward stochastic differential equations (BSDEs). First, applying the Adomian decomposition to the nonlinear generator of BSDEs, we introduce a new system of BSDEs. Then we theoretically prove that the solution of the CN scheme for BSDEs admits an asymptotic expansion with its coefficients the solutions of the new system of BSDEs. Based on the expansion, we propose Richardson extrapolation algorithms for solving BSDEs. Finally, some numerical tests are carried out to verify our theoretical conclusions and to show the stability, efficiency and high accuracy of the algorithms.

  • Keywords

Backward stochastic differential equations, Crank-Nicolson scheme, Adomian decomposition, Richardson extrapolation, asymptotic error expansion.

  • AMS Subject Headings

65C30, 60H10, 60H35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-21-268, author = {Xu , Yafei and Zhao , Weidong}, title = {Richardson Extrapolation of the Crank-Nicolson Scheme for Backward Stochastic Differential Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2024}, volume = {21}, number = {2}, pages = {268--294}, abstract = {

In this work, we consider Richardson extrapolation of the Crank-Nicolson (CN) scheme for backward stochastic differential equations (BSDEs). First, applying the Adomian decomposition to the nonlinear generator of BSDEs, we introduce a new system of BSDEs. Then we theoretically prove that the solution of the CN scheme for BSDEs admits an asymptotic expansion with its coefficients the solutions of the new system of BSDEs. Based on the expansion, we propose Richardson extrapolation algorithms for solving BSDEs. Finally, some numerical tests are carried out to verify our theoretical conclusions and to show the stability, efficiency and high accuracy of the algorithms.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2024-1011}, url = {http://global-sci.org/intro/article_detail/ijnam/23027.html} }
TY - JOUR T1 - Richardson Extrapolation of the Crank-Nicolson Scheme for Backward Stochastic Differential Equations AU - Xu , Yafei AU - Zhao , Weidong JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 268 EP - 294 PY - 2024 DA - 2024/04 SN - 21 DO - http://doi.org/10.4208/ijnam2024-1011 UR - https://global-sci.org/intro/article_detail/ijnam/23027.html KW - Backward stochastic differential equations, Crank-Nicolson scheme, Adomian decomposition, Richardson extrapolation, asymptotic error expansion. AB -

In this work, we consider Richardson extrapolation of the Crank-Nicolson (CN) scheme for backward stochastic differential equations (BSDEs). First, applying the Adomian decomposition to the nonlinear generator of BSDEs, we introduce a new system of BSDEs. Then we theoretically prove that the solution of the CN scheme for BSDEs admits an asymptotic expansion with its coefficients the solutions of the new system of BSDEs. Based on the expansion, we propose Richardson extrapolation algorithms for solving BSDEs. Finally, some numerical tests are carried out to verify our theoretical conclusions and to show the stability, efficiency and high accuracy of the algorithms.

Xu , Yafei and Zhao , Weidong. (2024). Richardson Extrapolation of the Crank-Nicolson Scheme for Backward Stochastic Differential Equations. International Journal of Numerical Analysis and Modeling. 21 (2). 268-294. doi:10.4208/ijnam2024-1011
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