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Volume 31, Issue 3
A Sparse-Grid Method for Multi-Dimensional Backward Stochastic Differential Equations

Guannan Zhang, Max Gunzburger & Weidong Zhao

DOI: 10.4208/jcm.1212-m4014

J. Comp. Math., 31 (2013), pp. 221-248.

Published online: 2013-06

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  • Abstract

A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathematical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.

  • Keywords

Backward stochastic differential equations, Multi-step scheme, Gauss-Hermite quadrature rule, Adaptive hierarchical basis, Sparse grids.

  • AMS Subject Headings

60H10, 60H35, 65C10, 65C20, 65C50.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-31-221, author = {}, title = {A Sparse-Grid Method for Multi-Dimensional Backward Stochastic Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2013}, volume = {31}, number = {3}, pages = {221--248}, abstract = {

A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathematical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1212-m4014}, url = {http://global-sci.org/intro/article_detail/jcm/9732.html} }
TY - JOUR T1 - A Sparse-Grid Method for Multi-Dimensional Backward Stochastic Differential Equations JO - Journal of Computational Mathematics VL - 3 SP - 221 EP - 248 PY - 2013 DA - 2013/06 SN - 31 DO - http://doi.org/10.4208/jcm.1212-m4014 UR - https://global-sci.org/intro/article_detail/jcm/9732.html KW - Backward stochastic differential equations, Multi-step scheme, Gauss-Hermite quadrature rule, Adaptive hierarchical basis, Sparse grids. AB -

A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathematical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.

Guannan Zhang, Max Gunzburger & Weidong Zhao. (2020). A Sparse-Grid Method for Multi-Dimensional Backward Stochastic Differential Equations. Journal of Computational Mathematics. 31 (3). 221-248. doi:10.4208/jcm.1212-m4014
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