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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 = {Guannan Zhang, Max Gunzburger and Weidong Zhao}, 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 AU - Guannan Zhang, Max Gunzburger & Weidong Zhao 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 and Weidong Zhao. (2013). 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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