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Volume 16, Issue 4
ODE-Based Multistep Schemes for Backward Stochastic Differential Equations

Shuixin Fang & Weidong Zhao

DOI: 10.4208/nmtma.OA-2023-0060

Numer. Math. Theor. Meth. Appl., 16 (2023), pp. 1053-1086.

Published online: 2023-11

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  • Abstract

In this paper, we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations (BSDEs). By the nonlinear Feynman-Kac formula, we reformulate the BSDE into a pair of reference ordinary differential equations (ODEs), which can be directly discretized by many standard ODE solvers, yielding the corresponding numerical schemes for BSDEs. In particular, by applying strong stability preserving (SSP) time discretizations to the reference ODEs, we can propose new SSP multistep schemes for BSDEs. Theoretical analyses are rigorously performed to prove the consistency, stability and convergency of the proposed SSP multistep schemes. Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.

  • Keywords

Backward stochastic differential equation, parabolic partial differential equation, strong stability preserving, linear multistep scheme, high order discretization.

  • AMS Subject Headings

65C30, 60H35, 65C20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-16-1053, author = {Fang , Shuixin and Zhao , Weidong}, title = {ODE-Based Multistep Schemes for Backward Stochastic Differential Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2023}, volume = {16}, number = {4}, pages = {1053--1086}, abstract = {

In this paper, we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations (BSDEs). By the nonlinear Feynman-Kac formula, we reformulate the BSDE into a pair of reference ordinary differential equations (ODEs), which can be directly discretized by many standard ODE solvers, yielding the corresponding numerical schemes for BSDEs. In particular, by applying strong stability preserving (SSP) time discretizations to the reference ODEs, we can propose new SSP multistep schemes for BSDEs. Theoretical analyses are rigorously performed to prove the consistency, stability and convergency of the proposed SSP multistep schemes. Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2023-0060 }, url = {http://global-sci.org/intro/article_detail/nmtma/22123.html} }
TY - JOUR T1 - ODE-Based Multistep Schemes for Backward Stochastic Differential Equations AU - Fang , Shuixin AU - Zhao , Weidong JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1053 EP - 1086 PY - 2023 DA - 2023/11 SN - 16 DO - http://doi.org/10.4208/nmtma.OA-2023-0060 UR - https://global-sci.org/intro/article_detail/nmtma/22123.html KW - Backward stochastic differential equation, parabolic partial differential equation, strong stability preserving, linear multistep scheme, high order discretization. AB -

In this paper, we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations (BSDEs). By the nonlinear Feynman-Kac formula, we reformulate the BSDE into a pair of reference ordinary differential equations (ODEs), which can be directly discretized by many standard ODE solvers, yielding the corresponding numerical schemes for BSDEs. In particular, by applying strong stability preserving (SSP) time discretizations to the reference ODEs, we can propose new SSP multistep schemes for BSDEs. Theoretical analyses are rigorously performed to prove the consistency, stability and convergency of the proposed SSP multistep schemes. Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.

Fang , Shuixin and Zhao , Weidong. (2023). ODE-Based Multistep Schemes for Backward Stochastic Differential Equations. Numerical Mathematics: Theory, Methods and Applications. 16 (4). 1053-1086. doi:10.4208/nmtma.OA-2023-0060
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