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Volume 40, Issue 4
General Full Implicit Strong Taylor Approximations for Stiff Stochastic Differential Equations

Kai Liu & Guiding Gu

J. Comp. Math., 40 (2022), pp. 541-569.

Published online: 2022-04

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  • Abstract

In this paper, we present the backward stochastic Taylor expansions for a Ito process, including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions. We construct the general full implicit strong Taylor approximations (including Ito-Taylor and Stratonovich-Taylor schemes) with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations (SSDE) by employing truncations of backward stochastic Taylor expansions. We demonstrate that these schemes will converge strongly with corresponding order $1,2,3,\ldots$ Mean-square stability  has been investigated for full implicit strong Stratonovich-Taylor scheme with order $2$, and it has larger mean-square stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order $2$. We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes. The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift  and the diffusion terms. Our numerical experiment shows these points.

  • AMS Subject Headings

65C30, 60H35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

752964253@qq.com (Kai Liu)

guiding@mail.shufe.edu.cn (Guiding Gu)

  • BibTex
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  • TXT
@Article{JCM-40-541, author = {Liu , Kai and Gu , Guiding}, title = {General Full Implicit Strong Taylor Approximations for Stiff Stochastic Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {4}, pages = {541--569}, abstract = {

In this paper, we present the backward stochastic Taylor expansions for a Ito process, including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions. We construct the general full implicit strong Taylor approximations (including Ito-Taylor and Stratonovich-Taylor schemes) with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations (SSDE) by employing truncations of backward stochastic Taylor expansions. We demonstrate that these schemes will converge strongly with corresponding order $1,2,3,\ldots$ Mean-square stability  has been investigated for full implicit strong Stratonovich-Taylor scheme with order $2$, and it has larger mean-square stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order $2$. We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes. The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift  and the diffusion terms. Our numerical experiment shows these points.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2011-m2019-0174}, url = {http://global-sci.org/intro/article_detail/jcm/20500.html} }
TY - JOUR T1 - General Full Implicit Strong Taylor Approximations for Stiff Stochastic Differential Equations AU - Liu , Kai AU - Gu , Guiding JO - Journal of Computational Mathematics VL - 4 SP - 541 EP - 569 PY - 2022 DA - 2022/04 SN - 40 DO - http://doi.org/10.4208/jcm.2011-m2019-0174 UR - https://global-sci.org/intro/article_detail/jcm/20500.html KW - Stiff stochastic differential equations, Approximations, Backward stochastic Taylor expansions, Full implicit Taylor methods. AB -

In this paper, we present the backward stochastic Taylor expansions for a Ito process, including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions. We construct the general full implicit strong Taylor approximations (including Ito-Taylor and Stratonovich-Taylor schemes) with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations (SSDE) by employing truncations of backward stochastic Taylor expansions. We demonstrate that these schemes will converge strongly with corresponding order $1,2,3,\ldots$ Mean-square stability  has been investigated for full implicit strong Stratonovich-Taylor scheme with order $2$, and it has larger mean-square stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order $2$. We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes. The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift  and the diffusion terms. Our numerical experiment shows these points.

Kai Liu & Guiding Gu. (2022). General Full Implicit Strong Taylor Approximations for Stiff Stochastic Differential Equations. Journal of Computational Mathematics. 40 (4). 541-569. doi:10.4208/jcm.2011-m2019-0174
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