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 - <p style="text-align: justify;">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&nbsp; 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&nbsp; and the diffusion terms. Our numerical experiment shows these points.</p>