TY - JOUR
T1 - Deterministic implicit two-step Milstein methods for stochastic differential equations
AU - Ren, Quanwei
AU - Tian, Hongjiong
AU - Tian, Tianhai
N1 - Funding Information:
The work of this author is supported by Natural Science Foundation of China (11801146), Foundation of Henan Educational Committee, PR China (No. 19A110013), The youth backbone teacher cultivation project of Henan University of Technology, PR China (21420123), The youth support project for basic research of Henan University of Technology, PR China (2018QNJH17) and the High-Level Personal Foundation of Henan University of Technology, PR China (2017BS023).The work of this author is supported in part by the National Natural Science Foundation of China under Grant Nos. 11671266 and 11871343, Science and Technology Innovation Plan of Shanghai, PR China under Grant No. 20JC1414200 and E-Institutes of Shanghai Municipal Education Commission, PR China under Grant No. E03004.
Publisher Copyright:
© 2021 Elsevier B.V.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
PY - 2021/12
Y1 - 2021/12
N2 - In this paper, we propose a class of deterministic implicit two-step Milstein methods for solving Itô stochastic differential equations. Theoretical analysis is conducted for the convergence and stability properties of the proposed methods. We derive sufficient conditions such that these methods have the mean-square(M-S) convergence of order one, as well as sufficient and necessary conditions for linear M-S stability of the implicit two-step Milstein methods. Stability analysis shows that our proposed implicit two-step Milstein methods have much better stability property than those of the corresponding two-step explicit or semi-implicit Milstein methods. Numerical results using two test equations confirm our theoretical analysis results.
AB - In this paper, we propose a class of deterministic implicit two-step Milstein methods for solving Itô stochastic differential equations. Theoretical analysis is conducted for the convergence and stability properties of the proposed methods. We derive sufficient conditions such that these methods have the mean-square(M-S) convergence of order one, as well as sufficient and necessary conditions for linear M-S stability of the implicit two-step Milstein methods. Stability analysis shows that our proposed implicit two-step Milstein methods have much better stability property than those of the corresponding two-step explicit or semi-implicit Milstein methods. Numerical results using two test equations confirm our theoretical analysis results.
KW - Deterministic implicit method
KW - Mean-square convergence
KW - Mean-square stability
KW - Stochastic differential equation
KW - Two-step Milstein method
UR - http://www.scopus.com/inward/record.url?scp=85112838358&partnerID=8YFLogxK
U2 - 10.1016/j.spl.2021.109208
DO - 10.1016/j.spl.2021.109208
M3 - Article
AN - SCOPUS:85112838358
SN - 0167-7152
VL - 179
JO - Statistics and Probability Letters
JF - Statistics and Probability Letters
M1 - 109208
ER -