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The improved split‐step θ methods for stochastic differential equation
Author(s) -
Guo Qian,
Li Hongqun,
Zhu Ying
Publication year - 2014
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2972
Subject(s) - mathematics , stochastic differential equation , convergence (economics) , numerical analysis , stability (learning theory) , two step , property (philosophy) , numerical stability , differential equation , mathematical analysis , computer science , philosophy , epistemology , machine learning , economics , economic growth
Two improved split‐step θ methods, which, respectively, named split‐step composite θ method and modified split‐step θ ‐Milstein method, are proposed for numerically solving stochastic differential equation of Itô type. The stability and convergence of these methods are investigated in the mean‐square sense. Moreover, an approach to improve the numerical stability is illustrated by choices of parameters of these two methods. Some numerical examples show the accordance between the theoretical and numerical results. Further numerical tests exhibit not only the Hamiltonian‐preserving property of the improved split‐step θ methods for a stochastic differential system but also the positivity‐preserving property of the modified split‐step θ ‐Milstein method for the Cox–Ingersoll–Ross model. Copyright © 2013 John Wiley & Sons, Ltd.