The improved split‐step θ methods for stochastic differential equation |
| |
Authors: | Qian Guo Hongqun Li Ying Zhu |
| |
Institution: | 1. Department of Mathematics, Shanghai Normal University, , Shanghai, 200234 China;2. Humanities and Communication College, Shanghai Normal University, , Shanghai, 200234 China |
| |
Abstract: | 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. |
| |
Keywords: | stochastic differential equation numerical method mean‐square stability Hamiltonian‐preserving positivity‐preserving Cox– Ingersoll– Ross model |
|
|