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Mathematical Problems in Engineering
Volume 2015, Article ID 810160, 11 pages
http://dx.doi.org/10.1155/2015/810160
Research Article

A Compensated Numerical Method for Solving Stochastic Differential Equations with Variable Delays and Random Jump Magnitudes

Department of Statistics, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Received 29 September 2014; Revised 13 January 2015; Accepted 14 January 2015

Academic Editor: Chin-Chia Wu

Copyright © 2015 Ying Du and Changlin Mei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Stochastic differential equations with jumps are of a wide application area especially in mathematical finance. In general, it is hard to obtain their analytical solutions and the construction of some numerical solutions with good performance is therefore an important task in practice. In this study, a compensated split-step method is proposed to numerically solve the stochastic differential equations with variable delays and random jump magnitudes. It is proved that the numerical solutions converge to the analytical solutions in mean-square with the approximate rate of 1/2. Furthermore, the mean-square stability of the exact solutions and the numerical solutions are investigated via a linear test equation and the results show that the proposed numerical method shares both the mean-square stability and the so-called A-stability.