Volume 10, Issue 4, pp 1365--1376
Publication Date: 2017-04-20
Yeol Je Cho
- Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea.
Young-Ho Kim - Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea.
In this paper, we show the difference between an approximate solution and an accurate solution for a stochastic differential delay equation, where the approximate solution, which is called by Carathéodory, is constructed by successive approximation. Furthermore, we study the p-th moment continuity of the approximate solution for this delay equation.
Hölder’s inequality, moment inequality, Carathéodory approximation procedure, stochastic differential delay equation.
 Y. J. Cho, S. S. Dragomir, Y.-H. Kim,/ A note on the existence and uniqueness of the solutions to SFDES,/ J. Inequal. Appl.,/ 2012 (2012), 11 pages.
 S. Janković, G. Pavlović,/ Moment decay rates of stochastic differential equations with time-varying delay,/ Filomat,/ 24 (2010), 115–132.
 Y.-H. Kim,/ A note on the solutions of neutral SFDEs with infinite delay,/ J. Inequal. Appl.,/ 2013 (2013), 11 pages.
 Y.-H. Kim,/ The difference between the approximate and the accurate solution to stochastic differential delay equation,/ Proc. Jangjeon Math. Soc.,/ 18 (2015), 165–175.
 X.-R. Mao,/ Stochastic differential equations and applications,/ Second edition, Horwood Publishing Limited, Chichester,/ (2008).
 M. Milošević,/ On the approximations of solutions to stochastic differential delay equations with Poisson random measure via Taylor series,/ Filomat,/ 27 (2013), 201–214.
 Y. Ren, S.-P. Lu, N.-M. Xia,/ Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay,/ J. Comput. Appl. Math.,/ 220 (2008), 364–372.
 Y. Ren, N.-M. Xia,/ Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay,/ Appl. Math. Comput.,/ 210 (2009), 72–79.
 M. Vasilova, M. Jovanović,/ Dynamics of Gilpin-Ayala competition model with random perturbation,/ Filomat,/ 24 (2010), 101–113.
 F.-Y.Wei, Y.-H. Cai, Existence,/ uniqueness and stability of the solution to neutral stochastic functional differential equations with infinite delay under non-Lipschitz conditions,/ Adv. Difference Equ.,/ 2013 (2013), 12 pages.
 F.-Y. Wei, K. Wang,/ The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay,/ J. Math. Anal. Appl.,/ 331 (2007), 516–531.