Carathéodory's approximate solution to stochastic differential delay equation
Yeol Je Cho
- Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea.
- 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.