Abstract
In this paper, we present the compensated stochastic θ method for stochastic age-dependent delay population systems (SADDPSs) with Poisson jumps. The definition of mean-square stability of the numerical solution is given and a sufficient condition for mean-square stability of the numerical solution is derived. It is shown that the compensated stochastic θ method inherits stability property of the numerical solutions. Finally, the theoretical results are also confirmed by a numerical experiment.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, L. Stochastic Differential Equations: Theory and Applications, Wiley, 1972
Baker, C.T.H., Buckwar, E. On Halanay-type analysis of exponential stability for the θ-Mar-uyama method for stochastic delay differential equations. Stoch. Dyn.,a 5: 201–209 (2005)
Jacod, J., Shiryaev, A.N. Limit Theorems for Stochastic Process, 2nd ed, Springer Verlag, New York, 2002
Li, Q., Gan, S. Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps. Abst. Appl. Anal., 2: 183–194 (2012)
Li, R., Leung, P., Pang, W. Convergence of numerical solutions to stochastic age-dependent population equations with Markovian switching. Appl. Math. Comput., 233: 1046–1055 (2009)
Ma, W., Zhang, Q., Han, C. Numerical analysis for stochastic age-dependent population equations with fractional Brownian motion. Commun. Nonlinear. Sci. Numer. Simul., 17: 1884–1893 (2012)
Mao, X. Exponential stability of stochastic differential equations. Marcel Dekker, New York, 1994.
Mao, X. Stochastic Differential Equations and Applications, 2nd ed. Horwood Publishing, Chichester, 2007
Pang, W., Li, R., Liu, M. Convergence of the semi-implicit Euler method for stochastic age-dependent population equations. Appl. Math. Comput., 195: 466–474 (2008)
Sobczyk, K. Stochastic Differential Equations with Applications to Physics and Engineerin. Kluwer Academic, Dordrecht, The Netherlands, 1991
Wang, W., Chen, Y. Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations. Appl. Numer. Math., 61: 696–701 (2011)
Wang, L., Wang, X. Convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps. Appl. Math. Comput., 34: 2034–2043 (2010)
Zhang, Q., Liu, W., Nie, Z. Existence, uniqueness and exponential stability for stochastic age-dependent population. Appl. Math. Comput., 154: 183–201 (2004)
Acknowledgements
The authors thank the anonymous referee and the editors for their detailed comments and helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by Major Innovation Projects for Building First-class Universities in China’s Western Region (No. ZKZD2017009)(China).
Rights and permissions
About this article
Cite this article
Li, Q., Zhang, Qm. & Cao, Bq. Mean-square stability of stochastic age-dependent delay population systems with jumps. Acta Math. Appl. Sin. Engl. Ser. 34, 145–154 (2018). https://doi.org/10.1007/s10255-018-0732-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-018-0732-3
Keywords
- stochastic age-dependent delay population systems
- compensated stochastic θ method
- poisson jumps
- mean-square stability