Abstract
This paper discusses the asymptotic behavior of the solution for a class of perturbed nonlocal stochastic functional differential equations (SFDEs, in short). By comparing it with the solution of the corresponding unperturbed one, we derive the conditions under which their solutions are close. Firstly, the results are established on finite time-intervals. Then, we also show the results hold when the length of time-interval tends to infinity as small perturbations tend to zero.
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References
Hu, Y., Wu, F.: A class of stochastic differential equations with expectations in the coefficients. Nonlinear Anal. 81, 190–199 (2013)
Janković, S., Jovanović, M.: Perturbed stochastic hereditary differential equations with integral contractors. Comput. Math. Appl. 42, 871–881 (2001)
Janković, S., Jovanović, M.: Generalized stochastic perturbation-depending differential equations. Stoch. Anal. Appl. 20, 1281–1307 (2002)
Janković, S., Jovanović, M.: On perturbed nonlinear Itô type stochastic integrodifferential equations. J. Math. Anal. Appl. 269, 301–316 (2002)
Janković, S., Jovanović, M.: Functionally perturbed stochastic differential equations. Math. Nachr. 279, 1808–1822 (2006)
Janković, S., Jovanović, M.: Neutral stochastic functional differential equations with additive perturbations. Appl. Math. Comput. 213, 370–379 (2009)
Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichestic, UK (1997)
Peter, K., Thomas, L.: A Peano-like theorem for stochastic differential equations with nonlocal sample dependence. Stoch. Anal. Appl. 31, 19–30 (2013)
Ren, Y., Lu, S., Xia, N.: Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay. J. Comput. Appl. Math. 220, 364–372 (2008)
Ren, Y., Xia, N.: Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay. Appl. Math. Comput. 210, 72–79 (2009)
Ren, Y., Chen, L.: A note on the neutral stochastic functional differential equations with infinite delay and Poisson jumps in an abstract space. J. Math. Phys. 50, 082704 (2009)
Sheinkman, J., LeBaron, B.: Nonlinear dynamics and stock returns. J. Busines 62, 311–337 (1989)
Stoica, G.: A stochastic delay financial model. Proc. Am. Math. Soc. 133, 1837–1841 (2005)
Thomas, L.: Nonlocal stochastic differential equations: existence and uniqueness of solutions. Bol. Soc. Esp. Mat. Apl. SeMA 51, 99–107 (2010)
Wu, F., Hu, S.: On a class of nonlocal stochastic functional differential equations with infinite delay. Stoch. Anal. Appl. 29, 713–721 (2011)
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The authors are deeply grateful to the editor and anonymous referees for the careful reading, valuable comments and correcting some errors, which have greatly improved the quality of the paper.
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This work is supported by the National Natural Science Foundation of China (11871076)
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Zhang, Q., Ren, Y. Perturbed Nonlocal Stochastic Functional Differential Equations. Qual. Theory Dyn. Syst. 19, 82 (2020). https://doi.org/10.1007/s12346-020-00421-1
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DOI: https://doi.org/10.1007/s12346-020-00421-1