Publication Cover
Stochastics
An International Journal of Probability and Stochastic Processes
Volume 88, 2016 - Issue 7
217
Views
1
CrossRef citations to date
0
Altmetric
Articles

Asymptotically almost automorphic solutions to stochastic differential equations driven by a Lévy process

&
Pages 980-1011 | Received 27 Sep 2014, Accepted 12 Apr 2016, Published online: 02 May 2016

References

  • S. Abbas, Pseudo almost automorphic solutions of fractional order neutral differential equation, Semigroup Forum 81 (2010), pp. 393–404.
  • S. Abbas, Pseudo almost automorphic solutions of some nonlinear integro-differential equations, Comput. Math. Appl. 62 (2011), pp. 2259–2272.
  • S. Abbas, Existence and attractivity of k-pseudo almost automorphic sequence solution of a model of bidirectional neural networks, Acta Appl. Math. 119 (2012), pp. 57–74.
  • S. Abbas and D. Bahuguna, Almost periodic solutions of neutral functional differential equations, Comput. Math. Appl. 55 (2008), pp. 2593–2601.
  • D. Applebaum, Lévy Process and Stochastic Salculus, Cambridge University Press, Cambridge, 2009.
  • P. Bezandry, Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations, Stat. Probab. Lett. 78 (2008), pp. 2844–2849.
  • P. Bezandry and T. Diagana, Square-mean almost periodic solutions nonautonomuos stochastic differential equations, Electron. J. Differ. Equ. 2007 (2007), pp. 1–10.
  • P. Bezandry and T. Diagana, Almost Periodic Stochastic Processes, Springer, New York, 2011.
  • S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Natl. Acad. Sci. USA 52 (1964), pp. 907–910.
  • D. Bugajewski and G.M. N’Guérékata, On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces, Nonlinear Anal. 59 (2004), pp. 1333–1345.
  • J.F. Cao, Q.G. Yang, and Z.T. Huang, Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations, Stochastics 83 (2011), pp. 259–275.
  • Y.K. Chang, Z.H. Zhao, G.M. N’Guérékata, and R. Ma, Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations, Nonlinear Anal. RWA 12 (2011), pp. 1130–1139.
  • Z. Chen and W. Lin, Square-mean pseudo almost automorphic process and its applications to stochastic evolution equations, J. Funct. Anal. 261 (2011), pp. 69–89.
  • Z. Chen and W. Lin, Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations, J. Math. Pures Appl. 100 (2013), pp. 476–504.
  • T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, 2013.
  • T. Diagana, E. Hernández, and J.P.C. dos Santos, Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations, Nonlinear Anal. 71 (2009), pp. 248–257.
  • T. Diagana, G.M. Mophou, and G.M. N’Guérékata, Existence of weighted pseudo almost periodic solutions to some classes of differential equations with Sp-weighted pseudo almost periodic coefficients, Nonlinear Anal. 72 (2010), pp. 430–438.
  • H.S. Ding, J. Liang, and T.J. Xiao, Weighted pseudo almost automorphic functions and WPAA solutions to semilinear evolution equations, J. Math. Anal. Appl. 409 (2014), pp. 409–427.
  • H.S. Ding, W. Long, and G.M. N’Guérékata, Almost automorphic solutions of nonautonomous evolution equations, Nonlinear Anal. 70 (2009), pp. 4158–4164.
  • H.S. Ding, T.J. Xiao, and J. Liang, Asymptotically almost automorphic solutions for some integro-differential equations with nonlocal initial conditions, J. Math. Anal. Appl. 338 (2008), pp. 141–151.
  • R.M. Dudley, Real Analysis and Probability, 2nd ed., Cambridge University Press, Cambridge, 2003.
  • K. Ezzinbi, S. Fatajou, and G.M. N’Guérékata, Pseudo almost automorphic solutions to some neutral partial functional differential equations in Banach space, Nonlinear Anal. 70 (2009), pp. 1641–1647.
  • M.M. Fu and Z.X. Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Amer. Math. Soc. 138 (2010), pp. 3689–3701.
  • H. Henríquez and C. Cuevas, Almost automorphy for abstract neutral differential equation via control theory, Ann. Mat. 192 (2013), pp. 393–405.
  • Z.X. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal. 266 (2014), pp. 1115–1149.
  • G.M. N’Guérékata, Sur les solutions presqu’utomorphes d’équations différentielles abstraites [On almost automorphic solutions of abstract differential equations], Ann. Sci. Math. Québec 5 (1981), pp. 69–79.
  • G.M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic, New York, 2001.
  • G.M. N’Guérékata, Topics in Almost Automorphy, Springer, New York, 2005.
  • G.M. N’Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal. 68 (2008), pp. 2658–2667.
  • B. Øksendal Bernt, Stochastic Differential Equations, Springer, Berlin, 2005.
  • S. Peszat and J. Zabczyk, Stochastic partial differential equations with Lévy noise, Cambridge University Press, 2007.
  • Y. Ren, Reflected backward doubly stochastic differential equations driven by a Lévy process, C. R. Acad. Sci. Paris, Ser. I 348 (2010), pp. 439–444.
  • Y. Ren and R. Sakthivel, Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps, J. Math. Phys. 53 (2012), p. 073517.
  • R. Sakthivel, Y. Ren, and N.I. Mahmudov, Approximate controllability of second order stochastic differential equations with impulsive effects, Mod. Phys. Lett. B 24 (2010), pp. 1559–1572.
  • R. Sakthivel, P. Revathi, and S. Marshal Anthoni, Existence of pseudo almost automorphic mild solutions to stochastic fractional diffenertial equations, Nonlinear Anal. 75 (2012), pp. 3339–3347.
  • R. Sakthivel, P. Revathi, and Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Anal. 81 (2013), pp. 70–86.
  • J.P.C. dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. Math. Lett. 23 (2010), pp. 960–965.
  • K.I. Sato, Lévy Processes and Infinite Divisibility, Cambridge University Press, Cambridge, 1999.
  • C. Tang and Y.K. Chang, Stepanov-like weighted asymptotic behavior of solutions to some stochastic differential equations in Hilbert spaces, Appl. Anal. 93 (2014), pp. 2625–2646.
  • Y. Wang and Z. Liu, Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity 25 (2012), pp. 2803–2821.
  • Z.N. Xia, Weighted pseudo almost automorphic solutions of hyperbolic semilinear integrodifferential equations, Nonlinear Anal. 95 (2014), pp. 50–65.
  • Z.N. Xia and M. Fan, Weighted Stepanov-like pseudo almost automorphy and applications, Nonlinear Anal. 75 (2012), pp. 2378–2397.
  • T.J. Xiao, J. Liang, and J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach space, Semigroup Forum 76 (2008), pp. 518–524.
  • Z.H. Zhao, Y.K. Chang, and J.J. Nieto, Square-mean asymptotically almost automorphic process and its application to stochastic integro-differential equations, Dynam. Syst. Appl. 22 (2013), pp. 269–284.
  • J.Q. Zhao, Y.K. Chang, and G.M. N’Guérékata, Existence of asymptotically almost automorphic solutions to nonlinear delay integral equations, Dynam. Syst. Appl. 21 (2012), pp. 339–350.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.