Abstract
In the paper, stationary solutions of stochastic differential equations driven by Lévy processes are considered. And the existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. Moreover, under a one-sided Lipschitz continuity condition and a temperedness condition, Itô and Marcus stochastic differential equations driven by Lévy processes are proved to have stationary solutions. Besides, continuous dependence of stationary solutions on drift coefficients of these equations is presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009)
Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)
Behme, A., Lindner, A., Maller, R.: Stationary solutions of the stochastic differential equation \( \text{ d } V_t = V_{t-} \text{ d } U_t + \text{ d } L_t\) with Lévy noise. Stoch. Process. Appl. 121, 91–108 (2011)
Caraballo, T., Kloeden, P.E., Schmalfuss, B.: Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Appl. Math. Optim. 50, 183–207 (2004)
Feng, C., Zhao, H., Zhou, B.: Pathwise random periodic solutions of stochastic differential equations. J. Differ. Equ. 251, 119–149 (2011)
Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier Stokes equation with multiplicative noise. Stoch. Stoch. Rep. 59, 21–45 (1996)
Gushchin, A.A., Küchler, U.: On stationary solutions of delay differential equations driven by a Lévy process. Stoch. Process. Appl. 88, 195–211 (2000)
Imkeller, P., Schmalfuss, B.: The conjugacy of stochastic and random differential equations and the existence of global attractors. J. Dyn. Differ. Equ. 13, 215–249 (2001)
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland/Kodanska, Amsterdam/Tokyo (1989)
Khasminskii, R.: Stochastic Stability of Differential Equations. Springer, Berlin Heidelberg (2012)
Lindner, A., Maller, R.: Lévy integrals and the stationarity of generalised Ornstein–Uhlenbeck processes. Stoch. Process. Appl. 115, 1701–1722 (2005)
Marcus, S.I.: Modelling and approximations of stochastic differential equations driven by semimartingales. Stochastics 4, 223–245 (1981)
Qiao, H.: Homeomorphism flows for Non-Lipschitz SDEs driven by Lévy processes. Acta Math. Sci. 32, 1115–1125 (2012)
Qiao, H., Duan, J.: Topological equivalence for discontinuous random dynamical systems andapplications. Stoch. Dyn. 14, 1350007 (28 pp) (2014)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Schmalfuss, B.: Lyapunov functions and non-trivial stationary solutions of stochastic differential equations. Dyn. Syst. 19, 303–317 (2001)
Situ, R.: Theory of Stochastic Differential Equations with Jumps and Applications. Springer, Berlin (2005)
Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 1, 2nd ed. Pub. Berkeley (1979)
Acknowledgments
The author would like to thank Professor Xicheng Zhang for his valuable discussions. And the author would also wish to thank the anonymous referee for giving useful suggestions to improve this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by NSF of China (Nos. 11001051, 11371352) and the Fundamental Research Funds for the Central Universities.
Rights and permissions
About this article
Cite this article
Qiao, H. Stationary Solutions for Stochastic Differential Equations Driven by Lévy Processes. J Dyn Diff Equat 29, 1195–1213 (2017). https://doi.org/10.1007/s10884-015-9513-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-015-9513-3
Keywords
- Stationary solutions
- Conjugacy or topological equivalence
- Temperedness
- Random attractors
- Continuous dependence