Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-22T02:45:55.623Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise

Part of: Stability Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

David Applebaum  and
Michailina Siakalli
David Applebaum*
Affiliation:
University of Sheffield
Michailina Siakalli*
Affiliation:
University of Sheffield
*
∗Postal address: Department of Probability and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, UK. Email address: d.applebaum@sheffield.ac.uk
∗∗Email address: michailina27@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using key tools such as Itô's formula for general semimartingales, Kunita's moment estimates for Lévy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Lévy noise are stable in probability, almost surely and moment exponentially stable.

Keywords

Stochastic differential equationLévy noisePoisson random measureBrownian motionalmost-sure asymptotic stabilitymoment exponential stabilityLyapunov exponent

MSC classification

Secondary: 60H10: Stochastic ordinary differential equations 60G51: Processes with independent increments; L'evy processes 93D20: Asymptotic stability 93D05: Lyapunov and other classical stabilities (Lagrange, Poisson, $L^p, l^p$, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1116 - 1129
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press.Google Scholar
[2] Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press.Google Scholar
[3] Brockwell, P. J. (2001). Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 53, 113–124.Google Scholar
[4] Burkill, J. C. (1962). A First Course in Mathematical Analysis. Cambridge University Press.Google Scholar
[5] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
[6] Grigoriu, M. (1992). Lyapunov exponents for nonlinear systems with Poisson white noise. Phys. Lett. A 217, 258–262.Google Scholar
[7] Grigoriu, M. and Samorodnitsky, G. (2004). Stability of the trivial solution for linear stochastic differential equations with Poisson white noise. J. Phys. A 37, 8913–8928.Google Scholar
[8] Hardy, G. H., Littlewood, J. E. and Pólya, G. (1934). Inequalities, 2nd edn. Cambridge University Press.Google Scholar
[9] Has'minskii, R. Z. (1980). Stochastic Stability of Differential Equations. Sojtjoff and Noordhoff.Google Scholar
[10] Kozin, F. (1969). A survey of stability of stochastic systems. Automatica 5, 95–112.CrossRefGoogle Scholar
[11] Kunita, H. (2004). Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms. In Real and Stochastic Analysis, New Perspectives, ed. Rao, M. M., Birkhäuser, Boston, MA, pp. 305–373.Google Scholar
[12] Li, C. W., Dong, Z. and Situ, R. (2002). Almost sure stability of linear stochastic equations with Jumps. Prob. Theory Relat. Fields 123, 121–155.Google Scholar
[13] Mao, X. (1991). Stability of Stochastic Differential Equations with Respect to Semimartingales (Pitman Research Notes Math. Ser. 251). Longman Scientific and Technical, Harlow.Google Scholar
[14] Mao, X. (1994). Exponential Stability of Stochastic Differential Equations. Marcel Dekker, New York.Google Scholar
[15] Mao, X. (1994). Stochastic stabilization and destabilization. Systems Control Lett. 23, 279–290.Google Scholar
[16] Mao, X. (1997). Stochastic Differential Equations and Their Applications. Horwood Publishing, Chichester.Google Scholar
[17] Mao, X. and Rodkina, A. E. (1995). Exponential stability of stochastic differential equations driven by discontinuous semimartingales. Stoch. Stoch. Reports 55, 207–224.Google Scholar
[18] Øksendal, B. and Sulem, A. (2004). Applied Stochastic Control of Jump Diffusions. Springer, Berlin.Google Scholar
[19] Patel, A. and Kosko, B. (2008). Stochastic resonance in continuous and spiking neutron models with Lévy noise. IEEE Trans. Neural Networks 19, 1993–1993.Google Scholar
[20] Siakalli, M. (2009). Stability properties of stochastic differential equations driven by Lévy noise. , University of Sheffield.Google Scholar