We consider a system of ordinary differential equations possessing an asymptotically stable equilibrium position and examine the stability of this equilibrium under permanent stochastic whitenoise-type perturbations. The perturbed system is considered in the form of Stratonovich stochastic differential equations. We also describe restrictions on the parameters of perturbations that guarantee the trajectory-wise stability of the equilibrium with probability 1.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price includes VAT (USA)
Tax calculation will be finalised during checkout.
L. Arnold, E. Oeljeklaus, and E. Pardoux, “Almost sure and moment stability for linear Ito equations,” Lect. Notes Math., 1186, 129–159 (1986).
A. S. Asylgareyev and F. S. Nasyrov, “On comparison theorems and theorems on stability with probability 1 for one-dimensional stochastic differential equations,” Sib. Mat. Zh., 57, No. 5, 969–977 (2016).
A. N. Borodin and P. Salminen, Handbook of Brownian Motion. Facts and Formulae, Birkhäuser, Basel–Boston–Berlin (2002).
M. M. Hapaev, Averaging in Stability Theory: A Study of Resonance Multi-Frequency Systems, Kluwer, Dordrecht–Boston (1993).
I. Ya. Kats and A. A. Martynyuk, Stability and Stabilization of Nonlinear Systems with Random Structures, Taylor and Francis, New York–London (2002).
R. Khasminskii, Stochastic Stability of Differential Equations, Springer-Verlag, Berlin–Heidelberg (2012).
F. Kozin and S. Prodromou, “Necessary and sufficient conditions for almost sure sample stability of linear Ito equations,” SIAM J. Appl. Math., 21, No. 3, 413–424 (1971).
H. J. Kushner, Stochastic Stability and Control, Academic Press, New York–London (1967).
X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester (1997).
F. S. Nasyrov, Local Times, Symmetric Integrals, and Stochastic Analysis [in Russian], Fizmatlit, Moscow (2011).
F. S. Nasyrov, “On integration of systems of stochastic differential equations,” Mat. Tr., 19, No. 2, 158–169 (2016).
B. Øksendal, Stochastic Differential Equations. An Introduction With Applications, Springer- Verlag, New York–Heidelberg–Berlin (1998).
O. A. Sultanov, “Stochastic stability of a dynamical system perturbed by white noise,” Mat. Zametki, 101, No. 1, 130–139 (2017).
O. A. Sultanov, “Stochastic perturbations of stable dynamical systems: the trajectory approach,” Itogi Nauki Tekh. Sovr. Mat. Prilozh. Temat. Obzory, 139, 91–103 (2017).
O. Sultanov, “White noise perturbation of locally stable dynamical systems,” Stochast. Dynam., 17, No. 1, 1750002 (2017).
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018.
About this article
Cite this article
Sultanov, O.A. On the Almost Sure Stability of Dynamical Systems with Respect to White Noise. J Math Sci 252, 242–246 (2021). https://doi.org/10.1007/s10958-020-05157-6
Keywords and phrases
- dynamical system
- white noise
AMS Subject Classification