Abstract
The main result is reduction of the asymptotic stability problem for a stochastic differential equation (SDE) with sufficiently rapid Markovian switching to the analogous wellstudied problem for the “averaged” SDE without switching. Applications to the switching stabilization problem and to ordinary differential equations (ODE) with switching are also considered.
Similar content being viewed by others
References
Mao, X., Stability of Stochastic Differential Equations with Markovian Switching, Stochastic Process. Appl., 1999, vol. 79, no. 1, pp. 45–67.
Mao, X. and Yuan, C., Stochastic Differential Equations with Markovian Switching, London: Imperial College Press, 2006.
Khasminskii, R., Zhu, C., and Yin, G., Stability of Regime-Switching Diffusions, Stochastic Process. Appl., 2007, vol. 117, no. 8, pp. 1037–1051.
Yin, G. and Zhu, C., Hibrid Switching Diffusions, New York: Spriger, 2010.
Khasminskii, R.Z., Ustoichivost’ sistem differentsial’nykh uravnenii pri sluchainykh vozmushcheniyakh ikh parametrov (Stability of Systems of Differential Equations under Random Perturbations of Their Parameters), Moscow: Nauka, 1969. Translated under the title Stochastic Stability of Differential Equations, New York: Springer, 2011, 2nd ed.
Mariton, M., Jump Linear Systems in Automatic Control, New York: Dekker, 1990.
Khasminskii, R., Necessary and Sufficient Conditions for the Asymptotic Stabilty of Linear Stochastic Systems, Teor. Veroyatnost. i Primenen., 1967, vol. 12, no. 1, pp. 167–172 [Theory Probab. Appl. (Engl. Transl.), 1967, vol. 12, no. 1, pp. 144–147].
Arnold, L. and Kloeden, P., Lyapunov Exponents and Rotation Number of Two-Dimensional Systems with Telegraphic Noise, SIAM J. Appl. Math., 1989, vol. 49, no. 4, pp. 1242–1274.
Mao, X., Yin, G., and Yuan, C., Stabilization and Destabilization of Hybrid Systems of Stochastic Differential Equations, Automatica J. IFAC, 2007, vol. 43, no. 2, pp. 264–273.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © R.Z. Khasminskii, 2012, published in Problemy Peredachi Informatsii, 2012, vol. 48, No. 3, pp. 70–82.
Rights and permissions
About this article
Cite this article
Khasminskii, R.Z. Stability of regime-switching stochastic differential equations. Probl Inf Transm 48, 259–270 (2012). https://doi.org/10.1134/S0032946012030064
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0032946012030064