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Stochastic asymptotic exponential stability of stochastic integral equations

Published online by Cambridge University Press:  14 July 2016

Chris P. Tsokos  and
M. A. Hamdan
Chris P. Tsokos
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia
M. A. Hamdan
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia
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Abstract

The object of this paper is to study the stochastic asymptotic exponential stability of a stochastic integral equation of the form

A random solution of the stochastic integral equation is considered to be a second order stochastic process satisfying the equation almost surely. The random solution, y(t, ω) is said to be. stochastically asymptotically exponentially stable if there exist some β > 0 and a γ > 0 such that for t∈ R+.

The results of the paper extend the results of Tsokos' generalization of the classical stability theorem of Poincaré-Lyapunov.

Keywords

RANDOM SOLUTION OF STOCHASTIC INTEGRAL EQUATIONSSTOCHASTIC ASYMPTOTIC EXPONENTIAL STABILITYPROBABILITY MEASURE SPACELp-SPACESADMISSIBILITY THEORYBANACH SPACESSTRONG CONVERGENCECONTRACTION MAPPING PRINCIPLE
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 1 , March 1972 , pp. 169 - 177
Copyright
Copyright © Applied Probability Trust 1972 

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References

[1] Ahmed, N. U. (1969) A class of stochastic nonlinear integral equations on Lp spaces and its application to optimal control. Information and Control 14, 512–523.CrossRefGoogle Scholar
[2] Anderson, M. W. (1966) Stochastic Integral Equations. Ph.D. Dissertation, University of Tennessee.Google Scholar
[3] Bharucha-Reid, A. T. (1960) On random solutions of integral equations in Banach spaces. Trans. Second Prague Conf. Information Theory, Statistical Decision Function, and Random Processes, Academic Press, New York, 27–48.Google Scholar
[4] Bharucha-Reid, A. T. (1964) On the theory of random equations. Proc. Symp. Appl. Math. 16, 40–69. American Mathematical Society, Providence, Rhode Island.Google Scholar
[5] Distefano, N. (1968) A Volterra integral equation in the stability of some linear hereditary phenomena. J. Math. Anal. Appl. 23, 365–383.CrossRefGoogle Scholar
[6] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[7] Dunford, N. and Schwartz, J. (1958) Linear Operations, Part I. Interscience, New York.Google Scholar
[8] Fortet, R. (1956) Random distribution with applications to telephone engineering. Proc. Third Berkeley Symp. Math. Statist. and Prob. University of California Press, Berkeley, 81–88.Google Scholar
[9] Morozan, T. (1969) Stabilities Sistemelor cu Parametri Aleatori. Editura Academiei Republicii Socialiste Românio, Bucarest.Google Scholar
[10] Padgett, W. J. and Tsokos, C. P. (1971) On a stochastic integral equation of the Volterra type in telephone traffic theory. J. Appl. Prob. 8, 269–275.CrossRefGoogle Scholar
[11] Padgett, W. J. and Tsokos, C. P. (1971) On the existence of a solution of a stochastic integral equation in turbulence theory. J. Math. Phys. 12, 210–212.CrossRefGoogle Scholar
[12] Padgett, W. J. and Tsokos, C. P. (1970) On a semi-stochastic model arising in a biological system. Math. Biosciences 9, 105–117.CrossRefGoogle Scholar
[13] Padgett, W. J. and Tsokos, C. P. (1970) A stochastic model for chemotherapy: Computer simulation. Math. Biosciences 9, 119–133.CrossRefGoogle Scholar
[14] Tsokos, C. P. (1969) On a nonlinear differential system with a random parameter. Internat. Conf. on Systems Sciences, IEEE Proc., Honolulu, Hawaii.Google Scholar
[15] Tsokos, C. P. (1969) On some stochastic differential systems. IEEE Proc., Third Annual Princeton Conf. on Inf. of Sciences and Systems, 228–234.Google Scholar
[16] Tsokos, C. P. (1969) On the classical stability theorem of Poincaré-Lyapunov with a random parameter. Proc. Japan Acad. 45, 780–785.Google Scholar
[17] Tsokos, C. P. and Hamdan, M. A. (1970) Stochastic nonlinear integro-differential systems with time lag. J. Natur. Sci. and Math. 10, 293–303.Google Scholar