Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 8 (2003) > Paper 22 open journal systems 


Exponential Asymptotic Stability of Linear Ito-Volterra Equation with Damped Stochastic Perturbations

John A. D. Appleby, Dublin City University, Ireland
Alan Freeman, Dublin City University, Ireland


Abstract
This paper studies the convergence rate of solutions of the linear It^o - Volterra equation begin{equation} label{ref:iveqn} dX(t) = left(AX(t) + int_{0}^{t} K(t-s)X(s),dsright),dt + Sigma(t),dW(t) end{equation} where $K$ and $Sigma$ are continuous matrix--valued functions defined on $mathbb{R}^{+}$, and $(W(t))_{t geq 0}$ is a finite-dimensional standard Brownian motion. It is shown that when the entries of $K$ are all of one sign on $mathbb{R}^{+}$, that (i) the almost sure exponential convergence of the solution to zero, (ii) the $p$-th mean exponential convergence of the solution to zero (for all $p>0$), and (iii) the exponential integrability of the entries of the kernel $K$, the exponential square integrability of the entries of noise term $Sigma$, and the uniform asymptotic stability of the solutions of the deterministic version of (0.1) are equivalent. The paper extends a result of Murakami which relates to the deterministic version of this problem.


Full text: PDF

Pages: 1-22

Published on: December 27, 2003
Bibliography
  1. J.A.D. Appleby. Almost sure asymptotic stability of linear Ito-Volterra equations with damped stochastic perturbations. Electron. Commun. Probab., Paper no. 22, 7:223--234, 2002.  Math Review link

  2. J.A.D. Appleby and D.W. Reynolds. Subexponential solutions of linear Volterra integro-differential equations and transient renewal equations. Proc. Roy. Soc. Edinburgh. Sect. A, 132:521--543, 2002.   Math Review link

  3. J.A.D. Appleby and D.W. Reynolds. Non-exponential stability of scalar stochastic Volterra equations. Statist. Probab. Lett., 62:335--343, 2003.  Math Review link

  4. M.A. Berger and V.J. Mizel. Volterra equations with Ito integrals I. J. Integral Equations, 2(3):187--245, 1980.  Math Review link

  5. P.Clément, G.Da Prato, and J. Prüss. White noise perturbation of the equations of linear parabolic viscoelasticity. Rend. Inst. Mat. Univ. Trieste, 29(1-2):207--220, 1997.  Math Review link

  6. C. Corduneanu and V. Lakshmikantham. Equations with unbounded delay: a survey. Nonlinear Anal., 4:831--877, 1980.  Math Review link

  7. R. D. Driver. Existence and stability of solutions of a delay differential system. Arch. Rational Mech. Anal., 10:401--426, 1962.  Math Review link

  8. I.Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113. Springer, New York, 1991.  Math Review link

  9. U. Küchler and B.Mensch. Langevin's stochastic differential equation extended by a time-delay term. Stochastics Stochastics Rep., 40(1-2):23--42, 1992.  Math Review link

  10. K. Liu and X. Mao. Exponential stability of non-linear stochastic evolution equations. Stochastic Process. Appl., 78:173--193, 1998. Math Review link

  11. X. Mao. Exponential stability in mean-square for stochastic differential equations. Stochastic Anal. Appl., 8(1):91--103, 1990. Math Review link

  12. X. Mao. Almost sure polynomial stability for a class of stochastic diferential equations. Quart. J. Math. Oxford Ser. (2), 43(171):339--348, 1992. Math Review link

  13. X. Mao. Exponential Stability of Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 182. Marcel Dekker, Inc., New York, 1994 Math Review link

  14. X. Mao. Stochastic stabilization and destabilization. System Control Lett., 23:279--290, 1994. Math Review link

  15. X. Mao. Razumikhin-type theorems on exponential stability of stochastic stochastic functional-differential equations. Stochastic Process. Appl., 65(2):233--250, 1996. Math Review link

  16. X. Mao. Stochastic Differential Equations and Applications. Horwood Publishing Series in Mathematics & Applications. Horwood Publishing Limited, Chichester, 1997. Math Review link

  17. X. Mao. Stability of stochastic integro-differential equations. Stochastic Anal. Appl., 18(6):1005--1017, 2000. Math Review link

  18. X. Mao. Almost sure exponential stability of delay equations with damped stochastic perturbation. Stochastic Anal. Appl., 19(1):67--84, 2001. Math Review link

  19. X. Mao and X. Liao. Almost sure exponential stability of neutral differential difference equations with damped stochastic perturbations. Electron. J. Probab., 1, Paper no. 8, 16pp. (electronic), 1996. Math Review link

  20. X. Mao and A. Shah. Exponential stability of stochastic delay differential equations. Stochastics Stochastics Rep., 60(1):135--153, 1997. Math Review link

  21. R. K. Miller. Asymptotic stability properties of linear Volterra integrodifferential equations. J. Differential Equations, 10:485--506, 1971. Math Review link

  22. V. J. Mizel and V. Trutzer. Stochastic heredity equations: existence and asymptotic stability. J. Integral Equations, 7:1--72, 1984. Math Review link

  23. S.-E. A. Mohammed. Stochastic Functional Differential Equations, Research Notes in Mathematics 99. Pitman, London, 1984. Math Review link

  24. S. Murakami. Exponential stability for fundamental solutions of some linear functional differential equations. In T.Yoshizawa and J.Kato, editors, Proceedings of the international symposium: Functional differential equations, pages 259--263, Singapore, 1990. World Scientific.   Math Review number not available.

  25. S. Murakami. Exponential asymptotic stability for scalar linear Volterra equations. Differential Integral Equations, 4(2):519--525, 1991.Math Review link

















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489