Electronic Journal of Probability

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

John Appleby and Alan Freeman

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This paper studies the convergence rate of solutions of the linear Itô-Volterra equation $$ dX(t) = \left(AX(t) + \int_{0}^{t} K(t-s)X(s),ds\right)\,dt + \Sigma(t)\,dW(t) \tag{1} $$ 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 $p \gt 0$), 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 (1) are equivalent. The paper extends a result of Murakami which relates to the deterministic version of this problem.

Article information

Electron. J. Probab., Volume 8 (2003), paper no. 22, 22 p.

First available in Project Euclid: 23 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H20: Stochastic integral equations 34K20: Stability theory
Secondary: 45D05: Volterra integral equations [See also 34A12]


Appleby, John; Freeman, Alan. Exponential Asymptotic Stability of Linear Itô-Volterra Equation with Damped Stochastic Perturbations. Electron. J. Probab. 8 (2003), paper no. 22, 22 p. doi:10.1214/EJP.v8-179. https://projecteuclid.org/euclid.ejp/1464037595

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