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.
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