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

   
@article{EJP179,
	author = {John Appleby and Alan Freeman},
	title = {Exponential Asymptotic Stability of Linear Ito-Volterra Equation with Damped Stochastic Perturbations},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {8},
	year = {2003},
	keywords = {},
	abstract = {This paper studies the convergence rate of solutions of the linear Ito-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 $p>0$), 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.},
	pages = {no. 22, 1-22},
	issn = {1083-6489},
	doi = {10.1214/EJP.v8-179},    
        url = {http://ejp.ejpecp.org/article/view/179}}