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EXPONENTIAL ASYMPTOTIC STABILITY OF LINEAR IT^O-VOLTERRA EQUATIONS WITH DAMPED STOCHASTIC
 

Summary: EXPONENTIAL ASYMPTOTIC STABILITY OF LINEAR
IT^O-VOLTERRA EQUATIONS WITH DAMPED STOCHASTIC
PERTURBATIONS
JOHN A. D. APPLEBY AND ALAN FREEMAN
Abstract. This paper studies the convergence rate of solutions of the linear
It^o - Volterra equation
(0.1) dX(t) = AX(t) +
t
0
K(t - s)X(s) ds dt + (t) dW(t)
where K and are continuous matrix­valued functions defined on R+, and
(W(t))t0 is a finite-dimensional standard Brownian motion. It is shown that
when the entries of K are all of one sign on 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 , 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.
1. Introduction

  

Source: Appleby, John - School of Mathematical Sciences, Dublin City University

 

Collections: Mathematics