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Exponential asymptotic stability of linear ItĂ´-Volterra equations with damped stochastic perturbations. (English) Zbl 1065.60060
This paper investigates the solutions of a \(d\)-dimensional linear stochastic integro-differential equation of the form \[ dX(t)= \Biggl(AX(t)+ \int^t_0 K(t- s)X(s)\,ds\Biggr)\,dt+ \Sigma(t) dW(t), \] where \(W\) is an \(r\)-dimensional Brownian motion with independent components. Theorems are proved concerning conditions leading to the exponential convergence of these solutions and concerning the implications of such convergence.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34K20 Stability theory of functional-differential equations
34K50 Stochastic functional-differential equations
60H20 Stochastic integral equations
60H99 Stochastic analysis
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