Abstract
In this paper we study the a.s. convergence of all solutions of the Itô-Volterra equation \[ dX(t) = (AX(t) + \int_{0}^{t} K(t-s)X(s),ds)\,dt + \Sigma(t)\,dW(t) \] to zero. $A$ is a constant $d\times d$ matrix, $K$ is a $d\times d$ continuous and integrable matrix function, $\Sigma$ is a continuous $d\times r$ matrix function, and $W$ is an $r$-dimensional Brownian motion. We show that when \[ x'(t) = Ax(t) + \int_{0}^{t} K(t-s)x(s)\,ds \] has a uniformly asymptotically stable zero solution, and the resolvent has a polynomial upper bound, then $X$ converges to 0 with probability 1, provided \[ \lim_{t \rightarrow \infty} |\Sigma(t)|^{2}\log t= 0. \] A converse result under a monotonicity restriction on $|\Sigma|$ establishes that the rate of decay for $|\Sigma|$ above is necessary. Equations with bounded delay and neutral equations are also considered.
Highlights
A literature has appeared over the last fifteen years on the properties of linear stochastic functional equations with additive noise
See for instance, Kuchler and Mensch [7], Mohammed and Scheutzow [12], and work in Mohammed [11]. Such evolutions have Gaussian solutions which can be represented in a variation of parameters form. The similarity of this representation to that of the solutions of linear stochastic differential equations with deterministic volatility indicates that a steady state will be reached by the solutions of such equations under similar conditions that bring about the a.s. stability of the solution of the SDE
Under a monotonicity constraint on the norm of the matrix Σ, one can show that the a.s. asymptotic stability of solutions of the lin√ear Ito-Volterra equation implies that the norm of the volatility matrix decays faster than 1/ log t, as t → ∞, which was earlier seen to be sufficient for almost sure asymptotic stability
Summary
A literature has appeared over the last fifteen years on the properties of linear stochastic functional equations with additive noise (often supplied by Brownian motion). We ask how large damped stochastic perturbations can be, while still allowing convergence at any speed to the equilibrium The motivation in these papers, as here, is to ask whether an asymptotically stable deterministic system will remain (almost surely) asymptotically stable when it is subjected to an external random perturbation whose intensity decays over time. We return to consider necessary and sufficient conditions under which linear and nonlinear Ito-Volterra equations with damped stochastic perturbations exhibit a.s. exponential stability. To obtain sufficient conditions for convergence, we suppose that the underlying deterministic system has a uniformly asymptotically stable equilibrium, and a polynomially bounded√resolvent It transpires that if the coefficient of the volatility matrix decay to 0 faster than 1/ log t as t → ∞, the solution of the Ito-Volterra equation converges to 0 a.s. The result of the theorem is shown to be quite sharp. We mention that the results may be extended to linear equations with bounded delay, and to neutral equations
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