Almost Sure Stability of Linear Ito-Volterra Equations with Damped Stochastic Perturbations (original) (raw)
Related papers
Electronic Journal of Probability, 2003
This paper studies the convergence rate of solutions of the linear Itô-Volterra equation dX(t) = AX(t) + t 0 K(t − s)X(s) ds dt + Σ(t) dW (t) (0.1) where K and Σ are continuous matrix-valued functions defined on R + , and (W (t)) t≥0 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 Σ, 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.
Polynomial asymptotic stability of damped stochastic differential equations
The paper studies the polynomial convergence of solutions of a scalar nonlinear Itô stochastic differential equation dX(t) = −f (X(t)) dt + σ(t) dB(t) where it is known, a priori, that lim t→∞ X(t) = 0, a.s. The intensity of the stochastic perturbation σ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function f obeys lim x→0 sgn(x)f (x)/|x| β = a, for some β > 1, and a > 0. We study two asymptotic regimes: when σ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when σ ≡ 0). When σ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for σ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.
Stochastic Analysis and Applications, 2006
In this note we address the question of how large a stochastic perturbation an asymptotically stable linear functional differential system can tolerate without losing the property of being pathwise asymptotically stable. In particular, we investigate noise perturbations which are either independent of the state or influenced by the current and past states. For perturbations independent of the state, we prove that the assumed rate of fading for the noise is optimal.
Almost Sure Asymptotic Stability of Itˆ O-Volterra Equations
This paper provides some sucient conditions un- der which the zero solution of an Ito-Volterra equation is almost surely asymptotically stable. Under Lipschitz continuity and linear growth assumptions, it is established that if the solution is almost surely integrable, and also almost surely square integrable, then the solution is almost surely asymptotically stable. Additionally, if the solution is p-th mean asymptotically stable, and the p-th mean of the solution is integrable, for p = 2, the solution is almost surely asymptotically stable. Extensions of this result are provided for p 2. We present examples and give sucient conditions under
Almost sure subexponential decay rates of scalar Ito-Volterra equations
Proceedings of The 7'th Colloquium on the Qualitative Theory of Differential Equations (July 14--18, 2003, Szeged, Hungary) edited by: László Hatvani and Tibor Krisztin, 2003
The paper studies the subexponential convergence of solutions of scalar Itô-Volterra equations. First, we consider linear equations with an instantaneous multiplicative noise term with intensity σ. If the kernel obeys lim t→∞ k (t)/k(t) = 0, and another nonexponential decay criterion, and the solution X σ tends to zero as t → ∞, then lim sup t→∞ log |X σ (t)| log(tk(t)) = 1 − Λ(|σ|), a.s. where the random variable Λ(|σ|) → 0 as σ → ∞ a.s. We also prove a decay result for equations with a superlinear diffusion coefficient at zero. If the deterministic equation has solution which is uniformly asymptotically stable, and the kernel is subexponential, the decay rate of the stochastic problem is exactly the same as that of the underlying deterministic problem.
Decay Rates of Solutions of Linear Stochastic Volterra Equations
Electronic Journal of Probability, 2008
The paper studies the exponential and non-exponential convergence rate to zero of solutions of scalar linear convolution Itô-Volterra equations in which the noise intensity depends linearly on the current state. By exploiting the positivity of the solution, various upper and lower bounds in first mean and almost sure sense are obtained, including Liapunov exponents .