Almost Sure Asymptotic Stability of Stochastic Volterra Integro-Differential Equations with Fading Perturbations

Abstract

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 that 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.

Keywords:

• Almost sure asymptotic stability
• Exponential asymptotic stability
• Itô-Volterra equations
• Simulated annealing
• Stochastic delay differential equations
• Stochastic functional differential equations
• Uniform asymptotic stability

1991 Mathematics Subject Classification:

• Primary 60H10, 60H20, 34K20
• Secondary 45D05

ACKNOWLEDGMENT

The authors thank the anonymous referees for their assistance in improving the article. In particular, we thank them for indicating to us some of the relevant literature. The first author was partially supported by an Albert College Fellowship, awarded by the Dublin City University Research Advisory Panel.