Almost Sure Stability of Linear Ito-Volterra Equations with Damped Stochastic Perturbations
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | Almost Sure Stability of Linear Ito-Volterra Equations with Damped Stochastic Perturbations |
| 2. | Creator | Author's name, affiliation, country | John A. D. Appleby; Dublin City University |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | Stochastic functional-differential equations, Ito-Volterra equations, uniform asymptotic stability, almost sure stability, pathwise stability, simulated annealing. |
| 3. | Subject | Subject classification | Primary 60H10, 60H20, 34K20; Secondary 45D05. |
| 4. | Description | Abstract | In this paper we study the a.s. convergence of all solutions of the Ito-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. |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | |
| 7. | Date | (YYYY-MM-DD) | 2002-08-21 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ecp.ejpecp.org/article/view/1063 |
| 10. | Identifier | Digital Object Identifier | 10.1214/ECP.v7-1063 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Communications in Probability; Vol 7 |
| 12. | Language | English=en | |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
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