International Journal of Differential Equations

On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations

John A. D. Appleby

Full-text: Open access

Abstract

We consider the zero crossings and positive solutions of scalar nonlinear stochastic Volterra integrodifferential equations of Itô type. In the equations considered, the diffusion coefficient is linear and depends on the current state, and the drift term is a convolution integral which is in some sense mean reverting towards the zero equilibrium. The state dependent restoring force in the integral can be nonlinear. In broad terms, we show that when the restoring force is of linear or lower order in the neighbourhood of the equilibrium, or if the kernel decays more slowly than a critical noise-dependent rate, then there is a zero crossing almost surely. On the other hand, if the kernel decays more rapidly than this critical rate, and the restoring force is globally superlinear, then there is a positive probability that the solution remains of one sign for all time, given a sufficiently small initial condition. Moreover, the probability that the solution remains of one sign tends to unity as the initial condition tends to zero.

Article information

Source
Int. J. Differ. Equ., Volume 2010, Special Issue (2010), Article ID 508217, 25 pages.

Dates
Received: 1 November 2009
Accepted: 14 January 2010
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485399836

Digital Object Identifier
doi:10.1155/2010/508217

Mathematical Reviews number (MathSciNet)
MR2601228

Zentralblatt MATH identifier
1203.45008

Citation

Appleby, John A. D. On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations. Int. J. Differ. Equ. 2010, Special Issue (2010), Article ID 508217, 25 pages. doi:10.1155/2010/508217. https://projecteuclid.org/euclid.ijde/1485399836


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