On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential EquationsReport as inadecuate




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International Journal of Differential EquationsVolume 2010 2010, Article ID 508217, 25 pages

Research ArticleEdgeworth Centre for Financial Mathematics, School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

Received 1 November 2009; Accepted 14 January 2010

Academic Editor: Elena Braverman

Copyright © 2010 John A.
D.
Appleby.
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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





Author: John A.
D.
Appleby


Source: https://www.hindawi.com/



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