Edgeworth Centre for Financial Mathematics, School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
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.
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.