On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations
Author
Source
International Journal of Differential Equations
Issue
Vol. 2010, Issue 2010 (31 Dec. 2010), pp.1-25, 25 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2010-03-10
Country of Publication
Egypt
No. of Pages
25
Main Subjects
Abstract EN
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.
American Psychological Association (APA)
Appleby, John A. D.. 2010. On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations. International Journal of Differential Equations،Vol. 2010, no. 2010, pp.1-25.
https://search.emarefa.net/detail/BIM-477208
Modern Language Association (MLA)
Appleby, John A. D.. On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations. International Journal of Differential Equations No. 2010 (2010), pp.1-25.
https://search.emarefa.net/detail/BIM-477208
American Medical Association (AMA)
Appleby, John A. D.. On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations. International Journal of Differential Equations. 2010. Vol. 2010, no. 2010, pp.1-25.
https://search.emarefa.net/detail/BIM-477208
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-477208