An Osgood criterion for integral equations with applications to stochastic differential equations with an additive noise

• Published in: Statistics & probability letters Vol. 81; no. 4; pp. 470 - 477
• Main Authors: León, Jorge A., Villa, José
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.04.2011; Elsevier
• Series: Statistics & Probability Letters
• Subjects: Bifractional Brownian motion; Bifractional Brownian motion Comparison theorem Feller test Osgood criterion; Comparison theorem; Exact sciences and technology; Feller test; General topics; Mathematics; Osgood criterion; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Stochastic analysis; Stochastic processes; Feller test; 60H10; Comparison theorem; 34F05; Bifractional Brownian motion; Osgood criterion; Additive noise; Self-similar process; Measurable function; Probability theory; Brownian motion; Statistical method; Integral equation; Law of iterated logarithm
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/j.spl.2010.12.001

In this paper we use a comparison theorem for integral equations to show that the classical Osgood criterion can be applied to solutions of integral equations of the form Xt=a+∫0tb(Xs)ds+g(t),t≥0. Here, g is a measurable function such that lim supt→∞(inf0≤h≤1g(t+h))=∞, and b is a positive and non-decreasing function. Namely, we will see that the solution X explodes in finite time if and only if ∫⋅∞dsb(s)<∞. As an example, we use the law of the iterated logarithm to see that the bifractional Brownian motion and some increasing self-similar Markov processes satisfy the above condition on g. In other words, g can represent the paths of these processes.