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

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 X-t = a + integral(1)(0) b(X-s)ds + g(t), t >= 0. Here, g is a measurable function such that lim sup(t ->infinity) (inf(0 <= h <= 1) g(t +h)) = infinity, 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 integral(infinity) ds/b(s) < infinity.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. (c) 2010 Elsevier B.V. All rights reserved.