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 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.