Almost sure explosion of solutions to stochastic differential equations

This paper is concerned with the problem of explosive solutions for a class of stochastic differential equations. Our main results are presented as two theorems. Theorem 1 is concerned with the existence of explosive solutions with positive probability under certain sufficient conditions. With some additional mild conditions, it is shown in Theorem 2 that the explosion will occur almost surely. The methods of auxiliary functions and cycles are used in the proofs. Several remarks about their applications are given.