A Comparison Theorem for Stochastic Equations with Volterra Drifts

Abstract

A comparison theorem is proved for one-dimensional stochastic equations driven by continuous semimartingales and having Volterra-type drifts. A counterexample which shows that the coefficient of the continuous martingale term cannot be Volterra-type is given. Then the comparison result is used in order to obtain the existence of strong solutions when the Lipschitz condition is replaced by a Holder-type one.