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.
Constantin Tudor. "A Comparison Theorem for Stochastic Equations with Volterra Drifts." Ann. Probab. 17 (4) 1541 - 1545, October, 1989. https://doi.org/10.1214/aop/1176991173