On a class of Ito stochastic differential equations

We provide sufficient conditions on the drift and on the diffusion coefficients of Ito-type equations for the existence of stochastic-process solutions, constructed by successive approximations and which converge almost surely or in the mean-square. We also obtain a result for the uniqueness of the solutions, extending the classical theorem of Ito and consistent with respect to more recent pathwise uniqueness results. In particular, a relaxation of the Lipschitz condition is given by allowing a suitably controlled growth in the time-variable.