It is shown that the Sliding Mode Control (SMC) technique can be successfully applied to stochastic systems governed by the stochastic differential equations of the Ito type which contain additive stochastic unbounded white noise perturbations. The existence of a strong solution to the corresponding stochastic differential inclusion is discussed. To do this approach workable the gain control parameter is suggested to be done state-dependent on norms of system states. It is demonstrated that under such modification of the conventional SMC we can guarantee the exponential convergence of the averaged squared norm of the sliding variable to mu-zone (around the sliding surface) which is proportional to the diffusion parameter sigma in the model description and inversely depending on the gain parameter k(0).