This article deals with the recursive filtering problem for nonlinear time-varying stochastic systems subject to possible measurement outliers. In order to mitigate the effects from possible abnormal measurements, we construct a filter with a saturation constraint imposed on the innovations where the saturation level is adaptively determined according to the estimation errors. Two performance indices, namely, the finite-horizon <inline-formula><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> specification and the envelope-constraint criterion with a prescribed probability, are put forward to describe the transient characteristics of the filtering error dynamics over a specified time interval. The purpose of the addressed problem is to design a filter capable of guaranteeing both the finite-horizon <inline-formula><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance index and the probability-guaranteed envelope-constraint. Sufficient conditions are derived for the existence of the desired filter via certain convex optimization algorithms. Finally, an illustrative numerical example is proposed to demonstrate the effectiveness of the developed algorithm.