In this paper, a new finite-horizon H-infinity, control problem is considered for a class of time-varying systems with stochastic nonlinearities, measurements degradation and chance constraints. The purpose of the addressed problem is to design the time-varying controller such that the closed-loop system satisfies the prespecified H-infinity disturbance attenuation requirement and certain chance constraints on the controlled output vector z(k) (i.e., the probability of the controlled output z(k) belonging to a given set is larger than a prescribed value). A modified maximum-volume-inscribed-ellipsoid (MVIE) method is employed to convert the chance constraint into some tractable inequalities, which could be conveniently handled by the recursive linear matrix inequality (RLMI) approach. Then, by using stochastic control analysis method, sufficient conditions are derived for the existence of the desired multi-objective controller and, furthermore, the gains of the desired controllers are characterized by means of RLMIs. Finally, two illustrative examples and a practical system are proposed to highlight the effectiveness and applicability of the presented controller design technology. (C) 2019 Elsevier Ltd. All rights reserved.