Uncertainty quantification is the state-of-the-art framework dealing with uncertainties arising in all kind of real-life problems. One of the framework's functions is to propagate uncertainties from the stochastic input factors to the output quantities of interest, hence the name uncertainty propagation. To this end, polynomial chaos expansions (PCE) have been effectively used in a wide variety of practical problems. However, great challenges are hindering the use of PCE for time-dependent problems. More precisely, the accuracy of PCE tends to decrease in time. In this paper, we develop an approach based on a stochastic time-transform, which allows one to apply low-order PCE to complex time-dependent problems.