We use physical arguments and statistical analysis to formulate stochastic models of rainfall intensity in time during intense convective storms. Special attention is given to the issue of multiplicative versus additive model structure and to the type of self-similarity that can be displayed by rainfall under the constraints of stationarity and nonnegativity. We show that some multiscaling models proposed in the past do not satisfy these constraints. Using a set of six high-resolution records, we find that the best fitting models are multiplicative, with a log (rain rate) spectrum of the segmented power type, i.e., of the form |ω|-β, with β that varies in different frequency ranges. Four spectral regimes are identified between scales from a few seconds to several hours. Nowhere do we find a log (rain rate) spectrum of the |ω|-1 type or scaling of the moments, which would be consistent with a conserved multifractal model. Other moment and spectral analyses as well as theoretical arguments lead us to reject also nonconserved multiscaling representations. The stochastic model we finally propose is lognormal with a segmented log spectrum and is not scaling. Two versions of the model are considered, one stationary for the central portion of the storm and the other nonstationary to include the buildup and decay phases. Compared to existing alternatives, the model is very easy to fit to data and to simulate. ¿ American Geophysical Union 1996