A statistical model of the nonlinear response of silicon photomultipliers is presented. It includes losses of both the photodetection efficiency and the gain during pixel recovery periods as well as the effect of correlated and uncorrelated noise. The model provides either the mean output charge of a SiPM for incident light pulses of arbitrary shape or the output current for continuous light. The dependence of the SiPM response on both the overvoltage and the pulse shape is also properly described. The model has been validated for two different silicon photomultipliers using scintillation light pulses from a LYSO crystal as well as continuous light from a LED. Good agreement is found with experimental data at moderate nonlinearity.