The evolution of magnetospheric indices on temporal scales shorter than that of substorms is characterized by bursty, intermittent events that may arise from turbulence intrinsic to the magnetosphere or that may reflect solar wind-magnetosphere coupling. This leads to a generic problem of distinguishing between the features of the system and those of the driver. We quantify scaling properties of short-term (up to few hours) fluctuations in the geomagnetic indices AL and AU during solar minimum and maximum, along with the parameter $epsilon$ that is a measure of the solar wind driver. We find that self-similar statistics provide a good approximation for the observed scaling properties of fluctuations in the geomagnetic indices, regardless of the solar activity level, and in the $epsilon$ parameter at solar maximum. This self-similarity persists for fluctuations on timescales at least up to about 1--2 hours. The scaling exponent of AU index fluctuations show dependence on the solar cycle, and the trend follows that found in the scaling of fluctuations in $epsilon$. The values of their corresponding scaling exponents, however, are always distinct. Fluctuations in the AL index are insensitive to the solar cycle, as well as being distinct from those in the $epsilon$ parameter. This approximate self-similar scaling leads to a Fokker-Planck model which, we show, captures the probability density function of fluctuations and provides a stochastic dynamical equation (Langevin equation) for time series of the geomagnetic indices.