  We derive self-consistent expressions of current and noise for
single-electron transistors driven by time-dependent perturbations. We take
into account effects of the electrical environment, higher-order co-tunneling,
and time-dependent perturbations under the two-charged state approximation
using the Schwinger-Kedysh approach combined with the generating functional
technique. For a given generating functional, we derive exact expressions for
tunneling currents and noises and present the forms in terms of transport
coefficients. It is also shown that in the adiabatic limit our results
encompass previous formulas. In order to reveal effects missing in static
cases, we apply the derived results to simulate realized radio-frequency
single-electron transistor. It is found that photon-assisted tunneling affects
largely the performance of the single-electron transistor by enhancing both
responses to gate charges and current noises. On various tunneling resistances
and frequencies of microwaves, the dependence of the charge sensitivity is also
discussed.
