A fast simulator for electron beam lithography exposure, based on the Boltzmann transport equation is proposed. Using LITHOS (LITHOgraphy Simulator) it is possible to calculate various important parameters which are useful in performing proximity corrections in e-beam lithography and to predict the resist profile after development. This method is proposed as an alternative to the more common Monte Carlo approach as being much faster since it is based on the calculation of analytical expressions. The results obtained by the analytical model are compared to existing experimental results and to those obtained by other methods. The case of a multilayer sample is considered as being of importance in electron beam patterning. All important phenomena (backscattering, secondary electrons) are included in the calculations.