The Lyman-&agr; auroral emission is characterized by a broad line profile whose shape depends on the energy and pitch angle distributions of the initial proton beam, whereas its total brightness reflects the proton energy flux precipitated into the auroral upper atmosphere. Global remote sensing of the proton aurora through its ultraviolet signature makes it is increasingly important to relate the characteristics of the Lyman-&agr; emission to the physical properties of the precipitated proton flux. We present a numerical model of proton and hydrogen flux transport and kinetics based on the direct simulation Monte Carlo method. In this approach, all elastic and inelastic processes are stochastically simulated as well as is the production of Lyman-&agr; photons with the associated Doppler velocity component. The model also includes collisional, geomagnetic, and geometric spreading of the proton-hydrogen beam. We show that consideration of the stochastic character of the H atom velocity redistribution after collisions produces line profiles different from those obtained in the strictly forward or mean scattering angle approximations previously used in proton transport codes. In particular, the predicted fraction of photons due to backscattered particles is considerably larger when stochastic collision scattering is considered than in the strictly forward or mean scattering angle approximations. In contrast to the median wavelength, the position of the peak in the line profile shows a weak inverse dependence on the proton energy. The efficiency of the Lyman-&agr; photon production per unit incident energy flux significantly drops as the mean proton energy increases. The line profile and the amount of blue-shifted (for downward viewing) emission depends in a complex way on the initial energy and pitch angle distribution of the protons. The line profiles expected for the noon cusp and midnight proton aurora are shown to be significantly different.