Investigations of high-voltage open discharge in helium have shown a possibility of generation of current pulses with subnanosecond front rise, due to ultra-fast breakdown development. The open discharge is ignited between two planar cathodes with mesh anode in the middle between them. For gas pressure 6 Torr and 20 kV applied voltage, the rate of current rise reaches 500 A/(cm2 ns) for current density 200 A/cm2 and more. The time of breakdown development was measured for different helium pressures and a kinetic model of breakdown in open discharge is presented, based on elementary reactions for electrons, ions and fast atoms. The model also includes various cathode emission processes due to cathode bombardment by ions, fast atoms, electrons and photons of resonant radiation with Doppler shift of frequency. It is shown, that the dominating emission processes depend on the evolution of the discharge voltage during the breakdown. In the simulations, two cases of voltage behavior were considered: (i) the voltage is kept constant during the breakdown; (ii) the voltage is reduced with the growth of current. For the first case, the exponentially growing current is maintained due to photoemission by the resonant photons with Doppler-shifted frequency. For the second case, the dominating factor of current growth is the secondary electron emission. In both cases, the subnanosecond rise of discharge current was obtained. Also the effect of gas pressure on breakdown development was considered. It was found that for 20 Torr gas pressure the time of current rise decreases to 0.1 ns, which is in agreement with experimental data.