Based on the theory of quasi-three-level rate equations modified by amplified spontaneous emission, the stored energy density and the small signal gain of the cryogenic Yb:YAG regenerative amplifier for a given geometry for pulsed pumping in three dimensions are theoretically studied using the Monte Carlo simulation. The present model provides a straightforward procedure to design the Yb:YAG parameters and the optical coupling system for optimization when running at cryogenic temperature. A fiber-coupled laser diode end-pumped cryogenic Yb:YAG regenerative amplifier running at 1 030 nm is demonstrated with a maximum output energy 10.2 mJ at a repetition rate of 10 Hz. A very good agreement between the experiments and the theoretical model is achieved.