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The photon correlation of photon emission from a single quantum dot with cw excitation and pulsed excitation is investigated in details. To calculate the second-order correlation function for optical pumping, we deduce rate equations with a simplified two-level model under cw excitation and present the master equation approach in the interaction picture to the study of evolution of a three-level system under pulsed excitation. In addition, we report photon correlation measurements on a single self-assembled In0.5 Ga0.5As quantum dot, which show strong antibunching behaviour under both the conditions of cw and pulsed excitations. The calculated results are in agreement with the experimental measurements.