The stochastic theory of grain growth proposed previously [C. S. Pande, Acta metall. 35, 2671 (1987)] is developed further, in this paper. It is shown that in addition to lognormality and power law kinetics, dynamic scaling (self similarity) can also be derived from the model. Effects of an arbitrary initial grain size distribution and specimen size are also examined. Finally the question of volume conservation is re-examined in the light of recent criticisms.