In this paper we introduce a stochastic model for the A + 1/2B2 --> 0 reaction on a square lattice. Reaction between an A and a B particle occurs if they are nearest neighbors on the lattice. To this system which includes adsorption and reaction steps we add the effect of A-diffusion and A-desorption. We describe the model in terms of master equations using the Markovian behavior of the system. The equations are truncated at a certain level via a modified Kirkwood approximation. The reaction is in this paper introduced between particles which are nearest neighbors on the lattice. This approach which is different from a previous article [J. Mai et al, J. Chem. Phys. 98, 10017 (1993)] requires a special treatment of the stochastic equations and the correlation functions. In particular the Kirkwood superposition approximation, which is used to truncate the hierarchy of equations, has to be modified. The resulting system of lattice equations is solved in a small region around a reference point. The solution is connected to continuous functions which describe the system behavior for larger distances. This system shows kinetic phase transitions which separate the reactive regime from two nonreactive states where the lattice is completely covered by A or B. We study the location and the character of the phase transitions in detail. With the help of correlation functions we identify the different phases of particles on the lattice. Island formation and segregation of the particles on the lattice are found to be dominant processes. It is established that finite lattices which have to be used in simulations can be seriously inadequate and miss physical processes. This problem does not appear in the ansatz presented here.