A Markov chain model for spatially distributed autocatalytic systems with a quadratic reaction rate is considered. An approximate solution for the local probability distribution is obtained in the form of a perturbation expansion for the regimes where diffusion is relatively fast. Using this approximate distribution, properties of the chemical wave fronts found in these autocatalytic systems are studied, and deviations of the minimum propagation velocity and the concentration profile from deterministic predictions are analyzed. A comparison with numerical results from lattice-gas automaton simulations is also provided.