A diffusion limit for generalized correlated random walks

A generalized correlated random walk is a process of partial SUMS X-k = Sigma(k)(j=1) Y-J such that (X, Y) forms a Markov chain. For a sequence (X-n) of such processes in which each Y-J(n) takes only two values, we prove weak convergence to a diffusion process whose generator is explicitly described in terms of the limiting behaviour of the transition probabilities for the Y-n. Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.