On first-passage problems for asymmetric one-dimensional Diffusions

For a, b > 0, we consider a temporally homogeneous, one-dimensional diffusion process X(t) defined over I = (-b, a), with infinitesimal parameters depending on the sign of X(t). We suppose that, when X(t) reaches the position 0, it is reflected rightward to delta with probability p > 0 and leftward to -delta with probability 1 - p, where delta > 0. It is presented a method to find approximate formulae for the mean exit time from the interval (-b,a), and for the probability of exit through the right end a, generalizing the results of Lefebvre ([1]) holding, in the limit delta -> 0, for asymmetric Brownian motion with drift.