On first-passage problems for asymmetric one-dimensional diffusions

Abstract

For a, b > 0, we consider a temporally homogeneous, onedimensional diffusion process X(t) defined over 7 = (-6, 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 6 with probability p > 0 and leftward to -δ with probability 1 -p, where δ > 0. It is presented a method to find approximate formulae for the mean exit time from the interval ( - 6, a), and for the probability of exit through the right end a, generalizing the results of Lefebvre ([1]) holding, in the limit δ → 0, for asymmetric Brownian motion with drift. © Springer-Verlag Berlin Heidelberg 2007.