On Skew Brownian Motion

Abstract

We consider the stochastic equation X(t) = W(t) + ¥beta l_0^x(t), where W is a standard Wiener process and lox(.) is the local time at zero of the unknown process X. There is a unique solution X (and it is adapted to the fields of W) if 1,81 'I 1, but no solutions exist if 1,81 > 1. In the former case, setting a = (,B + 1)/2, the unique solution X is distributed as a skew Brownian motion with parameter a. This is a diffusion obtained from standard Wiener process by independently altering the signs of the excursions away from zero, each excursion being positive with probability a and negative with probability 1 - a. Finally, we show that skew Brownian motion is the weak limit (as n -> ¥infty) of n^{-1/2} S[nt], where S_n is a random walk with exceptional behavior at the origin