On Skew Brownian Motion

We consider the stochastic equation X(t) = W(t) + βlX0(t), where W is a standard Wiener process and lX0(⋅) 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 |β| ≤ 1, but no solutions exist if |β| > 1. In the former case, setting α = (β + 1)/2, the unique solution X is distributed as a skew Brownian motion with parameter α. 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 α and negative with probability 1−α. Finally, we show that skew Brownian motion is the weak limit (as n→∞) of n−1/2S[nt], where Sn is a random walk with exceptional behavior at the origin.