On Skew Brownian Motion

Abstract

We consider the stochastic equation $X(t) = W(t) + \beta l^X_0(t)$, where $W$ is a standard Wiener process and $l^X_0(\cdot)$ 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 $|\beta| \leq 1$, but no solutions exist if $|\beta| > 1$. In the former case, setting $\alpha = (\beta + 1)/2$, the unique solution $X$ is distributed as a skew Brownian motion with parameter $\alpha$. 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 $\alpha$ and negative with probability $1 - \alpha$. Finally, we show that skew Brownian motion is the weak limit (as $n \rightarrow \infty$) of $n^{-1/2}S_{\lbrack nt\rbrack}$, where $S_n$ is a random walk with exceptional behavior at the origin.