The Annals of Probability

On Skew Brownian Motion

J. M. Harrison and L. A. Shepp

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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.

Article information

Ann. Probab., Volume 9, Number 2 (1981), 309-313.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J55: Local time and additive functionals
Secondary: 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]

Skew Brownian motion diffusion processes local time


Harrison, J. M.; Shepp, L. A. On Skew Brownian Motion. Ann. Probab. 9 (1981), no. 2, 309--313. doi:10.1214/aop/1176994472.

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