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April, 1981 On Skew Brownian Motion
J. M. Harrison, L. A. Shepp
Ann. Probab. 9(2): 309-313 (April, 1981). DOI: 10.1214/aop/1176994472

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.

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J. M. Harrison. L. A. Shepp. "On Skew Brownian Motion." Ann. Probab. 9 (2) 309 - 313, April, 1981. https://doi.org/10.1214/aop/1176994472

Information

Published: April, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0462.60076
MathSciNet: MR606993
Digital Object Identifier: 10.1214/aop/1176994472

Subjects:
Primary: 60J55
Secondary: 60J60, 60J65

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 2 • April, 1981
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