Journal of Applied Probability

Boundary crossing probability for Brownian motion

Klaus Pötzelberger and Liqun Wang

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Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n2).

Article information

J. Appl. Probab. Volume 38, Number 1 (2001), 152-164.

First available in Project Euclid: 5 August 2001

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 65C05: Monte Carlo methods

first hitting time first passage time optimal stopping curved boundaries Wiener process random walk barrier options cumulative sums sequential analysis Monte Carlo simulation


Pötzelberger, Klaus; Wang, Liqun. Boundary crossing probability for Brownian motion. J. Appl. Probab. 38 (2001), no. 1, 152--164. doi:10.1239/jap/996986650.

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