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Boundary crossing probability for Brownian motion

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Klaus Pötzelberger  and
Liqun Wang
Klaus Pötzelberger*
Affiliation:
Vienna University of Economics and Business Administration
Liqun Wang*
Affiliation:
University of Manitoba
*
∗ Postal address: Institute of Statistics, Vienna University of Economics and Business Administration, Augasse 2–6, A-1090 Vienna, Austria.
∗∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2. Email address: [email protected]
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Abstract

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).

Keywords

first hitting timefirst passage timeoptimal stoppingcurved boundariesWiener processrandom walkbarrier optionscumulative sumssequential analysisMonte Carlo simulation

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 152 - 164
Copyright
Copyright © by the Applied Probability Trust 2001 

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