First Passage Density of Brownian Motion with Two-sided Piecewise Linear Boundaries

The first passage time has many applications in fields like finance, econometrics, statistics, and biology. However, explicit formulas for the first passage density have only been obtained for a few cases. This paper derives an explicit formula for the first passage density of Brownian motion with t...

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Bibliographic Details
Published inActa mathematica Sinica. English series Vol. 40; no. 6; pp. 1505 - 1520
Main Authors Yu, Zhen, Tian, Mao Zai
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 2024
Springer Nature B.V
Subjects
Approximation
Biology
Boundaries
Brownian motion
Density
Econometrics
Finance
Mathematics
Mathematics and Statistics
Monte Carlo simulation
Statistics
Brownian motion
Boundary non-crossing probability
60G05
first density passage density
two-sided piecewise continuous boundaries
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Summary:The first passage time has many applications in fields like finance, econometrics, statistics, and biology. However, explicit formulas for the first passage density have only been obtained for a few cases. This paper derives an explicit formula for the first passage density of Brownian motion with two-sided piecewise continuous boundaries which may have some points of discontinuity. Approximations are used to obtain a simplified formula for estimating the first passage density. Moreover, the results are also generalized to the case of two-sided general nonlinear boundaries. Simulations can be easily carried out with Monte Carlo method and it is demonstrated for several typical two-sided boundaries that the proposed approximation method offers a highly accurate approximation of first passage density.
ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-024-1090-0