Abstract
A closed form formula is provided for the probability, in a closed time interval, that an arithmetic Brownian motion remains under or above a sequence of three affine, one-sided boundaries (equivalently, for the probability that a geometric Brownian motion remains under or above a sequence of three exponential, one-sided boundaries). The numerical evaluation of this formula can be done instantly and with the accuracy required for all practical purposes. The method followed can be extended to sequences of absorbing boundaries of higher dimension. It is also applied to sequences of two-sided boundaries.
Highlights
The question of the crossing of a non-constant boundary by a diffusion process is of central importance in many mathematical sciences
By closed form results, we mean fully explicit formulae involving functions whose numerical evaluation can be carried out with the accuracy and the efficiency required for all practical purposes, in contrast to approximate analytical solutions that are quickly computed but inaccurate, and to numerical algorithms that can only produce the required standard of precision through heavy computational burden
Scheike (1992) provides a closed form solution for the survival probability of Brownian motion in infinite time when the boundary consists of two successive linear functions of time but cannot explicitly compute the corresponding integral in finite time
Summary
The question of the crossing of a non-constant boundary by a diffusion process is of central importance in many mathematical sciences.
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