Brownian Type Boundary Crossing Probabilities for Piecewise Linear Boundary Functions

Abstract

The present paper offers exact boundary crossing probabilities for the Brownian motion and the Brownian bridge when the boundary function is piecewise linear. They are mostly given for one-sided bounds. The probabilities are expressed by multivariate normal distributions given by a Sylvester type formula. Some applications are sketched in statistics for Kolmogorov–Smirnov type test statistics also when ties of the data are present. We offer error bounds for the crossing probabilities when the boundary function is approximated by another one.

Key Words:

• Boundary crossing
• Brownian motion with drift
• Brownian bridge
• Error bounds
• Kolmogorov–Smirnov test
• Power function

Mathematics Subject Classification:

• Primary 60G15
• Secondary 62G10

Acknowledgments