Abstract
The finite Markov chain imbedding technique is an emerging approach for calculating boundary crossing probabilities for high-dimensional Brownian motion and certain one-dimensional diffusion processes. In 1996, Erdös and Kac produced an infinite series for the crossing probability of Brownian motion over a two-sided constant boundary. We derive this classic result based on a unified formula from the finite Markov chain imbedding technique. Also, an eigenvalues-and-eigenvectors approximation is given for fast computation. The main purpose of this paper is to show the versatility of the finite Markov chain imbedding technique.
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Wu, TL. A Note on Erdös and Kac’s Identity: Boundary Crossing Probabilities of Brownian Motion Over Constant Boundaries. Methodol Comput Appl Probab 22, 161–171 (2020). https://doi.org/10.1007/s11009-018-9686-4
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DOI: https://doi.org/10.1007/s11009-018-9686-4
Keywords
- Boudnary crossing probabilities
- Brownian motion
- Finite Markov chain imbedding
- Random walks
- Erdös and Kac’s identity