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A Note on Erdös and Kac’s Identity: Boundary Crossing Probabilities of Brownian Motion Over Constant Boundaries

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  • Tung-Lung Wu

    (Mississippi State University)

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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  • Tung-Lung Wu, 2020. "A Note on Erdös and Kac’s Identity: Boundary Crossing Probabilities of Brownian Motion Over Constant Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 161-171, March.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:1:d:10.1007_s11009-018-9686-4
    DOI: 10.1007/s11009-018-9686-4
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    References listed on IDEAS

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    1. Sheldon Lin, X., 1998. "Double barrier hitting time distributions with applications to exotic options," Insurance: Mathematics and Economics, Elsevier, vol. 23(1), pages 45-58, October.
    2. Hélyette Geman & Marc Yor, 1996. "Pricing And Hedging Double‐Barrier Options: A Probabilistic Approach," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 365-378, October.
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