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An Arc-Sine Law for Last Hitting Points in the Two-Parameter Wiener Space

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  • Jeong-Gyoo KIM

    (School of Games, Hongik University, Sejong 30016, Korea)

Abstract

We develop the two-parameter version of an arc-sine law for a last hitting time. The existing arc-sine laws are about a stochastic process X t with one parameter t . If there is another varying key factor of an event described by a process, then we need to consider another parameter besides t . That is, we need a system of random variables with two parameters, say X s , t , which is far more complex than one-parameter processes. In this paper we challenge to develop such an idea, and provide the two-parameter version of an arc-sine law for a last hitting time. An arc-sine law for a two-parameter process is hardly found in literature. We use the properties of the two-parameter Wiener process for our development. Our result shows that the probability of last hitting points in the two-parameter Wiener space turns out to be arcsine-distributed. One can use our results to predict an event happened in a system of random variables with two parameters, which is not available among existing arc-sine laws for one parameter processes.

Suggested Citation

  • Jeong-Gyoo KIM, 2019. "An Arc-Sine Law for Last Hitting Points in the Two-Parameter Wiener Space," Mathematics, MDPI, vol. 7(11), pages 1-11, November.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1131-:d:288404
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    References listed on IDEAS

    as
    1. Dale, Charles & Workman, Rosemarie, 1980. "The arc sine law and the treasury bill futures market," MPRA Paper 46101, University Library of Munich, Germany.
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