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Hitting Distribution of a Correlated Planar Brownian Motion in a Disk

Author

Listed:
  • Manfred Marvin Marchione

    (Department of Statistical Sciences, Sapienza University of Rome, 00185 Rome, Italy)

  • Enzo Orsingher

    (Department of Statistical Sciences, Sapienza University of Rome, 00185 Rome, Italy)

Abstract

In this article, we study the hitting probability of a circumference C R for a correlated Brownian motion B ̲ ( t ) = B 1 ( t ) , B 2 ( t ) , ρ being the correlation coefficient. The analysis starts by first mapping the circle C R into an ellipse E with semiaxes depending on ρ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels.

Suggested Citation

  • Manfred Marvin Marchione & Enzo Orsingher, 2022. "Hitting Distribution of a Correlated Planar Brownian Motion in a Disk," Mathematics, MDPI, vol. 10(4), pages 1-12, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:4:p:536-:d:745161
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    References listed on IDEAS

    as
    1. Metzler, Adam, 2010. "On the first passage problem for correlated Brownian motion," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 277-284, March.
    2. Chuang, Chin-Shan, 1996. "Joint distribution of Brownian motion and its maximum, with a generalization to correlated BM and applications to barrier options," Statistics & Probability Letters, Elsevier, vol. 28(1), pages 81-90, June.
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