Hitting Distribution of a Correlated Planar Brownian Motion in a Disk

Manfred Marvin Marchione and Enzo Orsingher
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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

Mathematics, 2022, vol. 10, issue 4, 1-12

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.

Keywords: elliptic coordinates; Poisson kernel; Ghizzetti transformation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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