B S, Balakrishna (2013): On multi-particle Brownian survivals and the spherical Laplacian.
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Abstract
The probability density function for survivals, that is for transitions without hitting the barrier, for a collection of particles driven by correlated Brownian motions is analyzed. The analysis is known to lead one to a study of the spectrum of the Laplacian on domains on the sphere in higher dimensions. The first eigenvalue of the Laplacian governs the large time behavior of the probability density function and the asymptotics of the hitting time distribution. It is found that the solution leads one naturally to a generating function for the eigenvalues and multiplicities of the Laplacian. Analytical properties of the generating function suggest a simple scaling procedure for determining the eigenvalues readily applicable for a homogeneous collection correlated particles. Comparison of the first eigenvalue with the theoretical and numerical results of Ratzkin and Treibergs for some special domains shows remarkable agreement.
Item Type: | MPRA Paper |
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Original Title: | On multi-particle Brownian survivals and the spherical Laplacian |
Language: | English |
Keywords: | Brownian; Survival Probability; Hitting Time; Correlation; Laplacian; Spherical Domain; Eigenvalue |
Subjects: | C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing |
Item ID: | 43567 |
Depositing User: | S Balakrishna |
Date Deposited: | 04 Jan 2013 16:49 |
Last Modified: | 28 Sep 2019 01:53 |
References: | DeBlassie, R. D., (1987), ``Exit times from cones in R^n of Brownian motion'', Probability Theory and Related Fields. 74-1 (1987), 1-29. Carslaw, H. S. and Jaeger, J. C., (1959), ``Conduction of heat in solids'', Oxford University Press, Oxford, 1959. Cheeger, J., (1983), ``Spectral geometry of singular Riemannian spaces'', Journal of Differential Geometry 18-4 (1983), 575-657. Ratzkin, J. and Treibergs, A., (2009), ``A capture problem in Brownian motion and eigenvalues of spherical domains'', Transactions of the American Mathematical Society 361 (2009), 391-405. Sommerfeld, A., (1894), ``Zur analytischen Theorie der W\"{a}rmeleitung'', Mathematische Annalen 45-2 (1894), 263-277. Vassilevich, D. V., (2003), ``Heat kernel expansion: user's manual'', Physics Reports, 388 (2003), 279-360. Zhou, C., (2001), ``An analysis of default correlations and multiple defaults'', Review of Financial Studies, 14-2 (2001), 555-576. |
URI: | https://mpra.ub.uni-muenchen.de/id/eprint/43567 |
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