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The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

Part of: Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  30 January 2018

Adam Metzler
Adam Metzler*
Affiliation:
University of Western Ontario
*
∗ Postal address: Department of Mathematics, Wilfrid Laurier University, 534 Bricker Academic Building, 75 University Avenue, Waterloo, Ontario N2L 3C5, Canada. Email address: ametzler@wlu.ca
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Abstract

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In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

Keywords

Integrated geometric Brownian motionconfluent hypergeometric functionhitting timeLaplace transform

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60J60: Diffusion processes 60E10: Characteristic functions; other transforms
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 295 - 299
Copyright
Copyright © Applied Probability Trust 2013 

References

Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Carr, P. and Schröder, M. (2004). Bessel processes, the integral of geometric Brownian motion, and Asian options. Theory Prob. Appl. 48, 400–425.Google Scholar
Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 1990, 39–79.Google Scholar
Dufresne, D. (2004). Bessel processes and a functional of Brownian motion. Tech. Rep., Centre for Acturial Studies, University of Melbourne.Google Scholar
Geman, H. and Yor, M. (1993). Bessel proceses, Asian options, and perpetuities. Math. Finance 3, 349–375.Google Scholar
Kyprianou, A. E. and Pistorius, M. R. (2003). Perpetual options and Canadianization through fluctuation theory. Ann. Appl. Prob. 13, 1077–1098.Google Scholar
Lebedev, N. N. (1965). Special Functions and Their Applications. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Lewis, A. L. (1998). Applications of eigenfunction expansions in continuous-time finance. Math. Finance 8, 349–383.Google Scholar
Matsumoto, H. and Yor, M. (2005). Exponential functionals of Brownian motion. I. Probability laws at fixed time. Prob. Surveys 2, 312–347.Google Scholar
Matsumoto, H. and Yor, M. (2005). Exponential functionals of Brownian motion. II. Some related diffusion processes. Prob. Surveys 2, 348–384.Google Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press.Google Scholar