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The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

Part of: Distribution theory - Probability Markov processes

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

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 295 - 299
Copyright
Copyright © Applied Probability Trust 2013 

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