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On the First Passage time for Brownian Motion Subordinated by a Lévy Process

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

T. R. Hurd  and
A. Kuznetsov
T. R. Hurd*
Affiliation:
McMaster University
A. Kuznetsov*
Affiliation:
York University
*
∗Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada. Email address: [email protected]
∗∗Postal address: Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada.
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Abstract

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In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

Keywords

Brownian motionfirst passagetime changeLévy subordinatorstopping timefinancial models

MSC classification

Secondary: 60J75: Jump processes 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 181 - 198
Copyright
Copyright © Applied Probability Trust 2009 

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