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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: hurdt@mcmaster.ca
∗∗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 

References

[1] Alili, L and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes. Ann. Appl. Prob. 15, 2062–2080.CrossRefGoogle Scholar
[2] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus (Camb. Stud. Adv. Math. 93). Cambridge University Press.CrossRefGoogle Scholar
[3] Asmussen, S. (2000). Ruin Probabilities (Adv. Ser. Statist. Appl. Prob. 2). World Scientific, River Edge, NJ.CrossRefGoogle Scholar
[4] Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.CrossRefGoogle Scholar
[5] Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distribution and stochastic volatility modelling. Scand. J. Statist. 24, 1–13.CrossRefGoogle Scholar
[6] Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
[7] Borodin, A. N. and Salminen, P. (1996). Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel.CrossRefGoogle Scholar
[8] Cherny, A. S. and Shiryaev, A. N. (2002). Change of time and measures for Lévy processes. (Lectures from the Summer School ‘From Lévy processes to semimartingales: recent theoretical developments and applications to finance’ (Aarhus, August 2002)).Google Scholar
[9] Cont, R. and Tankov, P. (2004). Financial Modeling with Jump Processes. Chapman & Hall, Boca Raton, FL.Google Scholar
[10] Cont, R. and Voltchkova, E. (2005). A finite difference scheme for option pricing in Jump diffusion and exponential Lévy models. SIAM J. Numer. Anal. 43, 1596–1626.CrossRefGoogle Scholar
[11] Gradshteyn, I. S. and Ryzhik, I. M. (2000). Tables of Integrals, Series, and Products, 6th edn. Academic Press, San Diego, CA.Google Scholar
[12] Hurd, T. R. (2007). Credit risk modelling using time-changed Brownian motion. Working paper. Available at http://www.math.mcmaster.ca/tom/HurdTCBMRevised.pdf.Google Scholar
[13] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.CrossRefGoogle Scholar
[14] Kou, S. G and Wang, H. (2003). First passage times of a Jump diffusion process. Adv. Appl. Prob. 35, 504–531.CrossRefGoogle Scholar
[15] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[16] Lewis, A. L. and Mordecki, E. (2008). Wiener–Hopf factorization for Lévy processes having positive Jumps with rational transforms. J. Appl. Prob. 45, 118–134.CrossRefGoogle Scholar
[17] Madan, D. and Seneta, E. (1990). The VG model for share market returns. J. Business 63, 511–524.CrossRefGoogle Scholar
[18] Mordecki, E. (2002). The distribution of the maximum of a Lévy process with positive Jumps of phase-type. Theory Stoch. Process. 8, 309–316.Google Scholar
[19] O'Cinneide, C. A. (1990). Characterization of phase-type distributions. Stoch. Models 6, 1–57.CrossRefGoogle Scholar
[20] Percheskii, E. A. and Rogozin, B. A. (1969). On the Joint distribution of random variables associated with fluctuations of a process with independent increments. Theory Prob. Appl. 14, 410–423.CrossRefGoogle Scholar
[21] Rogers, L. C. G. (1984). A new identity for real Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 20, 21–34.Google Scholar