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On Maxima and Ladder Processes for a Dense Class of Lévy Process

Part of: Stochastic processes Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Martijn Pistorius
Martijn Pistorius*
Affiliation:
King's College London
*
∗ Postal address: Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK. Email address: [email protected]
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Abstract

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In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.

Keywords

Lévy processfirst passageladder processWiener-Hopf factorizationphase-type distributionMarkov additive processnonlinear iteration

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60K15: Markov renewal processes, semi-Markov processes 60J25: Continuous-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 208 - 220
Copyright
© Applied Probability Trust 2006 

References

Asmussen, S. (1989). Exponential families generated by phase-type distributions and other Markov lifetimes. Scand. J. Statist. 16, 319–334.Google Scholar
Asmussen, S. (1992). Phase-type representations in random walk and queueing problems. Ann. Prob. 20, 772–789.Google Scholar
Asmussen, S. (2000). Ruin Probabilities. World Scientific, River Edge, NJ.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.CrossRefGoogle Scholar
Asmussen, S., Madan, D. P. and Pistorius, M. R. (2005). Pricing equity default swaps under the CGMY Lévy model. Submitted.Google Scholar
Barndorff-Nielssen, O. (2000). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.Google Scholar
Carr, P., Geman, H., Madan, D. P. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation. J. Business 75, 305–332.CrossRefGoogle Scholar
Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli 1, 281–299.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.Google Scholar
Kou, S. G. (2002). A Jump diffusion model for option pricing. Manag. Sci. 48, 1086–1101.CrossRefGoogle Scholar
London, R. R., McKean, H. P., Rogers, L. C. G. and Williams, D. (1982). A martingale approach to some Wiener–Hopf problems. I, II. In Seminar on Probability, XVI (Lecture Notes Math. 920), Springer, Berlin, pp. 41–90.Google Scholar
Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493.CrossRefGoogle Scholar
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
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Neuts, M. F. (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York.Google Scholar