Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-07-05T12:44:37.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Applications of factorization embeddings for Lévy processes

Part of: Stochastic processes Mathematical economics

Published online by Cambridge University Press:  01 July 2016

A. B. Dieker
A. B. Dieker*
Affiliation:
CWI Amstredam and University of Twente
*
∗ Current address: University College Cork, BCRI, 17 South Bank, Crosses Green, Cork, Ireland. Email address: t.dieker@proba.ucc.ie
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give three applications of the Pecherskii-Rogozin-Spitzer identity for Lévy processes. First, we find the joint distribution of the supremum and the epoch at which it is ‘attained’ if a Lévy process has phase-type upward jumps. We also find the characteristics of the ladder process. Second, we establish general properties of perturbed risk models, and obtain explicit fluctuation identities in the case that the Lévy process is spectrally positive. Third, we study the tail asymptotics for the supremum of a Lévy process under different assumptions on the tail of the Lévy measure.

Keywords

First factorization identityLévy processperturbed risk modelphase-type jumpruin probability

MSC classification

Secondary: 60G70: Extreme value theory; extremal processes 91B30: Risk theory, insurance
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 3 , September 2006 , pp. 768 - 791
Copyright
Copyright © Applied Probability Trust 2006 

References

Arjas, E. and Speed, T. P. (1973). A note on the second factorization identity of A. A. Borovkov. Theory Prob. Appl. 18, 576–578.CrossRefGoogle Scholar
Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Stoch. Models 11, 21–49.Google Scholar
Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354–374.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S., Avram, F. and Pistorius, M. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.CrossRefGoogle Scholar
Asmussen, S., Foss, S. and Korshunov, D. (2003). Asymptotics for sums of random variables with local subexponential behaviour. J. Theoret. Prob. 16, 489–518.CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J. and Doney, R. A. (1994). Cramér's estimate for Lévy processes. Statist. Prob. Lett. 21, 363–365.Google Scholar
Bertoin, J. and Doney, R. A. (1996). Some asymptotic results for transient random walks. Adv. Appl. Prob. 28, 207–226.CrossRefGoogle Scholar
Boucherie, R. J., Boxma, O. J. and Sigman, K. (1997). A note on negative customers, GI/G/1 workload, and risk processes. Prob. Eng. Inf. Sci. 11, 305–311.Google Scholar
Braverman, M., Mikosch, T. and Samorodnitsky, G. (2002). Tail probabilities of subadditive functionals of Lévy processes. Ann. Appl. Prob. 12, 69–100.CrossRefGoogle Scholar
Chan, T. (2004). Some applications of Lévy processes in insurance and finance. Finance (Revue Assoc. Française Finance) 25, 71–94.Google Scholar
Choudhury, G. (2003). Some aspects of an M/G/1 queueing system with optional second service. Top 11, 141–150.CrossRefGoogle Scholar
Dȩbicki, K., Dieker, A. B. and Rolski, T. (2006). Quasi-product forms for Lévy-driven fluid networks. To appear in Math. Operat. Res. Google Scholar
Doney, R. A. (1987). On Wiener–Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Prob. 15, 1352–1362.CrossRefGoogle Scholar
Doney, R. A. (2004). Stochastic bounds for Lévy processes. Ann. Prob. 32, 1545–1552.CrossRefGoogle Scholar
Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16, 91–106.CrossRefGoogle Scholar
Foss, S. and Zachary, S. (2002). Asymptotics for the maximum of a modulated random walk with heavy-tailed increments. In Analytic Methods in Applied Probability, American Mathematical Society, Providence, RI, pp. 37–52.Google Scholar
Furrer, H. J. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Actuarial J.. 1998, 59–74.CrossRefGoogle Scholar
Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14, 1378–1397.CrossRefGoogle Scholar
Iglehart, D. (1972). Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43, 627–635.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.Google Scholar
Kennedy, J. (1992). A probabilistic view of some algebraic results in Wiener–Hopf theory for symmetrizable Markov chains. In Stochastics and Quantum Mechanics, World Scientific, River Edge, NJ, pp. 165–177.Google Scholar
Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801.Google Scholar
Korshunov, D. (1997). On distribution tail of the maximum of a random walk. Stoch. Process. Appl. 72, 97–103.Google Scholar
Kou, S. and Wang, H. (2003). First passage times of a Jump diffusion process. Adv. Appl. Prob. 35, 504–531.CrossRefGoogle Scholar
Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116, 156–177.Google 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
Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407–424.CrossRefGoogle Scholar
Pistorius, M. (2006). On maxima and ladder processes for a dense class of Lévy process. J. Appl. Prob. 43, 208–220.CrossRefGoogle Scholar
Prabhu, N. U. (1998). Stochastic Storage Processes. Springer, New York.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. L. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
Schmidli, H. (2001). Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion. Insurance Math. Econom. 28, 13–20.CrossRefGoogle Scholar