Skip to main content Accessibility help
×
×
Home

Functional laws for trimmed Lévy processes

  • Boris Buchmann (a1), Yuguang F. Ipsen (a2) and Ross Maller (a1)

Abstract

Two different ways of trimming the sample path of a stochastic process in 𝔻[0, 1]: global ('trim as you go') trimming and record time ('lookback') trimming are analysed to find conditions for the corresponding operators to be continuous with respect to the (strong) J 1-topology. A key condition is that there should be no ties among the largest ordered jumps of the limit process. As an application of the theory, via the continuous mapping theorem, we prove limit theorems for trimmed Lévy processes, using the functional convergence of the underlying process to a stable process. The results are applied to a reinsurance ruin time problem.

Copyright

Corresponding author

* Postal address: Research School of Finance, Actuarial Studies and Statistics, Australian National University, 26C Kingsley Street, Acton, ACT 2601, Australia.
** Email address: boris.buchmann@anu.edu.au
*** Email address: yuguang.ipsen@anu.edu.au
**** Email address: ross.maller@anu.edu.au

References

Hide All
[1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
[2] Buchmann, B., Fan, Y. and Maller, R. A. (2016). Distributional representations and dominance of a Lévy process over its maximal jump processes. Bernoulli 22, 23252371.
[3] Fan, Y. (2015). Convergence of trimmed Lévy process to trimmed stable random variables at 0. Stoch. Process. Appl. 125, 36913724.
[4] Fan, Y. (2017). Tightness and convergence of trimmed Lévy processes to normality at small times. J. Theoret. Prob. 30, 675699.
[5] Fan, Y. et al. (2017). The effects of largest claim and excess of loss reinsurance on a company's ruin time and valuation. Risks 5, 27pp. Available at http://www.mdpi.com/2227-9091/5/1/3.
[6] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.
[7] Ladoucette, S. A. and Teugels, J. L. (2006). Reinsurance of large claims. J. Comput. Appl. Math. 186, 163190.
[8] Maller, R. A. and Fan, Y. (2015). Thin and thick strip passage times for Lévy flights and Lévy processes. Preprint. Available at https://arxiv.org/abs/1504.06400.
[9] Maller, R. and Mason, D. M. (2010). Small-time compactness and convergence behavior of deterministically and self-normalised Lévy processes. Trans. Amer. Math. Soc. 362, 22052248.
[10] Maller, R. A. and Schmidli, P. C. (2017). Small-time almost-sure behaviour of extremal processes. Adv. Appl. Prob. 49, 411429.
[11] Silvestrov, D. S. and Teugels, J. L. (2004). Limit theorems for mixed max-sum processes with renewal stopping. Ann. Appl. Prob. 14, 18381868.
[12] Skorokhod, A. V. (1956). Limit theorems for stochastic processes. Theory Prob. Appl. 1, 261290.
[13] Teugels, J. L. (2003). Reinsurance actuarial aspects. Res. Rep. 2003-006, Eindhoven University of Technology.
[14] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed