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Functional laws for trimmed Lévy processes

Part of:Limit theoremsMathematical economicsStochastic processes

Published online by Cambridge University Press:  15 September 2017

Boris Buchmann,
Yuguang F. Ipsen and
Ross Maller
Boris Buchmann*
Affiliation:
Australian National University
Yuguang F. Ipsen*
Affiliation:
Australian National University and University of Melbourne
Ross Maller*
Affiliation:
Australian National University
*
* Postal address: Research School of Finance, Actuarial Studies and Statistics, Australian National University, 26C Kingsley Street, Acton, ACT 2601, Australia.
* Postal address: Research School of Finance, Actuarial Studies and Statistics, Australian National University, 26C Kingsley Street, Acton, ACT 2601, Australia.
* Postal address: Research School of Finance, Actuarial Studies and Statistics, Australian National University, 26C Kingsley Street, Acton, ACT 2601, Australia.
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Abstract

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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.

Keywords

Functional lawSkorokhod J 1-topologytrimming càdlàg functiontrimmed Lévy processreinsurance risk process

MSC classification

Primary:60G51: Processes with independent increments; L'evy processes60G52: Stable processes60G55: Point processes
Secondary:60G70: Extreme value theory; extremal processes60F17: Functional limit theorems; invariance principles91B30: Risk theory, insurance

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017, pp. 873 - 889
Copyright
Copyright © Applied Probability Trust 2017 

References

[1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York. Google Scholar
[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, 2325–2371. Google Scholar
[3] Fan, Y. (2015). Convergence of trimmed Lévy process to trimmed stable random variables at 0. Stoch. Process. Appl. 125, 3691–3724. Google Scholar
[4] Fan, Y. (2017). Tightness and convergence of trimmed Lévy processes to normality at small times. J. Theoret. Prob. 30, 675–699. Google Scholar
[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. Google Scholar
[6] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin. Google Scholar
[7] Ladoucette, S. A. and Teugels, J. L. (2006). Reinsurance of large claims. J. Comput. Appl. Math. 186, 163–190. CrossRefGoogle Scholar
[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. Google Scholar
[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, 2205–2248. Google Scholar
[10] Maller, R. A. and Schmidli, P. C. (2017). Small-time almost-sure behaviour of extremal processes. Adv. Appl. Prob. 49, 411–429. Google Scholar
[11] Silvestrov, D. S. and Teugels, J. L. (2004). Limit theorems for mixed max-sum processes with renewal stopping. Ann. Appl. Prob. 14, 1838–1868. CrossRefGoogle Scholar
[12] Skorokhod, A. V. (1956). Limit theorems for stochastic processes. Theory Prob. Appl. 1, 261–290. Google Scholar
[13] Teugels, J. L. (2003). Reinsurance actuarial aspects. Res. Rep. 2003-006, Eindhoven University of Technology. Google Scholar
[14] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York. Google Scholar