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Functional laws for trimmed Lévy processes

Part of: Stochastic processes Mathematical economics Limit theorems

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

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) J1-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 J1-topologytrimming càdlàg functiontrimmed Lévy processreinsurance risk process

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60G52: Stable processes 60G55: Point processes
Secondary: 60G70: Extreme value theory; extremal processes 60F17: Functional limit theorems; invariance principles 91B30: Risk theory, insurance
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 873 - 889
Copyright
Copyright © Applied Probability Trust 2017 

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