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Ruin probabilities for competing claim processes

Part of:Mathematical economicsSpecial processesMarkov processesStochastic processes

Published online by Cambridge University Press:  14 July 2016

Miljenko Huzak,
Mihael Perman,
Hrvoje Šikić and
Zoran Vondraček
Miljenko Huzak*
Affiliation:
University of Zagreb
Mihael Perman*
Affiliation:
University of Ljubljana
Hrvoje Šikić*
Affiliation:
University of Zagreb
Zoran Vondraček*
Affiliation:
University of Zagreb
*
∗ Postal address: Department of Mathematics, University of Zagreb, Bijenička c. 30, 10000 Zagreb, Croatia
∗∗ Postal address: Institute for Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
∗ Postal address: Department of Mathematics, University of Zagreb, Bijenička c. 30, 10000 Zagreb, Croatia
∗ Postal address: Department of Mathematics, University of Zagreb, Bijenička c. 30, 10000 Zagreb, Croatia
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Abstract

Let C 1, C 2,…,C m be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators C i . Formulae for the probability that ruin is caused by C i are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

Keywords

Risk theoryruin probabilityPollaczek–Khinchin formulasubordinatorspectrally negative Lévy process

MSC classification

Primary:60J25: Continuous-time Markov processes on general state spaces
Secondary:60G51: Processes with independent increments; L'evy processes60J75: Jump processes60K30: Applications (congestion, allocation, storage, traffic, etc.)91B30: Risk theory, insurance

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004, pp. 679 - 690
Copyright
Copyright © Applied Probability Trust 2004 

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