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On the ruin probability of the generalised Ornstein–Uhlenbeck process in the cramér case

Published online by Cambridge University Press:  14 July 2016

Damien Bankovsky
Affiliation:
Australian National University, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia. Email address: [email protected]
Claudia Klüppelberg
Affiliation:
Technische Universität München, Center for Mathematical Sciences, and Institute for Advanced Study, Technische Universität München, 85748 Garching, Germany. Email address: [email protected]
Ross Maller
Affiliation:
Australian National University, Mathematical Sciences Institute, and School of Finance and Applied Statistics, Australian National University, Canberra, ACT 0200, Australia. Email address: [email protected]
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Abstract

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For a bivariate Lévy process (ξtt)t≥ 0 and initial value V0 define the generalised Ornstein–Uhlenbeck (GOU) process Vt:=eξt (V0+∫t0 es-s), t≥0, and the associated stochastic integral process Zt:=∫0t es-s, t≥0. Let Tz:=inf{t>0: Vt<0|V0=z} and ψ(z):=P(Tz<∞) for z≥0 be the ruin time and infinite horizon ruin probability of the GOU process. Our results extend previous work of Nyrhinen (2001) and others to give asymptotic estimates for ψ(z) and the distribution of Tz as z→∞, under very general, easily checkable, assumptions, when ξ satisfies a Cramér condition.

Type
Part 1. Risk Theory
Copyright
Copyright © Applied Probability Trust 2011 

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