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

Part of: Stochastic analysis Mathematical economics Markov processes

Published online by Cambridge University Press:  14 July 2016

Damien Bankovsky ,
Claudia Klüppelberg  and
Ross Maller
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 (ξt,ηt)t≥ 0 and initial value V0 define the generalised Ornstein–Uhlenbeck (GOU) process Vt:=eξt (V0+∫t0 e-ξs-dηs), t≥0, and the associated stochastic integral process Zt:=∫0t e-ξs-dηs, 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.

Keywords

Exponential functionals of Lévy processesgeneralised Ornstein–Uhlenbeck processruin probabilitystochastic recurrence equation

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60J25: Continuous-time Markov processes on general state spaces 91B30: Risk theory, insurance
Secondary: 60H25: Random operators and equations
Type
Part 1. Risk Theory
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 15 - 28
Copyright
Copyright © Applied Probability Trust 2011 

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