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Insurance with borrowing: first- and second-order approximations

Part of: Probability theory on algebraic and topological structures Special processes

Published online by Cambridge University Press:  01 July 2016

A. A. Borovkov
A. A. Borovkov*
Affiliation:
Sobolev Institute of Mathematics, Novosibirsk
*
∗ Postal address: Sobolev Institute of Mathematics, Acad. Koptyuga prospect 4, Novosibirsk, 630090, Russia. Email address: borovkov@math.nsc.ru
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Abstract

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We consider the operation of an insurer with a large initial surplus x>0. The insurer's surplus process S(t) (with S(0)=x) evolves in the range S(t)≥ 0 as a generalized renewal process with positive mean drift and with jumps at time epochs T1,T2,…. At the time Tη(x) when the process S(t) first becomes negative, the insurer's ruin (in the ‘classical’ sense) occurs, but the insurer can borrow money via a line of credit. After this moment the process S(t) behaves as a solution to a certain stochastic differential equation which, in general, depends on the indebtedness, -S(t). This behavior of S(t) lasts until the time θ(x,y) at which the indebtedness reaches some ‘critical’ level y>0. At this moment the line of credit will be closed and the insurer's absolute ruin occurs with deficit -S(θ(x,y)). We find the asymptotics of the absolute ruin probability and the limiting distributions of η(x), θ(x,y), and -S(θ(x,y)) as x → ∞, assuming that the claim distribution is regularly varying. The second-order approximation to the absolute ruin probability is also obtained. The abovementioned results are obtained by using limiting theorems for the joint distribution of η(x) and -S(Tη(x)).

Keywords

Insurance with borrowingabsolute ruindeficit at absolute ruindebit interestregularly varying distributionssubexponential functionfirst passage time; excesscompound renewal processstochastic differential equationsecond-order approximation

MSC classification

Primary: 60K99: None of the above, but in this section 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 4 , December 2009 , pp. 1141 - 1160
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Work partially supported by the Russian President's Grant NSh-3695-2008.1 and the RFBR Grant 08-01-00962.

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