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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 T 1,T 2,…. At the time T η(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(T η(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)

Information

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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