Hostname: page-component-848d4c4894-89wxm Total loading time: 0 Render date: 2024-07-07T13:30:12.205Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Asmussen, S. (2007). Ruin Probabilities. World Scientific, River Edge, NJ.Google Scholar
Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Process. Appl. 64, 103–125.Google Scholar
Borovkov, A. A. and Borovkov, K. A. (2008). Asymptotic Analysis of Random Walks. Cambridge University Press.Google Scholar
Cai, J. (2004). Ruin probabilities and penalty functions with stochastic rates of interest. Stoch. Process. Appl. 112, 53–78.Google Scholar
Cai, J. (2007). On the time value of absolute ruin with debit interest. Adv. Appl. Prob. 39, 343–359.Google Scholar
Cai, J. and Dickson, D. C. M. (2002). On the expected discounted penalty function at ruin of a surplus process with interest. Insurance Math. Econom. 30, 389–404.Google Scholar
Dassios, A. and Embrechts, P. (1989). Martingales and insurance risk. Stoch. Models 5, 181–217.Google Scholar
Dickson, D. C. M. and Egı´dio dos Reis, A. D. (1997). The effect of interest on negative surplus. Insurance Math. Econom. 21, 1–16.Google Scholar
Embrechts, P. and Schmidli, H. (1994). Ruin estimation for a general insurance risk model. Adv. Appl. Prob. 26, 404–422.Google Scholar
Korshunov, D. A. (2001). Large deviation probabilities for maxima of sums of independent summands with negative mean and subexponential distribution. Theory Prob. Appl. 46, 355–365.CrossRefGoogle Scholar
Sundt, B. and Teugels, J. L. (1995). Ruin estimates under interest force. Insurance Math. Econom. 16, 7–22.CrossRefGoogle Scholar