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On the expectation of total discounted operating costs up to default and its applications

Part of: Mathematical economics

Published online by Cambridge University Press:  01 July 2016

Jun Cai ,
Runhuan Feng  and
Gordon E. Willmot
Jun Cai*
Affiliation:
University of Waterloo
Runhuan Feng*
Affiliation:
University of Wisconsin-Milwaukee
Gordon E. Willmot*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
∗∗∗ Postal address: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, PO Box 413, Milwaukee, WI 53202-0413, USA.
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
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Abstract

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In this paper we first consider the expectation of the total discounted claim costs up to the time of ruin, and then, more generally, we study the expectation of the total discounted operating costs up to the time of default, which is the first passage time of a surplus process downcrossing a given level. These two quantities include the expected discounted penalty function at ruin or the Gerber–Shiu function, the expected total discounted dividends up to ruin, and other interesting quantities as special cases among a class of risk processes. As an illustration, we consider a piecewise-deterministic compound Poisson risk model. This model recovers many risk models appearing in the literature such as the compound Poisson risk models with interest, absolute ruin, dividends, multiple thresholds, and their dual models. We derive and solve the integro-differential equation for the expected present value of the total discounted operating costs up to default. The solutions to the expected present value of the total discounted operating costs up to default can be used as a unified approach to solving many ruin-related quantities. As applications, we derive explicit solutions for the expected accumulated utility up to ruin, the absolute ruin probability with varying borrowing rates, the expected total discounted claim costs up to ruin, the Gerber–Shiu function with two-sided jumps, and the price for a perpetual American put option with two-sided jumps.

Keywords

Claim costoperating costtime of ruintime of absolute ruintime of defaultdividenddividend thresholddividend barrierutilitypiecewise-deterministic compound Poisson processdefective renewal equationcompound geometric distribution

MSC classification

Primary: 91B30: Risk theory, insurance
Secondary: 91B70: Stochastic models
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 495 - 522
Copyright
Copyright © Applied Probability Trust 2009 

References

Asmussen, S. (2000). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2). World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Avanzi, B., Gerber, H. U. and Shiu, E. S. W. (2007). Optimal dividends in the dual model. Insurance Math. Econom. 41, 111–123.CrossRefGoogle Scholar
Brémaud, P. (1981). Point Processes and Queues. Springer, New York.CrossRefGoogle Scholar
Cai, J. (2007). On the time value of absolute ruin with debit interest. Adv. Appl. Prob. 39, 343–359.CrossRefGoogle Scholar
Dassios, A. and Embrechts, P. (1989). Martingales and insurance risk. Commun. Statist. Stoch. Models 5, 181–217.CrossRefGoogle Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman and Hall, London.CrossRefGoogle Scholar
Dickson, D. C. M. and Hipp, C. (2001). On the time to ruin for Erlang(2) risk processes. Insurance Math. Econom. 29, 333–344.CrossRefGoogle Scholar
Embrechts, P. and Schmidli, H. (1994). Ruin estimation for a general insurance risk model. Adv. Appl. Prob. 26, 404–422.CrossRefGoogle Scholar
Gerber, H. U. (1979). An Introduction to Mathematical Risk Theory (S. S. Heubner Found. Monogr. Ser. 8). University of Pennsylvania, Philadelphia.Google Scholar
Gerber, H. U. and Pafumi, G. (1998). Utility functions: from risk theory to finance. N. Amer. Actuarial J. 2, 74–90.CrossRefGoogle Scholar
Gerber, H. U. and Shiu, E. S. W. (1996). Actuarial bridges to dynamics hedging and option pricing. Insurance Math. Econom. 18, 183–218.CrossRefGoogle Scholar
Gerber, H. U. and Shiu, E. S. W. (1998a). On the time value of ruin. N. Amer. Actuarial J. 2, 48–78.CrossRefGoogle Scholar
Gerber, H. U. and Shiu, E. S. W. (1998b). Pricing perpetual options for Jump processes. N. Amer. Actuarial J. 2, 101–112.CrossRefGoogle Scholar
Gerber, H. U. and Shiu, E. S. W. (2006). On optimal dividend strategies in the compound Poisson model. N. Amer. Actuarial J. 10, 76–93.CrossRefGoogle Scholar
Gerber, H. U. and Yang, H. (2007). Absolute ruin probabilities in a Jump diffusion risk model with investment. N. Amer. Actuarial J. 11, 159–169.CrossRefGoogle Scholar
Hipp, C. and Plum, M. (2003). Optimal investment for investors with state dependent income, and for insurers. Finance Stoch. 7, 299–321.CrossRefGoogle Scholar
Kou, S. G. and Wang, H. (2003). First passage times of a Jump diffusion process. Adv. Appl. Prob. 35, 504–531.CrossRefGoogle Scholar
Léveillé, G. and Garrido, J. (2001a). Moments of compound renewal sums with discounted claims. Insurance Math. Econom. 28, 217–231.CrossRefGoogle Scholar
Léveillé, G. and Garrido, J. (2001b). Recursive moments of compound renewal sums with discounted claims. Scand. Actuarial J. 2001, 98–110.CrossRefGoogle Scholar
Li, S. and Garrido, J. (2004). On a class of renewal risk models with a constant dividend barrier. Insurance Math. Econom. 35, 691–701.CrossRefGoogle Scholar
Lin, X. S. and Pavlova, K. P. (2006). The compound Poisson risk model with a threshold dividend strategy. Insurance Math. Econom. 38, 57–80.CrossRefGoogle Scholar
Lin, X. S. and Sendova, K. P. (2008). The compound Poisson model with multiple thresholds. Insurance Math. Econom. 42, 617–627.CrossRefGoogle Scholar
Lin, X. S. and Willmot, G. E. (1999). Analysis of a defective renewal equation arising in ruin theory. Insurance Math. Econom. 25, 63–84.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
Sundt, B. and Teugels, J. L. (1995). Ruin estimates under interest force. Insurance Math. Econom. 16, 7–22.CrossRefGoogle Scholar
Wang, C. and Yin, C. (2009). Dividend payments in the classical risk model under absolute ruin with debit interest. Appl. Stoch. Models Business Industry 25, 247–262.CrossRefGoogle Scholar
Wang, G.-J., Qian, S.-P and Wu, R. (2003a). Distribution of deficit at ruin for a PDMP insurance risk model. Acta Math. Appl. Sinica 19, 521–528.CrossRefGoogle Scholar
Wang, G.-J, Zhang, C.-S and Wu, R. (2003b). Ruin theory for the risk process described by PDMPs. Acta Math. Appl. Sinica 19, 59–70.CrossRefGoogle Scholar
Yang, H., Zhang, Z. and Lan, C. (2008). On the time value of absolute ruin for a multi-layer compound Poisson model under interest force. Statist. Prob. Lett. 78, 1835–1845.CrossRefGoogle Scholar
Yuan, H. and Hu, Y. (2008). Absolute ruin in the compound Poisson risk model with constant dividend barrier. Statist. Prob. Lett. 78, 2086–2094.CrossRefGoogle Scholar
Zhang, C. and Wu, R. (1999). On the distribution of the surplus of the D–E model prior to and at ruin. Insurance Math. Econom. 24, 309–321.CrossRefGoogle Scholar
Zhu, J. and Yang, H. (2008). Estimates for the absolute ruin probability in the compound Poisson risk model with credit and debit interest. J. Appl. Prob. 45, 818–830.CrossRefGoogle Scholar