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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 

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