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A unifying approach to the analysis of business with random gains. (English) Zbl 1277.60148
Summary: In this paper, we consider a stochastic model in which a business enterprise is subject to constant rate of expenses over time and gains which are random in both time and amount. Inspired by H. Albrecher and O. J. Boxma [Insur. Math. Econ. 35, No. 2, 245–254 (2004; Zbl 1079.91048)], it is assumed in general that the size of a given gain has an impact on the time until the next gain. Under such a model, we are interested in various quantities related to the survival of the business after default, which include: (i) the fair price of a perpetual insurance which pays the expenses whenever the available capital reaches zero; (ii) the probability of recovery by the first gain after default if money is borrowed at the time of default; and (iii) the Laplace transforms of the time of recovery and the first duration of negative capital. To this end, a function resembling the so-called Gerber-Shiu function [H. U. Gerber and E. S. W. Shiu, N. Am. Actuar. J. 2, No. 1, 48–78 (1998; Zbl 1081.60550)] commonly used in insurance analysis is proposed. The function’s general structure is studied via the use of defective renewal equations, and its applications to the evaluation of the above-mentioned quantities are illustrated. Exact solutions are derived in the independent case by assuming that either the inter-arrival times or the gains have an arbitrary distribution. A dependent example is also considered and numerical illustrations follow.

MSC:
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
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