an:06238151
Zbl 1277.60148
Cheung, Eric C. K.
A unifying approach to the analysis of business with random gains
EN
Scand. Actuar. J. 2012, No. 3, 153-182 (2012).
00319440
2012
j
60K20 62P05 91B30
random gains; dual risk model; delayed renewal process; defective renewal equation; perpetual insurance; time of recovery
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 \textit{H. Albrecher} and \textit{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 [\textit{H. U. Gerber} and \textit{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.
Zbl 1079.91048; Zbl 1081.60550