A unifying approach to the analysis of business with random gains

Abstract

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 Albrecher & Boxma (2004), 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 (Gerber & Shiu (1998)) 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.