New Methods for the Problem of Collective Ruin

Abstract

The problem of “collective ruin” arises in a number of different situations in operations research and is particularly well suited as a model of risk business such as an insurance company. The problem of collective ruin is formulated in terms of dynamical stochastic processes for a risk reserve $Z( t )$. The reserve grows according to a deterministic process $\beta ( {Z( t )} )$, the insurance premiums, and is decremented according to a compound stochastic process, claims. The integral-differential-difference equation is derived for the probability of survival to time t and a number of different methods for the solution of the stationary version of the equation, which corresponds to probability of surviving forever, are described. In particular, asymptotic techniques are developed based on the WKB method and its extensions for the solution of a broad class of risk problems. This greatly extends the classical work of Feller, Cramer, and others who were only able to treat the case in which $\beta ( {Z( t )}...