Ordering ruin probabilities for dependent claim streams.
An insurer has \(s\) classes of customers/business in a set \(E=\{1,2,\dots,s\}\); when a shock of type \(K \subseteq E\) (following a Poisson process) occurs, claims in \(K\) arrive; ruin occurs if at some point of time a total claims net of premia exceed the insurer's reserves. The question addressed by the paper is whether ruin probability decreases if the \(s\) become less positively correlated. The complication is that since, any claim \(j \in E\) arrives whenever any shock \(K\) containing it occurs, to evaluate the degree of dependence of a vector of \(s\) individual claim processes, one has to consider the collection of the \(2^s\) types of shocks indexed by \(K\). It turns out that the appropriate dependence ordering is one that roughly says that a system of \(2^s\;K\)-type shocks is more independent than another if for any \(K\) and any partition of it the arrival rates of the elements of the partition are higher while that of \(K\) is lower. The corresponding vector of \(s\) classes is then less positively correlated, and the corresponding total-claim random variable is smaller in the stochastic convex order (roughly same mean smaller variance). This, by an existing result, implies that the ruin probability is lower for any level of initial capital.
scientific article; zbMATH DE number 2004072