Ordering Ruin Probabilities Resulting from Layer-Based Claim Amounts for Surplus Process Perturbed by Diffusion

In this paper we study orders of pairs of ruin probabilities resulting from two claim severity random variables X and Y for a continuous time surplus process perturbed by diffusion, each of which is the underlying risk Z with or without a deductible and/or a policy limit imposed, called a layer of Z. The deductibles and policy limits for X and Y could be the same or different. Under some condition regarding the relative security loadings, we find that the layer with a policy limit and the layer with a deductible yield the lowest and highest ruin probabilities, respectively, provided that Z has a decreasing failure rate, and the layer with and the layer without both a deductible and a policy limit produce the smallest and largest ruin probabilities, respectively, provided that Z has an increasing failure rate. Numerical examples are also given to illustrate the results of the proposed theorems for ordering ruin probabilities resulting from layers of two random variables distributed as a single exponential and a mixture of two exponentials.