Some stochastic orders and reliability properties for compound geometric distributions, their convolutions, and other ruin-related quantities

For the Sparre Andersen model of risk theory, we obtain some stochastic order results, in terms of the Laplace transform order and the moment generating function order, for random variables associated with the event of ruin. These include an extension of the deficit at ruin. We also derive some reliability properties for zero-modified compound geometric distributions. Finally, we give a general result comparing the maximal aggregate losses of two (classical) risk processes perturbed by diffusion, which is shown to be valid for any order satisfying the convolution closure property.