Merging of bivariate compound Binomial processes with shocks

TL;DR

This paper models bivariate claim processes with shocks in a discrete-time Binomial risk framework, addressing simultaneous jumps through thinning, and analyzes ruin probabilities, claim distributions, and their representations.

Contribution

It introduces a novel approach to handle parallel Binomial processes with shocks using thinning, and derives detailed properties of the claim and risk reserve processes.

Findings

Derived characteristics of total claim amount processes.

Analyzed ruin probabilities and deficit at ruin.

Showed uncountably many representations for certain distributions.

Abstract

The paper investigates a discrete time Binomial risk model with different types of polices and shock events may influence some of the claim sizes. It is shown that this model can be considered as a particular case of the classical compound Binomial model. As far as we work with parallel Binomial counting processes in infinite time, if we consider them as independent, the probability of the event they to have at least once simultaneous jumps would be equal to one. We overcome this problem by using thinning instead of convolution operation. The bivariate claim counting processes are expressed in two different ways. The characteristics of the total claim amount processes are derived. The risk reserve process and the probabilities of ruin are discussed. The deficit at ruin is thoroughly investigated when the initial capital is zero. Its mean, probability mass function and probability generating function are obtained. We show that although the probability generating function of the global maxima of the random walk is uniquely determined via its probability mass function and vice versa, any compound geometric distribution with non-negative summands has uncountably many stochastically equivalent compound geometric presentations. The probability to survive in much more general settings, than those, discussed here, for example in the Anderson risk model, has uncountably many Beekman's convolution series presentations.