Some Applications to Risk Theory

Abstract

In this chapter we consider the calculation of ruin probabilities in four different situations. First we consider a risk reserve process with nonlinear increments between claim arrivals. This model is of interest, not only because of the generality it offers for the specification of the premium function, but because it shows a general method for dealing with nonlinear processes that have phase-type (or matrix-exponential) jumps. The second theme we present is the calculation of ruin probabilities when claims are heavy-tailed. To this end, we will use the NPH class (see Definition 3.5.1, p. 181) of the infinite-dimensional phase-type distributions. The section, however, provides also a broader treatment of renewal theory with heavy-tailed interarrival times, which in particular will provide us with the solution to the ruin problem. In the third section we consider standard Cramér–Lundberg and Sparre–Andersen models, but with the linear increments between claims replaced by a Brownian motion with a corresponding drift. The last model considers the calculation of the probability of ruin in finite time. The method is algorithmic and uses a so-called Erlangization scheme.