A risk model driven by Lévy processes

Abstract

SUMMARYWe present a general risk model where the aggregate claims, as well as the premium function, evolve by jumps. This is achieved by incorporating a L! e evy process into the model. This seeks to account for the discrete nature of claims and asset prices. We give several explicit examples of L! e evy processes that can be used to drive a risk model. This allows us to incorporate aggregate claims and premium fluctuations in the same process. We discuss important features of such processes and their relevance to risk modeling. We also extend classical results on ruin probabilities to this model.

Risk Theory with the Generalized Inverse Gaussian Lévy Process

“…However this integral might be computed numerically or as Dufresne, Gerber and Shiu (1991) had already pointed out for the gamma process, this property leads to upper and lower bounds for the ruin probability. We can also find in Morales and Schoutens (2003) extensions of the notion of Lundberg bounds to a more general risk model based on Lévy processes. These two approaches are further discussed for a GIG risk model in the following sections.…”

“…A standard reference on GIG distribution is Jørgensen (1982). The inverse Gaussian process and the gamma process of Dufresne, Gerber and Shiu (1991) are particular or limiting cases of the GIG Lévy process, which in turn is another example of the spectrally negative Lévy processes discussed in Yang and Zhang (2001) and of the general risk models discussed in Morales and Schoutens (2003) and Morales (2003).…”

“…The risk process defined in (13) belongs to the family of processes discussed in Morales and Schoutens (2003). For an account on the classical risk model we refer to Grandell (1991).…”

Multifractality of products of geometric Ornstein-Uhlenbeck-type processes

“…In particular, the generalised z-distribution Z(α, (α, β, δ, µ) is known as the Meixner distribution (Schoutens and Teugels (1998), Grigelionis (1999), Morales and Schoutens (2003)). The density function of a Meixner distribution is given by…”

“…Note that the generalised z-distributions and generalised hyperbolic distributions form nonintersecting sets. However, we can show that some Meixner distributions and corresponding Lévy processes can be obtained by subordination, that is, by random time change in the Brownian motion (see, for instance, Morales and Schoutens (2003)). (C8) Consider a mother process of the form…”