Multivariate subexponentiality and interplay of insurance and financial risks in a renwal risk model
Dimitrios G. Konstantinides, Charalampos D. Passalidis
TL;DR
This work analyzes a multivariate renewal risk model where claim vectors are heavy-tailed and the insurer’s investment returns follow a Lévy process. It derives asymptotic entrance probabilities for discounted aggregate claims into rare sets, revealing a multivariate linear single big jump principle under two regimes: (i) weak dependence with claim distributions in a multivariate class $\mathcal{A}_A^*$ and nonnegative investments, and (ii) arbitrarily dependent insurance and financial risks with product tails in $(\mathcal{D}\cap\mathcal{A})_A$ and a negative Laplace exponent. The authors introduce the distribution class $\mathcal{A}_A^*$ (containing MRV) and develop scale-mixture and convolution techniques to establish the main tail-entrance results, then connect these to ruin probabilities over infinite horizons. The results generalize existing one-dimensional and multidimensional heavy-tailed risk models by allowing inhomogeneous tails, weak dependence, and a common Lévy driver for financial risk. Practically, the findings provide precise tail approximations for joint ruin-related events and offer tractable conditions to verify in actuarial applications.
Abstract
In this paper we consider a multivariate risk model with common renewal process, while the logarithmic returns of the insurers investment portfolio, are described by a Levy process. In the two main results are established an asymptotic expression for the entrance probability of the discounted aggregate claims in some rare sets x A. This asymptotic expression highlights the multivariate linear single big jump principle in asymptotic behavior of these probabilities. In the first result, we are restricted in the case where the insurer makes risk free investments, and hence we consider a non-negative Levy process. We assume that the claim vectors follow a distribution from a class, introduced here, and represents a negligibly smaller subclass of multivariate subexponential distributions, since the additional requirement for positive lower Karamata index, looks as a mild condition. Further, we consider that the insurance and financial risks, satisfy a weak dependence structure. In the second result, we allow arbitrarily dependence between the two risks, and we assume that the distribution of their product, at each renewal epoch, belongs to the intersection of the class of multivariate subexponential positively decreasing distributions with multivariate dominatedly varying distributions. In this theorem we also permit risky investment, putting a condition to Laplace exponent of the Levy process. We also note that even in the special one-dimensional subcase the main results are new. Furthermore, we present two examples, where we demand only conditions for the marginal distributions of both risks and their dependence structure. Both examples, under the restriction on multivariate regularly varying distributions provide more explicit and elegant relations in relation with that established in the main results.