Infinite-time ruin probability of a multivariate renewal risk model with Brownian perturbations
Dimitrios G. Konstantinides
TL;DR
This work analyzes the infinite-time ruin probability for a $d$-dimensional renewal risk model with a common arrival process, multivariate heavy-tailed claims, and Brownian perturbations. The authors develop a framework based on multivariate tail classes ($\mathcal{S}_A$, $\mathcal{L}_A$, and $MRV$) and the integrated tail $F_I$ to derive asymptotics for $\psi_{\mathbf{b},\mathbf{L}}(x)$, showing it is dominated by the tail of the integrated distribution along a linear ruin cone $A=(\mathbf{b}-\mathbf{L})\in\mathscr{R}$ and a corrected drift $\mathbf{c}^* = \mathbf{c} + \vec{\delta} \odot \vec{\mu}$. The main result—valid under $\mathbf{E}[\mathbf{X}]<\infty$, $F_I\in \mathcal{S}_A$, and $\mathbf{c}^*>0$—gives $\psi_{\mathbf{b},\mathbf{L}}(x) \sim \int_0^{\infty} \mathbf{P}(\mathbf{W} \in xA + v\mathbf{c}^*)\,dv$, with $\mathbf{W}=\mathbf{Z}-\vec{\delta}\odot \mathbf{B}$ and $\mathbf{c}^* = \mathbf{c} + \vec{\delta} \odot \vec{\mu}$. The paper shows insensitivity of the ruin asymptotics to Brownian perturbations (extending Veraverbeke-type results to the multivariate renewal setting with $\vec{\delta}>0$) and provides a practical criterion: if $F\in(\mathcal{D}\cap\mathcal{L})_{\mathscr{R}}$ with finite mean, then $F_I\in\mathcal{S}_{\mathscr{R}}$, broadening the class of distributions for which the ruin asymptotics can be computed. These results unify and extend previous univariate and multivariate heavy-tailed ruin analyses by incorporating Brownian perturbations and renewal structure, with implications for risk management across multiple lines of business.
Abstract
We consider the multivariate risk model with common renewal process among the lines of business, and Brownian perturbations. Assuming that the integrated tail distribution of claims is multivariate subexponential, we establish an asymptotic relation for the infinite-time ruin probability. A more explicit expression is given in case of claim distribution from multivariate regular variation. The results indicate the insensitivity of the asymptotic behavior of the ruin probability with respect to Brownian perturbations. Furthermore, we show that a multivariate distribution with finite expectation, that belongs to the class of multivariate dominatedly varying and long-tailed distributions, possesses integrated tail distribution from the class of multivariate subexponential distributions, which makes easier the checking of conditions in the theorem.