Joint Tail Probability of Renewal Models of Dependent Heavy-Tailed Random Variables With Applications to Systemic Risk Measures

Abstract

Consider a non-standard renewal risk model in which claims arrive in pairs $\left\{\right(X_{1i},X_{2i});i\in \mathbb{N}\}$ and the stochastic discounting process is given by $\{e^{-\xi \left(t\right)};t\ge 0\}$, where $\xi (·)$ is a Lévy process. We are interested in the joint tail probability of $L_1\left(t\right)$ and $L_2\left(t\right)$, the aggregate discounted claims along the respective lines $k=1,2$. Assume that $\left\{\right(X_{1i},X_{2i});i\in \mathbb{N}\}$ is a sequence of independent and identically distributed random pairs with generic pair $(X_1,X_2)$. Further assume that $(X_1,X_2)$ follows a dependence structure encompassing both tail dependence and tail independence, and has regularly varying marginal tails. We derive asymptotic formulas for the joint tail probability of $L_1\left(t\right)$ and $L_2\left(t\right)$. These results are then applied to evaluate two systemic risk measures. Finally, we conduct numerical studies to illustrate the theoretical findings.