Uniform estimates for ruin probabilities in the renewal risk model with upper-tail independent claims and premiums
The paper focuses on renewal risk models, overcoming the restrictive hypothesis about the mutual independence of the claim sizes, on the basis of specific references to some applicative cases. The starting point consists of the well-known risk-model involving the surplus U: \[ U(t)=e^{R_t}\left (x+c\int_0^t e^{R_u}du-\sum_{k=1}^{N(t)}X_ke^{R_{\tau_k}}\right ),\qquad t\geq 0 \] where \(U(0)\) represents the initial capital, \(c\) the premium rate, \(N(t)\) the claim number, \(X_k\) the \(k\)-th claim size, the \(X_k\) variables being nonnegative, identically distributed but not necessarily independent; moreover the price process is depicted by a geometric Lévy process \(\{e^{R_t}, t\geq 0\}\), the \(X_k\), \(N(t)\) and \(R_t\) variables are mutually independent. The inter-claim times are nonnegative i.i.d. random variables. Within the above nonstandard model, where the claim sizes and premium sizes are identically distributed and upper-tail independent random variables, respectively, the dependence structures involving claim sizes and inter-claim times are studied; in the case of claim size distribution belonging to the extended-regular-varying class, asymptotic results for stochastically discounted aggregate claims are provided. Uniform estimates for the finite and infinite-time ruin probability are proved under specific assumptions about the the tail of the premium-size distribution.
scientific article; zbMATH DE number 5998541