Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation
In a renewal risk model or a Markov-modulated risk model \(\{X_t\}\), the time spent in the negative \(\tau(u,T) = \int_0^T \text{1}_{\{u + X_t < 0\}}\; d t\) and the integrated negative part \(I_T(u) = \int_0^T (-u-X_t)^+ \;d t\) are studied. For a claim size distribution with a regularly varying tail, the asymptotic behavior of \(E[\tau(u,\infty)]\), \(E[I_\infty(u)]\) and \(E[I_T(u)]\) is calculated. For \(E[I_T(u)]\), the result is obtained for subexponential claim size distributions. Then the light-tail case is considered and the asymptotics of \(E[\tau(u,\infty)]\) and \(E[I_\infty(u)]\) are obtained in the case where the Lundberg exponent exists. The main part of the article deals with two independent Markov modulated models. The question is to allocate the initial reserve \(u = u_1 + u_2\), such that \(I_T(u_1,u_2) = E[I_T^1(u_1)] + E[I_T^2(u_2)]\) becomes minimal, where \(I_T^j(u) = \int_0^T (-u-X^j_t)^+ \;d t\) is the time spent in the negative of the \(j\)-th process. The optimal allocation has to fulfill \[ \frac{\partial E[I_T^1(u_1)]}{\partial u_1} = \frac{\partial E[I_T^2(u_2)]}{\partial u_2} \;. \] Writing \(u_2 = \beta(u)\), the asymptotic behavior of \(\beta(u)\) is given in the regular varying case, if the distribution tail of the claim sizes of the first process is heavier. Here, \(T\) can be both finite or infinite. The disadvantage in the infinite horizon case is, that the coefficient of regular variation has to be larger than three. Since in practical situations, the coefficient often lies between one and two, the most interesting case is not covered. For the light-tailed case with \(T = \infty\), it is also shown that \(u_1/u_2\) tends to a constant. The proofs rely on the fact, that in the asymptotic expression for \(E[I_T^j(u_j)]\), limit and derivative can be interchanged.
heavy-tailed and light-tailed claim size distribution
scientific article; zbMATH DE number 5706224