Exact asymptotics of the ruin probability in the Sparre Andersen model

Abstract

For the Sparre Andersen non-life insurance model with investments in an arbitrary Lévy process, we establish the exact power-law asymptotics of the ruin probability $\Psi (u)\sim C^{\ast }{u}^{-\beta }$ as $u\to \infty$ with a positive finite constant $C^{\ast }$; the exponent $\beta$ is the Cramér root of the Laplace exponent of the Lévy process describing the logarithm of the risky asset price. This strengthens previously known two-sided estimates of the order -- the existence of an exact limit had remained an open question. The proof combines a reduction to discrete time, the one-dimensional Kesten-Goldie theorem for the stationary measure of the associated affine recursion, and Goldie's result on the asymptotics of the supremum of a perpetuity.

14 pages, 1 figures

References in corpus (1)

• Distributional equations and the ruin problem for the Sparre Andersen model with investments