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 L\'evy process, we establish the exact power-law asymptotics of the ruin probability $\Psi(u)\sim C^* u^{-\beta}$ as $u\to\infty$ with a positive finite constant $C^*$; the exponent $\beta$ is the Cram\'er root of the Laplace exponent of the L\'evy 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.