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Exact Asymptotics of the Ruin Probability in the Sparre Andersen Model for Non-Life Insurance with Investments in a Lévy Process
We establish the exact power-law asymptotics of the ruin probability, as a function of the initial capital, in the Sparre Andersen model for non-life insurance with investments in an arbitrary Lévy process. The main advance over previous work, which only established two-sided estimates, is a proof of the existence of an exact limiting equality with a positive finite constant. The method combines a reduction to discrete time, the one-dimensional Kesten–Goldie theorem for the stationary measure of the associated affine stochastic recursion, and Goldie’s result on the asymptotics of the supremum of a perpetuity. The limiting constant is bounded below by the integral Goldie constant of the stationary measure. This bound need not be tight, and an explicit expression for the limiting constant is currently available only in the Cramér–Lundberg submodel. The exponent coincides with the positive Cramér root of the Laplace exponent of the Lévy process given by the logarithm of the risky-asset price; the distribution of the inter-jump times of the business process affects only the constant, not the exponent. In the final section, the constant is compared with an explicit formula in terms of double confluent Heun functions, obtained for the Cramér–Lundberg submodel with proportional investment in a geometric Brownian motion; the agreement of the two approaches is illustrated numerically.