On The Randomized Schmitter Problem

Hansjörg Albrecher; José Carlos Araujo-Acuna

doi:10.1007/s11009-021-09910-5

On The Randomized Schmitter Problem

Methodol Comput Appl Probab. 2022;24(2):515-535. doi: 10.1007/s11009-021-09910-5. Epub 2021 Nov 8.

Authors

Hansjörg Albrecher¹, José Carlos Araujo-Acuna²

Affiliations

¹ Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, Switzerland and Swiss Finance Institute, Lausanne, Switzerland.
² Department of Mathematics and Statistics, Institute of Mathematical Statistics and Actuarial Science, University of Bern, Bern, Switzerland.

Abstract

We revisit the classical Schmitter problem in ruin theory and consider it for randomly chosen initial surplus level U. We show that the computational simplification that is obtained for exponentially distributed U allows to connect the problem to m-convex ordering, from which simple and sharp analytical bounds for the ruin probability are obtained, both for the original (but randomized) problem and for extensions involving higher moments. In addition, we show that the solution to the classical problem with deterministic initial surplus level can conveniently be approximated via Erlang(k)-distributed U for sufficiently large k, utilizing the computational advantages of the advocated randomization approach.

Keywords: Erlangization; Laplace transform; Ruin probability; Ruin theory; Schmitter problem; Stochastic ordering.