Ruin probabilities as functions of the roots of a polynomial

Santana, David J.; Rincón, Luis

doi:10.15559/23-VMSTA226

Modern Stochastics: Theory and Applications

Ruin probabilities as functions of the roots of a polynomial

Volume 10, Issue 3 (2023), pp. 247–266

David J. Santana

Luis Rincón

https://doi.org/10.15559/23-VMSTA226

Pub. online: 15 March 2023 Type: Research Article

Open Access

Received
18 December 2022

Revised
8 March 2023

Accepted
9 March 2023

Published
15 March 2023

Abstract

A new formula for the ultimate ruin probability in the Cramér–Lundberg risk process is provided when the claims are assumed to follow a finite mixture of m Erlang distributions. Using the theory of recurrence sequences, the method proposed here shifts the problem of finding the ruin probability to the study of an associated characteristic polynomial and its roots. The found formula is given by a finite sum of terms, one for each root of the polynomial, and allows for yet another approximation of the ruin probability. No constraints are assumed on the multiplicity of the roots and that is illustrated via a couple of numerical examples.

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Keywords

Ruin probability Cramér–Lundberg risk model Erlang mixture distribution recurrence sequences 91B05 91G05 11B37

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