Polynomials, random walks and risk processes: a multivariate framework

Abstract

This paper is concerned with two families of multivariate polynomials: the Appell polynomials and the Abel-Gontcharoff polynomials. Both families are well-known in the univariate case, but their multivariate version is much less standard. We first provide a simple interpretation of these polynomials through particular constrained random walks on a lattice. We then derive nice analytical results for two special cases where the parameters of the polynomials are randomized. Thanks to the interpretation and randomization of the polynomials, we can derive new results and give other insights for the study of two different risk problems: the ruin probability in a multiline insurance model and the size distribution in a multigroup epidemic.

Keywords:

• Multivariate Appell polynomials
• Abel-Gontcharoff polynomials
• constrained random walk
• multiline insurance model
• multigroup epidemic model

AMS Subject Classifications:

• 91B30
• 92D30
• 13M10

Acknowledgements

We are grateful to the referees and editors who provided insightful and constructive comments during the review process. C. L. received support from the Chair Generali Actuariat Responsable sponsored by the French Fondation du Risque, and the ARC Research project IAPAS of the Fédération Wallonie-Bruxelles.

Notes

No potential conflict of interest was reported by the authors.