# Discrete Compound Poisson Process with Curved Boundaries: Polynomial Structures and Recursions

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## Abstract

This paper provides a review of recent results, most of them published jointly with Ph. Picard, on the exact distribution of the first crossing of a Poisson or discrete compound Poisson process through a given nondecreasing boundary, of curved or linear shape. The key point consists in using an underlying polynomial structure to describe the distribution, the polynomials being of generalized Appell type for an upper boundary and of generalized Abel–Gontcharoff type for a lower boundary. That property allows us to obtain simple and efficient recursions for the numerical determination of the distribution.

## Keywords

first crossing time compound Poisson process order statistics generalized Appell polynomials generalized Abel–Gontcharoff polynomials recursive methods dam modelling risk theory## AMS 2000 Subject Classification

60G40 12E10 62P05## Preview

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