On Multivariate Panjer Recursions

Abstract

In the present paper we generalise Panjer's (1981) recursion for compound distributions to a multivariate situation where each claim event generates a random vector. We discuss situations within insurance where such models could be applicable, and consider some special cases of the general algorithm. Finally we deduce from the algorithm a multivariate extension of De Pril's (1985) recursion for convolutions.

Panjer recursion versus FFT for compound distributions

“…Panjer and Wang (1993) examined the stability of the Panjer recursion; Willmot (1988), Sundt (1999) and Hess et al (2002) have given generalizations of the Panjer recursion; Hermesmeier (1999, 2000) have investigated the propagation of discretization errors through compounding and established an improved FFT based procedure using an exponential change of measure. The latter contribution is quite substantial since it essentially eliminates the so called aliasing error, which is the fundamental deficit that arises through the use of the discrete Fourier transform.…”

Multivariate Compound Poisson Distributions and Infinite Divisibility

“…e.g. subsection 2C in Sundt (1999)) that the elements of a random vector with compound Poisson distribution with severity distribution on N m+ are non-negatively correlated, and then the corollary follows from Theorem 1.…”

“…Ospina & Gerber (1987) gave a new proof based on the recursions of Panjer (1980) for compound Poisson distributions and De Pril (1985) for w-fold convolutions. In the present paper we shall extend the proof of Ospina & Gerber (1987) to multivariate distributions, using Sundt's (1999) multivariate extension of the recursions of Panjer and De Pril.…”

“…Before turning to infinite divisibility in Section 3 we present the recursions of Sundt (1999) in Section 2.…”

Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions

“…The distribution of the vector X = (X 1 , X 2 ) is recovered through a recursion procedure in [4,24], whereas the distribution of the vector Y = (Y 1 , Y 2 ) is studied in [12,24,26]. Recursion based techniques rely on the existence of recurrence relationships between the probabilities of the claim frequencies which limits their use to certain kind of distributions.…”