Insurance: Mathematics and Economics 27 2000 123–136
The multivariate De Pril transform
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Bjørn Sundt
a,b,∗
a
University of Bergen, Bergen, Norway
b
University of Melbourne, Melbourne, Australia Received 17 September 1999; received in revised form 5 January 2000; accepted 20 January 2000
Abstract
In the present paper we extend the definition of the De Pril transform to a class of multivariate functions and discuss various properties of this multivariate De Pril transform. In particular, we show that, like in the univariate case, it is additive
for convolutions, and discuss De Pril transforms of compound functions and higher order functions. Finally, we introduce a multivariate Dhaene–De Pril transform, which we compare with the De Pril transform. © 2000 Elsevier Science B.V. All
rights reserved.
MSC: M10; M11 Keywords: Multivariate functions; Recursions; De Pril transforms; Convolutions; Compound distributions
1. Introduction
1A. During the last two decades an extensive literature on recursive methods for exact and approximate evaluation of aggregate claims distributions has grown up, starting with Panjer 1980, 1981. Most of it has been confined
to univariate distributions. However, multivariate aggregate claims distributions are also of interest in insurance, e.g. when considering the joint distribution of aggregate claims of bodily injury and material damage in motor
insurance. Further applications are discussed in Sundt 1999a. Multivariate extensions of Panjer’s 1981 recursion for compound distributions have been presented in two directions, for references and discussion, see Sundt 2000a.
1B. Inspired by in particular De Pril 1989 and Dhaene and De Pril 1994, Sundt 1995 defined and discussed the De Pril transform of a univariate distribution as a tool for recursive evaluation of aggregate claims distributions.
In Dhaene and Sundt 1998 the definition was extended to more general univariate functions in connection with approximate evaluation of distributions. Properties of the De Pril transform are also discussed in Sundt et al. 1998,
Sundt 1998 and Sundt and Ekuma 1999.
1C. In the present paper we extend the definition of the De Pril transform to multivariate functions. In Section 2 we recapitulate some definitions and results from the univariate case in a form that will simplify the multivariate
extension performed in Section 3. As a starting point for the multivariate extension we apply the multivariate
q
The present paper is dedicated to the memory of Nelson De Pril on whose research the paper heavily leans.
∗
Present address: Vital Forsikring ASA, PO Box 250, N-1326 Lysaker, Norway. Tel.: +47-67-83-44-71; fax: +47-67-83-45-01. E-mail address: bjoern.sundtvital.no B. Sundt.
0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 0 0 0 0 0 4 1 - X
124 B. Sundt Insurance: Mathematics and Economics 27 2000 123–136
extension of Sundt 1999a to the recursion of Sundt 1992. We show that, like in the univariate case, the multivariate De Pril transform is additive for convolutions, and discuss De Pril transforms of compound functions, higher order
functions, and infinitely divisible distributions. We also express the De Pril transform of the distribution of a vector of independent sub-vectors of random variables by the De Pril transforms of the marginal distributions of these
sub-vectors. Finally, in Section 4 we introduce the Dhaene–De Pril transform as a multivariate version of a function discussed by Dhaene and De Pril 1994 in the univariate case, and discuss its advantages and disadvantages
compared to the De Pril transform.
Sundt 1999b discusses some results related to the present paper. 1D. In the present paper we shall work on distributions in the representation of their probability functions.
Therefore, we shall for simplicity refer to probability functions as distributions.
2. The univariate De Pril transform
