On Esscher transforms of distribution functions defining a compound poisson process for large values of the parameter

Abstract

Abstract F. Esscher has recently generalized the transformation of the distribution functions defining a Poisson process, which he introduced in 1932 and which Ammeter in 1948 applied to a Polya process. This generalization was first achieved for a positive compound Poisson process with an unconditioned distribution function of the risk (in the sequel called the risk distribution) and with a conditioned distribution relative to the hypothesis that a change has occurred in the random function attached to the process, of the size of such a change (in the sequel called claim distribution), which both are independent of the parameter t. In the following section 2 and throughout this paper the investigation will be restricted in the same way. An extension to cases where the risk distribution and/or the claim distribution depend on t can easily be derived by using the results given by the present author in a report to the congress in London (1964). Also the theory may easily be extended to cases where the changes in the random function attached to the process may be positive or negative. Later Esscher extended the transformation to concern a general distribution function for which the corresponding characteristic function is known (1963). This general case will be considered in section 1 here below.