Application of Marked Poisson Processes in Insurance Mathematics

Specialeforsvar ved Debbie Kusch Falden

Titel: Application of Marked Poisson Processes in Insurance Mathematics

Abstract: The aim of this thesis is to extend the standard Cramér-Lundberg model, known from non-life insurance mathematics, to take the waiting time from occurrence to notification, and notification to settlement into account. Using this extended model and fixing a time of consideration, a prediction of the total outstanding liability of insurance claims that have occurred, is desired.
In order to describe and analyze the model, a theoretical framework is firstly build, where Poisson processes and Poisson integrals are the fundamental theory. Specifically useful in this thesis, is the property of independence, when considering a Poisson process or a Poisson integral of disjoint sets, the ability of marking a Poisson process and representing the Cramér-Lundberg model as a Poisson integral.
The claims are assumed to be generated by a marked Poisson process, where the marks represent the developments of the claims, and the total claim amount is modelled by a Poisson integral. Furthermore, by the time of consideration the claims are decomposed into the categories not incurred, incurred but not reported, reported but not settled and settled, with independence between the categories. The independence implies that the insurance companies are able to characterize the different types of claims, consider distributional properties and make predictions within each category.
Two different models of payment processes are considered in order to calculate the size of a reserve that covers the outstanding liabilities. One is clusters following the arrival times, and the other is Markov chains following the arrival times. The Markov chain model is inspired by the payment processes used for reserves in life insurance mathematics. Within both models, a prediction of the outstanding liabilities is made as the expectation of future discounted payments of claims that have occurred.

Vejleder: Thomas V Mikosch
Censor:    Yuri Goegebeur, SDU