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Poisson processes and actuarial sciences


The aim of these Lectures is to present the main probabilistic models at the basis of Risk analysis in several fields of application such as insurance and reliability theory.

The Poisson process is widely used to model random "points" in time and space (e.g. claim arrivals in insurance, or failure times in reliability theory).

Furthermore, this model is widely used to construct number of other, more complex, random processes (e.g. compound processes to model the risk process in actuarial science).

The aim of this lecture is to introduce such models and study their fundamental properties from both probabilistic and statistical aspects, with application to reliability theory and actuarial science.


This lecture is divided into two parts:


Part I: Big picture


Introduction of the theoretical tools for:

  • Homogeneous Poisson processes with application to reliability theory
  • Inhomogeneous Poisson processes
  • Compound Poisson processes with application to actuarial science and non-life insurance (Cramér-Lundberg model)
  • Statistics and simulation methods for Poisson processes
  • Study of the Cramér-Lundberg model in insurance
  • Introduction to risk measures


Part II: Mini-Project


Application and illustration on simulated data of different aspects of Poisson processes.


At the end of this module, the student should be able to:

  • Know and understand the Poisson process theory fundamentals.
  • Estimate the rate of a homogeneous Poisson process and construct confidence intervals and statistical tests for such rate (theoretically and in practice with the R Statistical Software).
  • State the main outlines of the compound Poisson process and Ruin theory.
  • Apply and analyze the previous probabilistic theories to reliability theory and actuarial science.

Needed prerequisite

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...