Processus de Poisson et méthodes actuarielles
Lecture notes and exercises:
- Lectures notes in English from Nina Gantert (Link)
- Lectures notes in French from Massimiliano Gubinelli
(Link Part 1 ,
- Exercises (Link)
- Annales 2015-2016 (Link)
- Element of correction of TD4 (Link)
- January 19: Motivations; Counting processes; Characterization of standard counting processes; Standard counting processes defined by a renewal process.
- January 24: Poisson, Exponential (characterization by the lack of memory) , Gamma Laws. Definition of a Poisson process. Characterization by the law of the inter-arrival times (first part of the proof).
- January 31: Characterization by the law of the inter-arrival times (end of the proof). Law of large numbers for Poisson processes.
- February 9: Natural filtration. Markov property. Joint law of the order statistics of iid random variables. Conditional joint law of the n first jump times given that the Point process is equal to n at t. Example of application.
- February 21: Mixed Poisson process. Properties of the increments. Conditional joint law of the n first jump times given that the Point process is equal to n at t.
- February 28: Definition of the total of claim process. Moments and moments generating function for t fixed. Cumulative distribution of the total of claims, example of explicit computations. Definition of a compound Poisson process, path properties.
- March 2: Laplace transform and law of large numbers for a compound Poisson process. Renewal function and renewal measure. SLLN for a renewal process. Elementary renewal theorem.
- March 28: Blackwell's and Key Renewal theorems.
- April 18: Renewal equation for the renewal function. General renewal equation and resolution.
- April 25: Example of application (law of the next time jump after t, for a renewal process). Risk process and ruin probability (definitions). Ruin probability when the net profit condition is not satisfied. Thin and heavy tailed distributions. Characterization with the generating function.
- April 27: Lundberg (adjustment) coefficient. Example of existence. Renewal equation for the survival probability (part 1 of the proof).
- May 2: Renewal equation for the survival probability (part 2 of the proof). Asymptotics in the case of small risks.
- May 9: Large risks and asymptotic (sub-exponential case)