Bruno Bouchard Professor of Mathematics CEREMADE (UMR CNRS 7534), Université Paris-Dauphine, PSL Research University Contact mylastname@ceremade.dauphine.fr Université Paris-Dauphine, Ceremade, bureau C618bis, place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

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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 , Part 2)
• Element of correction of TD4 (Link)
Lecture progression:
• 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)