Schema della sezione

  • Target skills and knowledge. The course aims to provide an in-depth knowledge of stochastic processes, and in particular of Markovian processes, both from a theoretical and an applicative point of view. On the one hand, some general probabilistic tools and subjects will be introduced, such as Poisson random measures,  Lèvy processes, convergence of measures. On the other hand, advanced applications related to processes on discrete and continuous spaces will be shown. This will include branching processes, genetic models (the Wright-Fisher diffusion), interacting particle systems, and some selected applications in the context of partial differential equation (the Dirichlet problem and the martingale problem).  

    Prerequisites. The course of Stochastic Analysis (I semester). In particular: basics of stochastic processes -- measurability, filtration, stopping times;  Brownian motion and its fundamental properties;  Discrete and continuous time martingales.

    Contents.

    1. Markov chains. Transition probabilities (in discrete spaces) and Markov property; Markov semigroups; invariant measure and convergence by coupling technique; transience and recurrence.
    Applications:  Random walks; Wright's evolution model;  Dirichlet problem on discrete spaces; Branching processes and survival probability; Gibbs sampler (Monte Carlo Method). 
    Main reference:

    • P. Brémaud, Probability Theory and Stochastic Processes. Springer International Publishing, 2020.

    See also:

    • T.M. Liggett, Continuous time Markov processes an introduction. Providence: American Mathematical Society, 2010.
    • J.R. Norris, Markov chains. Cambridge University Press, Cambridge,1998.
    • R. van der Hofstad, Random graphs and complex networks. Vol. 1. Cambridge University Press, Cambridge, 2017.

    2. Poisson Processes and Poisson random measure: Definition, equivalent constructions, and main properties. Inhomogeneous and compound Poisson process. Poisson random measures and characterization. 

    Main references:

    • A. Klenke, Probability Theory: a comprehensive course. Springer, 2008.  
    • E. Cinlar, Probability and stochastics. New York: Springer, 2011.
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    3. Markov chains in continuous time. Generator and semigroup; Poisson construction of the process. The trajectory space D([0,T]) and the Skorohod topology.

    Applications: Interacting particle systems; Voter model and contact process. 
    Main references:

    • P. Brémaud, Probability Theory and Stochastic Processes. Springer International Publishing, 2020.
    • T.M. Ligget: Interacting particle systems. Springer, New York, 1985.
    •  

    4. Markov/Feller processes and PDE. Transition probabilities (in 'general' metric space) and Markov property; Feller semigroup and generator and strong Markov property. 
    Applications: Solution of stochastic differential equations and relation with partial differential equations; Dirichlet problem on metric spaces.
    Main references:

    • A. Klenke, Probability Theory: a comprehensive course. Springer, 2008.  
    • D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer-Verlag, Berlin, 1999
    • P. Baldi, Stochastic calculus - An introduction through theory and exercises. Universitext. Springer, 2017


    5.     Lévy processes. Infinite divisible laws; stable laws; Levy-Khinchin formula; Lévy processes and the Lévy-Ito decomposition. Compound Poisson processes construction.
    Application: First passage times of the Brownian motion.
    Main references:

    • A. Klenke, Probability Theory: a comprehensive course. Springer, 2008.  
    • A.E. Kyprianou, Introductory lectures on fluctuations of Lévy processes with applications. Springer-Verlag, Berlin, 2006.
    • E. Cinlar, Probability and stochastics. New York: Springer, 2011.

    6. Convergence of processes. Preliminary tools (convergence of measures review); criteria for probability measures convergence in C([0,T]) and in D([0,T]). Elements of convergence of MCs to diffusion.  

    Applications: Convergence of random walks to Brownian motion (invariance principle) in C[0,T] and D[0,T]; Convergence of the Wright-Fisher model to the Wright-Fisher diffusion and of the Ehrenfest Chain to the Ornstein-Uhlenbeck process  in D[0,T].
    Main references:

    • A. Klenke, Probability Theory: a comprehensive course. Springer, 2008.
    • R. Durrett, Stochastic calculus a practical introduction. Boca Raton [etc: CRC press, 1996.
    • S.N. Ethier and T.G Kurtz, Markov processes characterization and convergence. New York, NY: John Wiley & Sons, 2009.

    Textbooks (on the whole program)

    • Brémaud, Pierre, Probability Theory and Stochastic Processes. Springer International Publishing, 2020
    • D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer-Verlag, Berlin, 1999
    • Durrett, Richard, Stochastic calculus a practical introduction. Boca Raton [etc: CRC press, 1996
    • Cinlar,Erhan, Probability and stochastics. New York: Springer, 2011
    • Klenke, Achim, Probability Theory: a comprehensive course. Springer, 2008.  

    Optional supplementary reading

    • Liggett, Thomas Milton. Continuous time Markov processes an introduction. Providence: American Mathematical Society, 2010.
    • R. van der Hofstad, Random graphs and complex networks. Vol. 1. Cambridge University Press, Cambridge, 2017.
    • P. Baldi, Stochastic calculus - An introduction through theory and exercises. Universitext. Springer, 2017
    • T.M. Ligget: Interacting particle systems. Springer, New York, 1985.
    • A.E. Kyprianou, Introductory lectures on fluctuations of Lévy processes with applications. Springer-Verlag, Berlin, 2006.
    • Ethier, Stewart N; Kurtz, Thomas G, Markov processes characterization and convergence. New York, NY: John Wiley & Sons, 2009