Section outline

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    SCQ2101559 - ADVANCED STOCHASTIC PROCESSES 2025-2026 - PROF. ALESSANDRA BIANCHI

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    LECTURER
     

    Alessandra Bianchi 

    WEEKLY TIMETABLE AND LECTURES

    Wed 10:30-12:30, Room 2BC60
    Thu 14:30-17:30, Room 2BC60

    The course takes place over 12 weeks with 5 hours each week - excluding holidays -  with roughly 2 hours per week devoted to discuss models or applications.
    The lessons will be held in presence but the notes of each lesson will be available on this page.
    At the end of the course, a final exercise sheet will be given to students. The resolution of an exercise chosen from this sheet may be requested during the oral exam (see Examination Methods below).

    EXAMINATION METHODS

    The exam consists of an oral test in which the candidate is asked to discuss some theoretical topics, and to carry out an exercise chosen from the following final exercise sheet.

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    Course presentation

    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

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    Week 1 (February 25-26, 2026)

    Lecture 1 (25.02.26) - Introduction: notation, terminology, objectives and organization of the course. Markov Chains: transition matrix and transition probabilities (in one or n steps); law of the process. 

    Lecture 2 (26.02.26) - Markovian semigroup and its action on real functions and on distributions. Equivalent formulations of Markov property and strong Markov property. Construction of Markov Chains. Examples: genetic models  (Wright's model); interacting particle systems (voter model).

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    Week 2 (March 04-05, 2026)

    Lecture 3 (04.03.26) - Genetic models (Wright's model): computation of the fixation probability (martingale methods). Interacting particle systems (Voter model on the torus): construction of the model; Markovian properties; computation of the total agreement probability (martingale methods).

    Lecture 4 (05.03.26) -  Invariant distributions: definition and comments on its existence and uniqueness; proof of existence in finite spaces (and counterexample in infinite space). Convergence to stationarity tools: irreducibility, aperiodicity and strong irreducibility; distance in total variation; coupling among distributions. Theorem of convergence to stationarity in finite spaces (statement).

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    Week 3 (March 11-12, 2026)

    Lecture 5 (11.03.26) -  Theorem of convergence to stationarity in finite spaces (proof).  
    Markov chain in general discrete space: first passage time, transience and recurrence (definition). 

    Lecture 6 (12.03.26) -  Local times (or number of visits), Green function and criterium for recurrence (theorem). Recurrence/transience and existence of invariant measures/distributions: main results and comments.
    Characterization of the invariant measures for irreducible recurrent MC (theorem).

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    Week 4 (March 18-19, 2026)

    Lecture 7 (18.03.26) -  Proof of theorem [charcaterization of invariant measures for irreducible recurrent MC]. Convergence to stationarity: main result (general space) and proof. 

    Lecture 8 (19.03.26) - Example: simple symmetric RWs on Z are null recurrent.
    Ergodic Theorem: statement, comments and proof.      Application of the Ergodic Theorem to Monte Carlo Markov Chains: reversible Markov chains; construction; examples (hard-core model; Metropolis dynamics for Ising model). 

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    Week 5 (March 25-26, 2026)

    Lecture 9 (25.03.26) - Example of MCMC: hard-core model; Metropolis dynamics for Ising model. 
    Lecture 10 (26.03.26) - Harmonic functions w.r.t. the transition matrix P. The Dirichlet problem related to P: existence of bounded solutions; uniqueness of bounded solutions (tool: the Lévy's martingale). Comment on the Poisson equation.

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    Week 6 (April 01-02, 2026)

    Lecture 11 (31.03.26) - Branching processes: definition; extinction probability (theorem with proof). Size of the population(1): exploration process (bijection with a random walk),
    Lecture 12 (01.04.26) - Size of the population(2):  Chernoff bounds. Poisson proceeses: definition and first properties.

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    Week 7 (April 08-09, 2026)

    Lecture 13 (08.04.26) -  Homogeneous Poisson process: equivalent definitions, properties, arrival and inter-arrival times. Superposition and decomposition  of Poisson processes. Inhomogeneous Poisson process. Compound  Poisson process: definition and main statistics (average, variance, Laplace trasform).
    Lecture 14 (09.04.26) - Poisson Point processes and (counting) random measures: definition, intensity measure, Laplace functional. Random measure and Laplace functionals. Characterization of Poisson Point processes by Laplace functionals: properties and examples.