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Stochastic processes


Stochastic processes


Academic year 2024/2025

Course ID
Laura Lea Sacerdote (Lecturer)
Cristina Zucca (Lecturer)
1st year
Teaching period
Second semester
D.M. 270 TAF B - Distinctive
Course disciplinary sector (SSD)
MAT/06 - probability and statistics
Class Lectures
Type of examination
Written and oral
It is very important that students possess a good knowledge of Probability Theory and Analysis (contents presented during the first semester)

Sommario del corso


Course objectives

The course is aimed at giving the students the skills to use diffusion processes to represent different realities of practical interest. The student should use the different techniques for carrying out the analysis of the models. The student will demonstrate both the ability of self-study of advanced topics, connected to the content of the course, and the ability to collaborate. 

A module of the course, included in the overall courseload, will be taught by visiting professor 


Results of learning outcomes

Knowledge and Understanding

Students will attain a basic knowledge of stochastic processes, in particular Brownian motion and diffusion processes.

Applying Knowledge and Understanding

Students will be able to study stochastic models of applied interest. They will know some of the important classes of stochastic processes and will be able to study their main functionals and features.

Making Judgements and Learning Skills

Students will be able to apply theoretical or applied techiques to solve problems connected with the modeling of stochastic processes.

Communication Skills

Students will be able to properly use English to present their knowledge both in written form in the homeworks and in oral form in the final exam.



  • Brownian Motion:
    • Historical notes-BM as a rescaling of a random walk
    • Gaussian random variables; Transformations on Gaussian r.v.s, BM as a Gaussian process; Brownian Bridge
    • Invariance properties of BM : reflection, renewal, time inversion, scaling projective reflection
    • Multidimensional Brownian motion. The three golden martingales and their use
    • BM and martingales: applications of optional stopping theorem and Wald identities
    • Markov property, strong Markov property (examples and counter-examples), First Passage Times; Maximum and first passage time distribution;
    • Brownian Motion (Einstein Wiener process and Langevin-Ornstein-Uhlenbeck process); Stationary independent increments vs Gaussian property. Lévy processes.
    • Existence/construction of BM: Kolmogorov's consistency theorem; Kolmogorov's continuity theorem. Five "golden" BMs. Law of large numbers for BM.
    • The law of iterated logarithm for BM.
    • BM is nowhere differentiable. BM is of infinite variation. Quadratic variation of BM is finite.
    • Markov processes. Hunt's lemma. Strong Markov processes. Feller processes are strong Markov. BM is a Feller process.
    • BM is a strong Feller process. Reflection principle for BM and related laws. Levy's triple law.
    • Arcsine laws for BM (the time of maximum; the last zero; the time spent above zero).
    • Brownian bridge. Boundary classification for BM (regular, entrance, exit, natural). Boundary conditions (Dirichlet, Neumann, Robin, Feller).

  • Diffusion processes
    • Continuity of sample paths and diffusion processes; infinitesimal moments; Dynkin condition;
    • alternative characterizations of diffusion processes (martingale characterization. Stroock Varadhan characterization, Stochastic Differential equations); functions of diffusion processes
    • Differential equations associated with functionals. Scale function, speed density, speed measure.
    • Kolmogorov Backward and Forward equations.
    • Theorem to transform a diffusion into a BM.
    • Stationary distribution of a diffusion process.
    • Boundary classification for regular diffusion processes;
    • First passage times for diffusion processes(Fortet equation and its solution for time homogeneous processes) Laplace transform of the first Passage Time Probability density function and its applications
    • Differential equations for the Laplace transform of the FPT probability density; further integral equations for FPT; Inverse FPT problem
    • Diffusion approximation: motion of particles in presence of a permeable membrane; a model for the spike activity of a neuron.

Course delivery

The course is composed of 48 hours of lectures. The lectures will be in presence with exceptions in accordance with university regulation  


Learning assessment methods

During the course homeworks are assigned. Solution of these exercises is part of the final exam. Teamwork is allowed for this part of the activity. Exam is oral. Students that do not make homeworks will solve exercises immediately before the oral exam.

For the exercises there is no pubblic mark, just an evaluation which can be: passed, not passed. Only students that got a passed mark in the exercises can give the oral exam. The final mark will be given by the oral exam.

The learning assessment method will consist in a written exam ( solution of exercises) followed by an oral exam. Those who solved the assigned exercises during the semester will be exempted from the written exam. The exemption holds until the September session, included.

Suggested readings and bibliography


Schilling, Partzch, "Brownian Motion", De Gruyter

Karlin, Taylor. "A first Course in Stochastic Processes", Academic Press.

Karlin, Taylor. "A second Course in Stochastic Processes", Academic Press. 

Mörters, Peres. "Brownian Motion", Cambridge University Press.

Kannan. "An introduction to stochastic processes", North Holland.

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The book by Kannan can be found in Internet Archive. On line reading is allowed for 15 days after signing  (




Class scheduleV

  • Closed
    Enrollment opening date
    01/09/2021 at 00:00
    Enrollment closing date
    30/06/2022 at 00:00
    Last update: 05/06/2024 12:58
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