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Stochastic processes


Stochastic processes


Academic year 2020/2021

Course ID
Teaching staff
Prof. Laura Sacerdote (Lecturer)
Prof. Cristina Zucca (Lecturer)
Goran Peskir (Lecturer)
1st year
Teaching period
Second semester
D.M. 270 TAF B - Distinctive
Course disciplinary sector (SSD)
MAT/06 - probabilita' e statistica matematica
Formal authority
Type of examination
Written and oral
Good knowledge of Probability and Analysis

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. Students should also use the software Mathematica to perform some assigned simulations.

A module of the course, included in the overall courseload, will be taught by visiting professor Goran Peskir (University of Manchester, UK) on Diffusion processes and boundary classification (cf. International visiting professorsopen_in_new).


Results of learning outcomes

Knowledge and Understanding

Students will attain a 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 functional 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 and to present their knowledge both in written form in the homeworks and in oral form in the final exam.


Course delivery

The course is composed of 48 hours of lectures which, for the AY 2020/2021, will be held remotely,  as live streaming. All lectures will be recorded and made available on Moodle, together with the related slides or notes and other course material.

Some additional activities to favour direct interaction between professors and students may be organised as online meetings 


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 work. 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 evaluation of homeworks is valid only for the Summer exam session. From September session students are required to solve exercises immediately before the oral exam.

During the Covid-19 emergency the learning assessment method will consist in a written exam ( solution of exercises) followed by an oral exam via Webex video conference system. Those who solved the assigned exercises during the semester will be exempted from the written exam. The exemption holds until the September session, included



  • 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 Langeven-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.
    • Historical remarks on Pacioli and Cardano. 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.

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.


Class schedule



ATTENTION: arrangement for the lectures in March:

During the first two weeks the lectures there will be 6 hours per week (Tuesday, Thursday and Friday at 9:15). 

Then there will be 16 hours given by Prof. Peskir (Visiting Professor). Details on the schedule will be given soon but the schedule should be Tuesday and Thursday at 9:15 weekly, plus one Friday.


The link is published in the page of the course on Moodle



After Easter holidays the lectures will follow the regular schedule (Tuesday and Thursday at 9:


Last update: 21/05/2021 18:01
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