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Department of Mathematics "Giuseppe Peano"

# Laurea Magistrale (M.Sc.) in Stochastics and Data Science

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## STOCHASTIC PROCESSES

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Course ID
MAT0037
Teaching staff
Prof. Laura Sacerdote
Prof. Goran Peskir
Prof. Federico Polito
Year
1st year
Teaching period
Second semester
Type
D.M. 270 TAF B - Distinctive
Credits/Recognition
6
Course disciplinary sector (SSD)
MAT/06 - probabilita' e statistica matematica
Delivery
Class Lecture
Language
English
Attendance
Mandatory
Type of examination
Written and oral
Prerequisites
Good knowledge of Probability and Analysis
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## 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.

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## Results of learning outcomes

At the end of the course, students will know several important methods 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.

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## Course delivery

Lessons (48 hours, 6 CFU) are given in lecture rooms.

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## 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.

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.

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## Program

Brownian Motion: Markov property, existence of the Brownian motion; maximum and first passage time distribution; arcosine law; iterated logarithm law; Reflected Brownian motion; Heat equation and Brownian motion; multidimensional Brownian motion.

Stationary Processes: mean square distance; autoregressive processes; ergodic theory and stationary processes; Gaussian processes

Diffusion Processes: differential equations associated with some functionals of the process; backward and forward equations; stationary measures; boundary classification for regular diffusion processes; conditioned diffusion processes; spectral representation of  the transition density for a diffusion; diffusion processes and stochastic differential equations;  jump-diffusion processes; first passage time problems for diffusion processes

Additionally, the course includes a module of 16 hours taught by visiting professor Goran Peskir (University of Manchester, UK) on the following topics:

Scale function, speed measure, killing, creation. Infinitesimal
generator. Additive functionals and time changes. Local time. The
Ito-Tanaka formula. A change-of-variable formula with local time on
curves and surfaces. Local time-space calculus.

Boundary classification for 1D diffusion processes: Exit, entrance,
regular (reflecting, sticky, elastic), natural boundaries.

MLS functionals and PIDE problems. Mayer functional and Dirichlet
problem. Lagrange functional and Dirichlet/Poisson problem.
Supremum functional and Neumann problem. MLS functionals and Cauchy
problem. Connection with the Kolmogorov backward equation.

Applications: (i) Optimal stopping (free-boundary problems); Optimal
stochastic control (the Hamilton-Jacobi-Bellman equation).

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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.

Peskir, Shiryaev. "Optimal stopping and free-boundary problems", Lectures in Mathematics, ETH Zurich, Birkhauser (2006).

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## Class schedule

DaysTimeClassroom
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Last update: 04/05/2016 23:04
Location: https://www.master-sds.unito.it/robots.html