# HIGHER PROBABILITY (524SM)

#### A.Y. 2020 / 2021

Professor
Period
First semester
Credits
6
Duration/Length
48
Type of Learning Activity
Programme specific subjects
Study Path
[PDS0-2018 - Ord. 2018] common
Mutuazione
Mutuato: SM34 - 433SM - TOPICS IN ADVANCED PROBABILITY
Syllabus
Teaching language

English

Learning objectives

KNOWLEDGE AND UNDERSTANDING

At the end of the course the students will know some modern techniques in probability. Moreover, they will be able to use the studied methodologies in order to solve problems coming from the applications.

MAKING JUDGMENTS

At the end of the course the students will be able to recognize the fundamental characteristics of the considered problems and will have the
ability to appropriately choose the methods to solve them.

LEARNING SKILLS
At the end of the course the students will be able to consult texts of advanced probability

Prerequisites

Basic probability and a good background in analysis

Contents

The purpose of the course is to present a comprehensive treatment of advanced probabilistic techniques for applications to stochastic modelling. In particular, Brownian motion, stochastic calculus and stochastic differential equations

Teaching format

Lectures

Extended Programme

1 Stochastic processes
Definition and general facts. Kolmogorov’s continuity theorem. Construction of stochastic processes.

2 Brownian Motion
Definition and general facts. Wiener measure. Regularity of the path.

3 Martingales
Definitions and general facts. Discrete time martingales. Doob’s inequality. Continuous time martingales.

4 Markov Processes
Definitions and general facts. Semigroups and generators.

5 The Stochastic Integral
Introduction. Elementary processes.

6 Stochastic Calculus
Ito’s formula

7 Stochastic Differential Equations
Some examples

End-of-course test

Oral exam

Other information

Any changes to the methods described here, which become necessary to
ensure the application of the safety protocols related to the COVID19
emergency, will be communicated on the websites of the course (moodle)

Texts/Books

Baldi - Stochastic Calculus
An Introduction Through Theory and Exercises. Springer, 2017

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