# 90480 - Stochastic Processes

• Docente: Pietro Rigo
• Credits: 6
• SSD: SECS-S/01
• Language: English
• Campus: Bologna
• Corso: Second cycle degree programme (LM) in Statistical Sciences (cod. 9222)
• from Feb 13, 2023 to Mar 17, 2023

## Learning outcomes

By the end of the course, the student knows the basic theory of stochastic processes and martingales. On the theoretical side, the student possesses the tools to prove the main results on existence and convergence of conditional expectations and martingales.

## Course contents

Brief review of a few basic concepts on probability theory

Conditional expectation

General notions about stochastic processes: Definition, paths, filtrations, stopping times, finite dimensional distributions

Existence of processes with given finite dimensional distributions

Martingales

Markov chains

Stationary and exchangeable sequences

Random walks

Brownian motion

Poisson process

Cinlar E. (2011) Probability and stochastic processes, Springer.

Grimmett G. and Stirzaker D. (2001) Probability and random processes, Oxford University Press.

## Teaching methods

Lectures and class exercises

## Assessment methods

Oral exam. During the exam, the student may be requested to discuss (not necessarily to solve) some simple exercises together with the teacher. Such exercises are obvious versions of exercises which have been solved in class.

The possible questions may concern each part of the course. Typically, the interview starts with a very general question (such as "Brownian motion" or "Martingale") and then, as the topic is introduced, they become more specific. In addition to knowledge of the topics discussed in the course, evaluation criteria are the skill to connect different arguments and the adequacy and consistency of the adopted language. A mnemonic exposition, as well as the inability to discuss with the teacher, are penalized. In other terms, it is important to be able to discuss with the teacher, to be interrupted, and possibly to address some simple objections.

The above remarks do not depend on whether the exam is online or in presence. However, for online interviews, it is desirable (even if not mandatory) that the camera is able to frame the sheet where the student is writing.

## Teaching tools

Notes and the text-books quoted above

## Office hours

See the website of Pietro Rigo