# STOCHASTIC ANALYSIS

## Learning outcomes of the course unit

The student will get a good theoretical understanding of stochastic processes. She will be able to study simple stochastic differential equations in a qualitative and quantitative way, both in the field of pure research and in industrial applications (for example in finance and in the modeling of noisy systems).

## Prerequisites

Measure spaces, probability spaces, Borel-Cantelli lemmas, random variables, mathematical expectation, modes of convergence for random variables, L^p spaces

## Course contents summary

In the first part of the course we introduce continuous-time stochastic processes and we deal with the new issues arising from this object. In particular, we develop the tools needed for the study of stochastic processes and we show the existence of the Brownian motion.

Second part is devoted to the construction of the stochastic integral and to the study of its properties, in particular through martingales.

In the third part we give a short introduction to stochastic differential equations.

## Course contents

Stochastic processes, Gaussian vectors, law of a process, Gaussian processes, modifications, equivalent processes, Kolmogorov's extension theorem, Doob's lemma, independence;

Brownian motion, Kolmogorov's regularity theorem, existence and uniqueness of BM, elementary properties and transformations, quadratic variation, BM is not BV, Hölder property, Stieljes integral and extensions, filtrations and adapted processes;

conditional expectation, existence and uniqueness, elementary properties;

progressively measurable processes, simple processes and their density in M², stochastic integral for M² processes, elementary properties, Itō isometry;

discrete-time and continuous-time martingales, stopping times, Doob's optional stopping theorem, maximal inequality, Doob's optional sampling theorem, continuity of the stochastic integral process, quadratic variation of the stochastic integral;

stochastic integral for M²_loc processes, continuity, integration up to a stopping time, local martingale;

Itō formula;

stochastic differential equations; geometric BM, Orstein-Uhlenbeck process; Itō processes; existence and uniqueness of strong solutions for SDE.

## Recommended readings

Francesco Caravenna - Moto browniano e analisi stocastica

Daniel Revuz, Marc Yor - Continuous Martingales and Brownian Motion

Ioannis Karatzas, Steven E. Shreve - Brownian Motion and Stochastic Calculus

David Williams - Probability with Martingales

Paolo Baldi - Equazioni differenziali stocastiche e applicazioni

Bernt Øksendal - Stochastic Differential Equations: An Introduction with Applications

## Teaching methods

Traditional classes (48 hours). Arguments are presented in a formal way, with proofs for most statements. Much stress is given to the motivations and we include some examples of applications. There are no exercise sessions scheduled, but homework is regularly assigned during lessons and students are encouraged to do it at home and possibly ask for solutions during the teacher office hours.

## Assessment methods and criteria

The oral examination consists of three parts. In the first part the student will solve a complex problem assigned some days before by the teacher. In the second part he will be given one or two simple exercises. In the last part he will be asked to state and prove one of the main results of the course.

To pass the exam the student should master the mathematical language and formalism. He must know the mathematical objects and the theoretical results of the course and he should be able to use them with ease. He should also be able to prove theorems by himself.

## Other informations

On the website lea.unipr.it the student can find video and blackboard trascriptions for each lesson, since the teaching is done through tablet PC