# MATS352 Stochastic Analysis (5 cr)

**Study level:**

Advanced studies

**Grading scale:**

0-5

**Language:**

English, Finnish

**Responsible organisation:**

Department of Mathematics and Statistics

**Curriculum periods:**

2024-2025, 2025-2026, 2026-2027, 2027-2028

## Description

The course introduces basics of stochastic analysis. One cornerstone is the Brownian motion, probably one of the most important stochastic processes. The course will cover:

- definition of the Brownian motion, its construction, and basic properties
- Stochastic integrals as an extension of Riemann-integrals
- Itô's formula as an extension of the Taylor formula from calculus

## Learning outcomes

The students understand basic properties of the Brownian motion and can verify some of them. They are familiar with the construction of stochastic integrals. The students are able to compute particular stochastic integrals and to apply Itô's formula in various situations.

The students have developed their ability of abstraction by understanding that the stochastic integral is in fact a limit of simple expressions and that many properties need to be shown only for the simple expression and carry over to the limit expression.

## Description of prerequisites

Course MATS262 Probability 2 or similar.

Recommended: MATS254 Stochastic processes or similar.

Recommended: MATS254 Stochastic processes or similar.

## Study materials

Lecture notes: S. Geiss. Stochastic differential equations (chapters 1-3)

## Literature

- Karatzas, Ioannis, Shreve, Steven: Brownian Motion and Stochastic Calculus, 1998, Springer; ISBN: 978-1-4612-0949-2

## Completion methods

### Method 1

**Evaluation criteria:**

The grade of the course is determined by the points aquired in the exam and in the exercises.

**Time of teaching:**

Period 1

Select all marked parts

### Method 2

**Evaluation criteria:**

The grade of the course is determined by the points aquired in the exam.

Select all marked parts

**Parts of the completion methods**

x

### Teaching (5 cr)

**Type:**

Participation in teaching

**Grading scale:**

0-5

**Language:**

English

#### Teaching

##### 9/3–10/27/2024 Lectures

x

### Exam (5 cr)

**Type:**

Exam

**Grading scale:**

0-5

**Language:**

English