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# Stochastic Analysis

## Module MA4405

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Module version of WS 2021/2 (current)

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
WS 2021/2SS 2021SS 2020SS 2014WS 2013/4

### Basic Information

MA4405 is a semester module in English language at Master’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

• Specialization Modules in Elite-Master Program Theoretical and Mathematical Physics (TMP)
270 h 90 h 9 CP

### Content, Learning Outcome and Preconditions

#### Content

Continuous time stochastic processes as a general framework. Brownian motion: construction and path properties. Filtrations, (local) Martingales and stopping times, optional sampling, maximal inequalities and reflection principle. Poisson processes and stable processes. Quadratic and cross variations, Levy’s representation and characterization Theorems, Semi-martingales. Stochastic integrals with respect to Brownian motion and general semi-martingales. Integration rules with emphasis on Itô's formula.
Stochastic differential equations, weak and strong solutions. Girsanov's Theorem and applications. Donsker's invariance principle.

#### Learning Outcome

After successful completion of the module, students are able to:
- define Brownian motion, prove basic facts about Brownian motion and apply basic calculations involving Brownian motion.
- understand fundamental results such as Lévy's Theorems, Donsker's invariance principle etc, and apply them in calculations as well as in proofs.
- understand the basics of stochastic integration with respect to various processes. Apply Itô's formula and other tools.
- understand the basics of stochastic differential equations, can verify the (in)existence of strong or weak solutions in standard examples.
- understand and apply change-of-measure techniques.

#### Preconditions

MA0003 - Analysis 3
MA2409 - Probability Theory

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

VO 4 Stochastic Analysis [MA4405] Berger Steiger, N. Tue, 10:15–11:45, BC2 BC2 3.5.06
Thu, 12:15–13:45, BC2 BC2 3.5.06
UE 1 Exercises for Stochastic Analysis [MA4405] Berger Steiger, N. Lux, K. dates in groups

#### Learning and Teaching Methods

lecture, exercise module
The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets will be available.
In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.

#### Media

blackboard, assignments

#### Literature

Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus, New York: Springer.
P. Mörters, Y. Peres (2010): Brownian Motion, Cambridge University Press, New York / Melbourne / Madrid / Cape Town / Singapore / Sao Paulo / Delhi / Dubai / Tokyo
Le Gall, J. F. (2016). Brownian motion, martingales, and stochastic calculus. Springer.
Revuz, D., & Yor, M. (2013). Continuous martingales and Brownian motion (Vol. 293). Springer Science & Business Media.

### Module Exam

#### Description of exams and course work

The module examination is based on a written exam (60-90 minutes).
Students have to know and understand the theoretical foundations of Brownian motion, the theory of continuous martingales and Lévy's representation and characterisation Theorems. They also need to be able to understand the basics of stochastic integration and stochastic differential equations and to apply Itô's formula and change-of-measure techniques.

#### Exam Repetition

The exam may be repeated at the end of the semester.

#### Current exam dates

Currently TUMonline lists the following exam dates. In addition to the general information above please refer to the current information given during the course.

Title
TimeLocationInfoRegistration
Stochastic Analysis
Mon, 2022-08-08, 10:45 till 12:15 till 2022-06-30 (cancelation of registration till 2022-08-01)
Fri, 2022-10-07, 10:45 till 12:15 till 2022-09-26 (cancelation of registration till 2022-09-30)
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