# 105.653 Stochastic analysis in financial and actuarial mathematics 1 This course is in all assigned curricula part of the STEOP.\$(function(){PrimeFaces.cw("Tooltip","widget_j_id_20",{id:"j_id_20",showEffect:"fade",hideEffect:"fade",target:"isAllSteop"});});This course is in at least 1 assigned curriculum part of the STEOP.\$(function(){PrimeFaces.cw("Tooltip","widget_j_id_22",{id:"j_id_22",showEffect:"fade",hideEffect:"fade",target:"isAnySteop"});}); 2021W 2020W 2019W 2018W 2017W 2016W 2015W 2014W 2013W 2012W

2020W, VO, 3.0h, 5.0EC

## Properties

• Semester hours: 3.0
• Credits: 5.0
• Type: VO Lecture
• Format: Distance Learning

## Learning outcomes

After successful completion of the course, students are able to...eExplain and use the definition and properties of multidimensional normal distribution and related distributions,

• to list the definition and elementary properties of Brownian motion and to sketch the proof of its existence and Hölder continuity by means of the of Kolmogorov-Chentsov continuity criterion,
• put filtrations, stopping times, progressive measurability and path properties of processes in relation to each other,
• explain martingales, sub- and supermartingales, uniform integrability and Vitali's convergence theorem,
• to apply Doob's classical results (maximum inequalities, L^p inequality, optional stopping theorem) and sketch their proofs,
• to discuss local martingales and to give examples for strict local martingales
• to integrate predictable step processes, introduce the quadratic variation and the covariation process for continuous local martingales and calculate these processes for some examples,
• derive the existence of the stochastic integral for continuous local martingales by means of the Kunita-Watanabe inequality and explain the generalization to continuous semimartingales.

## Subject of course

Definition and properties of multi-dimensional normal distribution, definition and elementary properties of Brownian motion, existence and Hölder continuity of Brownian motion using the Kolmogorov-Chentsov continuity criterion, filtrations, stopping times, progressive measurability, path properties, martingales, uniform integrability, Vitali's convergence theorem, sub- and supermartingales, maximum inequality, Doob's inequality for p-integrable submartingales, Doob's optional sampling theorem with applications, local martingales and examples, integration of predictable step processes, p-variation of functions, quadratic variation and covariation process of continuous local martingales, Kunita-Watanabe inequality, stochastic integration for continuous local martingales and generalization for continuous semimartingales

## Teaching methods

Presentation and derivation of the results by the lecturer on the blackboard, self-study of the lecture notes. Active participation in the accompanying exercises is strongly recommended; numerous exercises are included in the lecture notes.

Oral

## Course dates

DayTimeDateLocationDescription
Mon14:00 - 16:0005.10.2020 - 25.01.2021 Zoom / siehe TUWEL (LIVE).
Tue08:30 - 10:0013.10.2020 - 26.01.2021 Zoom / siehe TUWEL (LIVE).
Stochastic analysis in financial and actuarial mathematics 1 - Single appointments
DayDateTimeLocationDescription
Mon05.10.202014:00 - 16:00 Zoom / siehe TUWEL.
Mon12.10.202014:00 - 16:00 Zoom / siehe TUWEL.
Tue13.10.202008:30 - 10:00 Zoom / siehe TUWEL.
Mon19.10.202014:00 - 16:00 Zoom / siehe TUWEL.
Tue20.10.202008:30 - 10:00 Zoom / siehe TUWEL.
Tue27.10.202008:30 - 10:00 Zoom / siehe TUWEL.
Tue03.11.202008:30 - 10:00 Zoom / siehe TUWEL.
Mon09.11.202014:00 - 16:00 Zoom / siehe TUWEL.
Tue10.11.202008:30 - 10:00 Zoom / siehe TUWEL.
Mon16.11.202014:00 - 16:00 Zoom / siehe TUWEL.
Tue17.11.202008:30 - 10:00 Zoom / siehe TUWEL.
Mon23.11.202014:00 - 16:00 Zoom / siehe TUWEL.
Tue24.11.202008:30 - 10:00 Zoom / siehe TUWEL.
Mon30.11.202014:00 - 16:00 Zoom / siehe TUWEL.
Tue01.12.202008:30 - 10:00 Zoom / siehe TUWEL.
Mon07.12.202014:00 - 16:00 Zoom / siehe TUWEL.
Mon14.12.202014:00 - 16:00 Zoom / siehe TUWEL.
Tue15.12.202008:30 - 10:00 Zoom / siehe TUWEL.
Mon11.01.202114:00 - 16:00 Zoom / siehe TUWEL.
Tue12.01.202108:30 - 10:00 Zoom / siehe TUWEL.

Oral examination

## Course registration

Begin End Deregistration end
03.09.2020 00:00 02.07.2021 23:59 02.07.2021 23:59

## Literature

Registered students have access to an English script in electronic format with numerous references. The script will be updated on a continuing basis. It contains study assignments.

Olav Kallenberg: Foundations of Modern Probability. 3. Edition, Springer-Verlag, 2021, ISBN 978-3-030-61871-1.
Daniel Revuz and Marc Yor: Continuous Martingales and Brownian Motion, 3. Edition, Springer-Verlag, 1999, ISBN 3-540-64325-7.
Ioannis Karatzas und Steven E. Shreve: Brownian Motion and Stochastic Calculus. 2. Edition, Springer-Verlag, ISBN 0-38797-655-8.
Bernt Øksendal: Stochastic Differential Equations: An Introduction with Applications. 6. Edition, Springer-Verlag, 2007, ISBN 978-3-54004-758-2.

Foundations:
David Williams: Probability with Martingales. Cambridge University Press, 1991, ISBN 0-521-40605-6.
Heinz Bauer: Maß- und Integrationstheorie. 2. Edition, De Gruyter, 1992, ISBN 3-11013-626-0.
Heinz Bauer: Wahrscheinlichkeitstheorie. 5. Edition, De Gruyter, 2002, ISBN 3-11017-236-4.

## Language

if required in English