# 105.091 Stochastic analysis in financial and actuarial mathematics 2 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"});}); 2021S 2020S 2019S 2018S 2017S 2016S 2015S 2014S 2013S 2012S 2011S 2010S 2009S 2008S 2007S

2021S, VO, 2.0h, 4.0EC

## Properties

• Semester hours: 2.0
• Credits: 4.0
• Type: VO Lecture
• Format: Distance Learning

## Learning outcomes

After successful completion of the course, students are able to ...

• explain and apply the chain rule, integration by parts, and convergence theorems for stochastic integrals (w.r.t. continuous semimartingales),
• formulate and use Ito's multidimensional formula, Tanaka's formula, lto's local formula and Ito's formula for holomophic functions, present selected applications,
• introduce the stochastic exponential and the stochastic logarithm, explain basic properties and characterisations,
• use Lévy's characterization of Brownian motion,
• formulate Girsanov's theorem and apply it to adjust the drift of Brownian motion by a measure change,
• explain Doob's upcrossing inequality and derive Doob's convergence theorems for submartingales,
• explain and apply the predictable integral representation theorem for Brownian local martingales,
• check and derive conclusions from Kazamaki's and Novikov's criterion,
• describe and apply the ideas and methods used to prove tha main theorems of the course.

## Subject of course

Chain rule and convergence theorems for stochastic integrals (with respect to continuous semimartingales), integration by parts, multi-dimensional Ito formula with applications, Tanaka's formula, local Ito formula and Ito formula for holomorphic functions, stochastic exponential of continuous semimartingales, stochastic logarithm, Lévy's characterization of standard Brownian motion, Girsanov's theorem, change of drift using Girsanov's theorem, Doob's upcrossing inequality, Doob's convergence theorems for submartingales, representation of Brownian local martingales, Kazamaki's and Novikov's criterion

## Teaching methods

The basic contents and concepts are presented by the head of the LVA and illustrated and discussed with the help of examples.

Oral

## Course dates

DayTimeDateLocationDescription
Thu09:00 - 11:0004.03.2021 - 24.06.2021 Zoom / siehe TUWEL (LIVE).
Stochastic analysis in financial and actuarial mathematics 2 - Single appointments
DayDateTimeLocationDescription
Thu04.03.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu11.03.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu18.03.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu25.03.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu15.04.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu22.04.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu29.04.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu06.05.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu20.05.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu27.05.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu10.06.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu17.06.202109:00 - 11:00 Zoom / siehe TUWEL.
Thu24.06.202109:00 - 11:00 Zoom / siehe TUWEL.

## Examination modalities

The performance is assessed by an oral examination at the end of the semester.

## Course registration

Begin End Deregistration end
01.03.2021 00:00 30.06.2021 23:59 31.03.2021 23:59

## Literature

Registered students (to part 1 of the course) have access to an English script in electronic format with numerous references. The script will be updated on a continuing basis.

Olav Kallenberg: Foundations of Modern Probability. 2. Edition, Springer-Verlag, 2002, ISBN 0-387-953113-2.
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