105.653 Stochastic analysis in financial and actuarial mathematics 1
This course is in all assigned curricula part of the STEOP.
This course is in at least 1 assigned curriculum part of the STEOP.

2022W, VO, 3.0h, 5.0EC


  • Semester hours: 3.0
  • Credits: 5.0
  • Type: VO Lecture
  • Format: Presence

Learning outcomes

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

  • explain and use the definition and properties of multidimensional normal distribution and related distributions,
  • 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,
  • apply Doob's classical results (maximum inequalities, L^p inequality, optional stopping theorem) and sketch their proofs,
  • discuss local martingales and to give examples for strict local martingales
  • 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,
  • explain and apply the chain rule, integration by parts, and convergence theorems for stochastic integrals (w.r.t. 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, 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,

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.

Mode of examination




Course dates

Mon14:00 - 16:0003.10.2022 - 23.01.2023FH Hörsaal 3 - MATH .
Tue15:00 - 16:0004.10.2022 - 24.01.2023FH Hörsaal 3 - MATH .
Mon17:00 - 18:3007.11.2022 17:00 online via Zoom: https://tuwien.zoom.us/j/95570719565?pwd=Z0Z0cWtDR0ZtV1lKc0E2cWYvRENVdz09 (LIVE).
Mon14:00 - 16:0019.12.2022 (LIVE)(Nur) Online via Zoom - siehe TUWEL-Kurs
Tue15:00 - 16:0020.12.2022 (LIVE)(Nur) Online via Zoom - siehe TUWEL-Kurs
Stochastic analysis in financial and actuarial mathematics 1 - Single appointments
Mon03.10.202214:00 - 16:00FH Hörsaal 3 - MATH .
Tue04.10.202215:00 - 16:00FH Hörsaal 3 - MATH .
Mon10.10.202214:00 - 16:00FH Hörsaal 3 - MATH .
Tue11.10.202215:00 - 16:00FH Hörsaal 3 - MATH .
Mon17.10.202214:00 - 16:00FH Hörsaal 3 - MATH .
Tue18.10.202215:00 - 16:00FH Hörsaal 3 - MATH .
Mon24.10.202214:00 - 16:00FH Hörsaal 3 - MATH .
Tue25.10.202215:00 - 16:00FH Hörsaal 3 - MATH .
Mon31.10.202214:00 - 16:00FH Hörsaal 3 - MATH .
Mon07.11.202217:00 - 18:30 17:00 online via Zoom: https://tuwien.zoom.us/j/95570719565?pwd=Z0Z0cWtDR0ZtV1lKc0E2cWYvRENVdz09.
Tue08.11.202215:00 - 16:00FH Hörsaal 3 - MATH .
Mon14.11.202214:00 - 16:00FH Hörsaal 3 - MATH .
Mon21.11.202214:00 - 16:00FH Hörsaal 3 - MATH .
Tue22.11.202215:00 - 16:00FH Hörsaal 3 - MATH .
Mon28.11.202214:00 - 16:00FH Hörsaal 3 - MATH .
Tue29.11.202215:00 - 16:00FH Hörsaal 3 - MATH .
Mon05.12.202214:00 - 16:00FH Hörsaal 3 - MATH .
Tue06.12.202215:00 - 16:00FH Hörsaal 3 - MATH .
Mon12.12.202214:00 - 16:00FH Hörsaal 3 - MATH .
Tue13.12.202215:00 - 16:00FH Hörsaal 3 - MATH .

Examination modalities

Oral examination

Course registration

Begin End Deregistration end
01.09.2022 00:00 31.10.2022 23:59 31.10.2022 23:59



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. 

Additional literature:
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.

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.

Preceding courses

Accompanying courses

Continuative courses


if required in English