SS 25/26: Rough stochastic differential equations (2h)

Lecture will be given 4h/week in the first half of semester, opening into a student seminar with a related focus (there will be opportunities for BSc and MSc theses).

Times: Mo 12-14 (2h) und DO 13:00-14 (1h), Location: TUB (details TBA)

(WS 23/24, to be updated) https://moseskonto.tu-berlin.de/moses/verzeichnis/veranstaltungen/veranstaltung.html?veranstaltung=161491

(WS 25/26) https://isis.tu-berlin.de/course/view.php?id=44082

Perequisites: measure theory, continuous time processes and awareness of Itô stochastic differential equations (SDEs), as taught in FiMa2 (TU), WT3 (TU), or Stochastic Processes II (BMS).

Itô stochastic theory has a powerful analytic companion, the theory of rough paths, which has had a profound impact of modern stochastic analysis, see e.g. [1].

"Rough stochastic differential equations" (RSDEs) are a common generalisation of Itō SDEs and Lyons RDEs. Since their introduction in [2] they have emerged as a powerful tool in several areas of applied probability, including non-linear stochastic filtering, pathwise stochastic optimal control, volatility modelling in finance and mean-field analysis conditional on common noise. This lecture will offer an overview of motivating examples, including some classes of stochastic partial differential equations, before presenting the key ideas of the theory, starting with a primer on classical rough differential equations.

[1] Peter K. Friz and Martin Hairer (2020), A Course on Rough Paths With an introduction to regularity structures, Springer Universitext and https://www.hairer.org/notes/RoughPaths.pdf

[2] Peter K. Friz, Antoine Hocquet, Khoa Lê (2021), Rough stochastic differential equations, CPAM (2025 subj revision) and https://arxiv.org/abs/2106.10340

SS 24: WT3 (BMS Course Stochastic Processes II)

Prof. Friz and Dr. Hager, starts on 15-April

Lecture 22-Apr: ComplementLeGallChapter2_SS24

https://isis.tu-berlin.de/course/view.php?id=39045#section-0 

f

SS 24: Student Seminar: Signatures and Data Science

https://moseskonto.tu-berlin.de/moses/verzeichnis/veranstaltungen/veranstaltung.html?veranstaltung=183841

WS 23/24:

https://moseskonto.tu-berlin.de/moses/verzeichnis/veranstaltungen/volltextsuche.html?query=Friz&search=true

VL Rough stochastic differential equations (2h)

This lecture is devoted to some fundamental recent progress in stochastic analysis, a hybrid theory which seamlessly combines the advantages of both Itô's stochastic - and Lyons' rough path theory. We rely in particular on a new stochastic variant of controlled rough paths spaces, inspired by Khoa Lê's celebrated stochastic sewing lemma.  There are many applications,  included robust filtering, pathwise stochastic control, conditional analysis of financial models, and the analysis of mean field SDEs with common noise, as well as related classes of non-linear stochastic partial differential equation. Time and audience permitting we shall discuss this in later parts of the lecture.

Prerequisites: measure theory, awareness of Itô's stochastic differential equations, as taught in FiMa2 (TU), WT3 (TU), or Stochastic Processes II (BMS)

Time + location: information available on https://page.math.tu-berlin.de/~friz/ (Link Student Info)

Zeit 10:00 - 12:00, Mi., Ort E-N 189 ( Charlottenburg )

Mi. 18.10.23, Mi. 25.10.23,  Mi. 01.11.23, Mi. 08.11.23, Mi. 15.11.23, Mi. 22.11.23, Mi. 29.11.23,

Mi. 06.12.23, Mi. 13.12.23, Mi. 20.12.23, Mi. 10.01.24, Mi. 17.01.24, Mi. 24.01.24, Mi. 31.01.24, Mi. 07.02.24, Mi. 14.02.24

Substitute Lectures

Mi. 25.10.23  ⇒ L2 on Friday 27.10.23, 8:30 - 10:00 online (zoom)

Mi. 01.11.23⇒ L3 on Friday 3.11..23,  8:30 - 10:00 online (zoom)

 Mi. 29.11.23⇒ L6 on Thursday, 30.11.23, 8:30 - 10:00  E-N 189

Mi. 20.12.233⇒ L8 on Thursday, 21.12..23, 8:30 - 10:00 online (zoom)

(Extra) Makeup Lecture on  Thursday, 1.2.24, 8:30 - 10:00 online (zoom)

Mi. 07.02.24, regular time, 10:15 - 11:45, Lecture online (zoom)

Notes: L1, 18-Oct, L2, 27-Oct L3, 3-Nov L4, 8-Nov Exercise Class at 15-Nov

           L5, 22-Nov Discussion Class on 29-Nov, L6, 6-Dec, L7-to be posted, L8, 21-Dec

• Le Gall, Jean-François. Brownian motion, martingales, and stochastic calculus. springer publication, 2016.
• Friz, Peter K., and Martin Hairer. A course on rough paths. Springer International Publishing, 2020.
• Friz, Peter K., Antoine Hocquet, and Khoa Lê. "Rough stochastic differential equations." arXiv preprint arXiv:2106.10340 (2021).
• Lacker, Daniel. Lecture notes on  Mean field games and interacting particle systems https://www.columbia.edu/~dl3133/MFGSpring2018.pdf (material not examinable)

SE Optimal Transport and Mean Field Games (2h)

Optimal Transport and Mean Field Games are two interconnected mathematical frameworks with wide-ranging applications in economics, physics, and machine learning. Optimal Transport, pioneered by mathematician Leonid Kantorovich, addresses the problem of efficiently transporting goods from one location to another, minimizing the associated cost. It has applications in logistics, image processing, and even neuroscience.

Mean Field Games, developed by Jean-Michel Lasry and Pierre-Louis Lions, extend this concept to dynamic systems with a large number of agents. It models the strategic interactions of individuals in a society or economy, seeking to find equilibrium solutions. This approach has profound implications in economics, where it can model market behavior, traffic flow, and pricing strategies.

Together, these fields merge to tackle complex problems involving the collective behavior of agents and the optimal allocation of resources. They find applications in diverse areas, from urban planning and traffic management to understanding the dynamics of financial markets and the behavior of particles in physics. As research continues to advance, Optimal Transport and Mean Field Games promise innovative solutions to real-world challenges.

Time + location: information available on https://page.math.tu-berlin.de/~friz/ (Link Student Info)

Prerequisites: measure theory, analysis, probability theory (at least at the level of WT2, TUB)

First meeting Thu,. 19.10.23, 14:15 Raum FH 316,

Attention, meeting on Thu, 2.11.23, 8:30-10:00  (zoom)

Attention, meeting on Thu, 30.11.23, canceled due to sickness

Attention, meeting on Thu, 8.2.24, canceled.

As discussed in class on 1.2.24, make up sessions will be scheduled later in February. (Times tbd)

  Fraunhoferstraße 33-36, 10587 Berlin

References:

• Figalli, Alessio, and Federico Glaudo. An invitation to optimal transport, Wasserstein distances, and gradient flows. 2021. https://ems.press/books/etb/190?na
• Ambrosio et al. Lectures on Optimal Transport, https://link.springer.com/book/10.1007/978-3-030-72162-6 (OA)

• A. Figalli An Introduction to Optimal Transport and Wasserstein Gradient Flows, lectures notes available https://people.math.ethz.ch/~afigalli/lecture-notes-pdf/An-introduction-to-optimal-transport-and-Wasserstein-gradient-flows.pdf
• Notes on Mean Field Games (from P.-L. Lions’ lectures at College de France) Pierre Cardaliaguet, notes available

https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf

• Rene Carmona, Francois Delarue, Probabilistic Analysis of Mean-Field Games, https://arxiv.org/abs/1210.5780

• Francois_Delarue_LectureBerlin_I.pdf

Oberseminar Rough Paths, SPDEs and Related Topics (2h)

(Wie im Vorjahr)

Time: Thursday 11:00,

more information available on https://page.math.tu-berlin.de/~friz/ (Link Student Info)

Stochastik und Finanzmathematik / Anleitung wiss. Arbeiten (1h)

(Wie im Vorjahr)

SS22: Stochastic processes II (BMS), WT 3

Update June-23: As announced in class, the regular classes next week Mon/Wed will not take place.

Instead, we will have two zoom lectures on Friday mornings (also recorded)

Jun 24, 2022 08:00 AM  / Jul 1, 2022 08:00 AM /  (tbc Jul 8, 2022 08:00 AM)

https://tu-berlin.zoom.us/j/66245759670?pwd=WkFPS1YxRjdONzZnemowZkphaEV2Zz09

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Additional notes for 9-May Lecture: Example Sheet

https://drive.google.com/file/d/1Y2HLCJeBpL922uA2WSY3b2ZefSWScHfW/view?usp=sharing

Example Sheets: Sheet 1 (tbd on 2-May)

(see also ISIS)

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Title: Stochastic processes II: continuous time - Wahrscheinlichkeitstheorie III

Lecturer: Friz, TU

This is a BMS basic course, hence held in English

https://www.math-berlin.de/index.php/academics/courses/basic-courses

Lectures

Location: MA 141 and MA 751Time: MO 14-16 and WED 14-16 (First lecture, MO 25-April 2022, at 14:15 - 15:45, MA 651)

Exercise Session MO 16-18 in room MA 750 (start on MO 2-May 2022)

The lectures will be held in presence.

Literature

We will follow closely the book Le Gall - Brownian Motion, Martingales and Stochastic Calculus, which is available as e-book in the online catalogue of TUB, get it here.

You may also find useful this set of notes by James Norris: http://www.statslab.cam.ac.uk/~james/Lectures/ap.pdf

To  look up bits and pieces of (functional) analysis, real and complex analysis, measure theory, Fourier analysis, etc. try

https://mathweb.ucsd.edu/~bdriver/DRIVER/Book/anal.pdf

(written by the former editor of the Journal of Functional Analysis)).

First lecture, MO 25-April 2022, at 14:15 - 15:45, MA 742

SS21:  

Einladungslink

 https://tu-berlin.zoom.us/j/65911167984?pwd=L1JDdGF2bDlVbDAyU2RUYngvSlZIdz09

SS 20:

 Rough Stochastic Analysis I         

Topic: VO Rough Stochastic Analysis I (Friz)

                      Apr 24, 2020 10:15 AM     (-> recorded video)

                     May 15, 2020 10:15 AM

                     May 22, 2020 09:30 AM

            May 29, 2020 09:30 AM

            Jun 5, 2020 09:30 AM

            Jun 12 + 19, 2020 09:30 AM

                Jun 26, 2020 09:30 AM

            Jul 3, 2020 09:30 AM

            Jul 10, 2020 09:30 AM

            Jul 17, 2020 09:30 AM

Exercise Sheet 1 (TBD on Jun 12, 11:00 AM immediately after class by Yizheng Yuan <yuan@math.tu-berlin.de>)

Exercise Sheet 2 (TBD on Jul 17, time TBC)

Join Zoom Meeting

https://tu-berlin.zoom.us/j/94580874299?pwd=eFN5MVRNVGFMZDl0bHh0Q0liMThzQT09

Meeting ID: 945 8087 4299

Password: 918394

References: https://math.ethz.ch/fim/activities/nachdiplom-lectures/peter-friz.html

Friz-Hairer, A course on rough paths, Springer (2020)

Oberseminar Rough Paths, Stochastic Partial Differential Equations and Related Topics

Thursdays 11:00 - 12:00 AM

Please contact antoine.hocquet@tu-berlin.de to get on the mailing list.

SS 19: Maß- und Integrationstheorie:

1. MT Exam: 30 min, 3 Fragen aus verschiedenen Kapiteln

Um Ihnen bei der Vorbereitung zu helfen, anbei eine Liste von typischen Fragen! link

1. Termine: 11. und 12. Juli

2. Ergaenzungen zum Skriptum:

 7.3. Wiener Maß

11 P-Variation, Young Integration (Final 28. Juni)

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Yizheng Yuan (UE)  <yuan@math.tu-berlin.de>  Sprechstunde: Mittwoch, 16:00 - 17:00

Michele Coghi (VO) <coghi@wias-berlin.de> Sprechstunde: Donnerstags, 15:00 - 16:00

Peter Friz (VO) <friz@math.tu-berlin.de> Sprechstunde: nach Vereinbarung

Aktuelles: 15.4.2019: “einmalige Brückenkurs VO” statt UE um 08:00

Aenderung: Mo-VO ab 29.4.2019 in  H 1012. Fr-VO weiterhin in MA 005.

Vorlesung: Maß- und Integrationstheorie

Prüfung: Schriftlich am Fr 13.07, 10:00 - 12:00; weiterer Termin ist zu Beginn WS19/20. Voraussetzung ist bestandene Übung (s.u.)

Übung: 

*** Aktuelle Übungsblätter hier! ****

Anmeldung: Email bis zum 15.04.2019 an Herrn Yuan, Subject: MT SS19 / Anmeldung UE

Scheinkriterien: Es gibt Anwesenheitspflicht. Für jede Übungsstunde werden 5-10 Aufgaben gestellt. Jeder/Jede Studierende entscheidet selber, wieviele er/sie rechnet. Bis einige Minuten vor der Übung sind diejenigen Beispiele in einer Liste anzukreuzen, die vorbereitet wurden. Die genauen Modalitäten, wo, wann, wie anzukreuzen ist, erfahren Sie vom Übungsleiter in der ersten Übung.

Zu den angekreuzten Beispielen kann der/die Studierende dann in der Übung an die Tafel zum Vorrechnen geholt werden. Dieses Vorrechnen beinhaltet auch Fragen zum Stoff rund um das Beispiel, und wird bewertet.

Achtung: Am Ende des Semesters muss man zwei Drittel der aufgegebenen Beispiele angekreuzt haben, um einen Schein zu erhalten. Daneben muss die Gesamtbeurteilung der Tafelleistung ebenfalls positiv ausfallen.

Skriptum:

MT Skriptum, Sie koennen kommentieren!

https://www3.math.tu-berlin.de/Vorlesungen/SS09/MaI/skript_mass_int.pdf

(TUB Skriptum, J. Gaernter)

Zusätzliche Literatur:

http://www.statslab.cam.ac.uk/~james/Lectures/pm.pdf

(Cambridge University Lecture notes, J. Norris)

http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/PDE-Anal-Book/analpde1.pdf

(Part III Lebesgue Integration Theory, B. Driver)

https://people.math.ethz.ch/~salamon/PREPRINTS/measure.pdf

https://www.springer.com/de/book/9781461411345

________________________________________________

WS  18/19 : Rough Volatility

Exam dates:   Mo, 8-April 2019, 2pm+ ///  

2-13 Sep 2019 (TBC)

NB: Lecture times 10:07 - 11:52 to make up for some cancelled lectures (* below)

Some reading/watching (updated as we go)

Overview:

Jim Gatheral video lecture on rough vol 

Lectures 1 & 2

Refresher on Ito semimartingales (very optional background reading) 

FiMa2 type material (background reading), Frey’s lectures

Guyon & Henry-Labordere (introduction as background reading)

Lectures 3

Abstract Gaussian measures, Cameron-Martin spaces, large deviation

Theory and Applications of Fractional Differential Equations, Volume 204, A.A. Kilbas H. M. Srivastava J.J. Trujillo, 2006.

Fraction Brownian motion (ICM Talk 2006)

UCV/UT:

Kurtz Slides

Large Deviations:

Large Deviations and Applications (short notes by Peter Moerters)         

VO Grosse Abweichungen (W. Koenig)

A 2-page note on Varadhan Lemma

Plan (draft)

SS  18 : Seminar: "Quantitative Finance"  

Attention: new time/location:

08:30 - 10:00 on Mondays, MA 742 or 721

First meeting: Mon, 16-Apr-2018, focus: Asymptotic option pricing

Additional reading on Large Deviations: Lectures (Robertson), VO (König)

Large Deviations in Finance (Pham), Wentzell–Freidlin theory (Gentz)

Partial material for 7-May: Primer on Rough Paths

Future meetings (always Mondays), 23-Apr / 30-Apr / 7-May / 14-May (TBC)

28-May / 4-Jun (TBC) / 11-Jun / 18-Jun / 25-Jun / 2-Jul (TBC) & blocked talks (dates TBC)

Talks distribution

WS  17 / 18 : Seminar: "Rough Analysis and Quantitative Finance"   Mondays 12:00 - 14:00, MA 742

First meeting: Monday, 23-Oct-2017

If interested, fill out the Seminar participation form and send between 11-Oct and 25-Oct

as 1-Page PDF to Frau Downes, downes@math.tu-berlin.de

Some topics found here:

https://docs.google.com/document/d/136FeGICoeP7jp0ASlVdCDqhYrtOYZ81Jh-Y78SVMkn4/edit?usp=sharing