V4F1 - Summer semester 2022

Stochastic Analysis

V4F1 - SS22

Schedule: Tuesday 16.00-18.00 (c.t.) and Thursday 12.00-14.00 (c.t.), Kleiner Hörsaal, Wegelerstr. 10.  (In presence)

Tutorial classes: Wed 8-10 SemR 0.007, Wed 12-14 SemR 0.007.  (In presence)

Exam: Oral, July 26-28. To attend the exam is mandatory to have reached at least half of the total number of points in the exercise sheets.

Sample exam questions from the SS20 (pdf). They are an indication of the kind of questions asked in the exam. Still to be updated according to the content we will cover this year.

Topics of course

The course develops applications of stochastic calculus to the study of continuous time stochastic processes:

• SDEs: properties, weak/martingale solutions, transformations, large deviations, link with PDEs
• Local times, martingale representation theorem, variational properties of Brownian motion, Backward SDEs.

Prerequisites 

Ito calculus for continuous semi-martingales, see  e.g. Prof. Eberle's lecture notes on "Introduction to Stochastic Analysis" (pdf) and my course "Foundations of Stochastic Analysis" from the WS19/20 (link) or the course "Foundations of Stochastic Analysis" of Dr. De Vecchi in WS21/22 [pdf]

Lecture Notes

I will post here below the notes for the lectures. For the initial part of the course we will follow closely the Stochastic Analysis course I gave in SS20 (link). Other useful material are Prof. Eberle's lecture notes for Stochastic Analysis SS16 (pdf) (in particular Chapters 2,3 but excluding processes with jumps). However the material presented in the lecture and the lecture notes of this course constitute the main reference for preparing the exam.

• Note 1 - Stochastic differential equations - version 1 - 2022.5.3 [pdf]
• Note 2 - SDE techniques - version 2 - 2022.5.20 [pdf]
• Note 3 - The martingale representation theorem and applications - version 3 - 2022.7.7 [pdf]

Recordings of the lectures of SS20 are available on the eCampus page of the course [link].

Further References

• Rogers/Williams: Diffusions, Markov processes and martingales, Vol.2
• Revuz/Yor: Continuous Martingales and Brownian motion
• Bass: Stochastic processes
• Protter: Stochastic integration and differential equations

Problem sheets

• Sheet 0 (discussed in the tutorial) [pdf] (related material: the paper of Yamada-Watanabe [pdf], Cherny [pdf] and a note on Osgood's condition [pdf])
• Sheet 1 [pdf] (to be handed in April 14th)
• Sheet 2 [pdf] (to be handed in April 21st)
• Sheet 3 [pdf] (to be handled in April 28th)
• Sheet 4 [pdf] (to be handled in May 5th)
• Sheet 5 [pdf] (to be handled in May 12th)
• Sheet 6 [pdf] (to be handled in May 19th)
• Sheet 7 [pdf] (to be handled in May 27th, no lecture May 26th)
• Sheet 8 [pdf] (to be handled in June 2nd)
• (No tutorials the week of June 22nd)
• Sheet 9 [pdf] (to be handled in June 23rd)
• Sheet 10 [pdf] (to be handled in June 30th, this is the last exercise sheet)

Course Journal

• Lecture 1 (4/4) Introduction. Historical and conceptual remarks on stochastic analysis. Definition of weak solutions.
• Lecture 2 (7/4) Relations between notions of uniqueness and existence, various examples.
• Lecture 3 (12/4) A theorem of Cherny about Uniqueness of the joint law (X,B) for a weak solution. Yamada-Watanabe theorem with proof.
• Lecture 4 (14/4) The "dual" Yadama-Watanabe theorem of Cherny with proof. Levy's caracterisation of multidimensional BM.
• Lecture 5 (19/4) Ortogonal transformations of BM. Bessel process. Discussion of pathwise uniqueness for SDEs. 
• Lecture 6 (21/4) Martingale problems, formulation, relation with weak solutions with proof.
• Lecture 7 (26/4) Uniqueness of martingale problems  with proof.
• Lecture 8 (28/4) Time change for martingale problems, DDS Brownian motion, applications of time-change.
• Lecture 9 (3/5) One dimensional diffusions. Change of probability on a filtered measure space.
• Lecture 10 (5/5) Girsanov's theorem. Doob's h-transform.
• Lecture 11 (10/5) Conditioning of diffusions.
• Lecture 12 (12/5) Conditioning a diffusion to never exit a domain. Conditioning a Brownian motion to stay positive.
• Lecture 13 (17/5) (Lucio Galeati) Novikov condition, change of drift in SDEs
• Lecture 14 (19/5) (Lucio Galeati) Uniqueness in law via Girsanov's theorem, path integral formula, sampling of diffusions via the path integral, representation of the semigroup of a reversible diffusion with additive noise (pay attention to the revised lecture note 2)
• Lecture 15 (24/5) (Francesco De Vecchi) Ito-Tanaka formula and local times
• No lecture May 26th.
• Lecture 16 (31/5) (Francesco De Vecchi)  Regularity of local times, representation of local times 
• Lecture 17 (2/6) (Francesco De Vecchi) Local time of Brownian motion, Tanaka's SDE, reflected Brownian motion, relation of local time with supremum, Skorohod lemma.
• Lecture 18 (14/6) Brownian martingale representation theorem. the Markovian proof.
• Lecture 19 (21/6) Entropy on the Wiener space, Boué-Dupuis formula. Application to Gaussian tails of Lipschitz functionals. Proof of Boué-Dupuis formula.
• 23/6 and 28/6 : lectures canceled
• Lecture 20 (30/6) Conclusing of the proof of Boué-Dupuis formula. Large deviations / Laplace principle for families of Brownian functionals.
• 5/7 :  lecture canceled
• Lecture 21 (7/7) Large deviations / Laplace principle for families of Brownian functionals. Applications to Brownian motion (Schilder's theorem) and to small noise diffusions.