### Stochastic Analysis

Tuesday 11:15-12:50, Seminarraum 11 (OMP 1)

Thursday 11:15-12:50, Seminarraum 8 (OMP 1)

**Summary:** This course gives an
introduction to the theory of stochastic processes in
continuous time. The following topics will be discussed:

- Brownian motion
- Markov processes
- Stochastic calculus
- Levy processes
- Stochastic differential equations

**Exam questions:** [PDF]

**Lectures content**

- 7.10: Introduction, Gaussian r.v. [G 1.1]
- 9.10: Gaussian vectors, spaces and processes [G 1.2-1.3]
- 14.10: Orthogonality and independence in G. spaces, G. measures [G 1.3-1.4]
- 16.10: Pre-BM, its characterisation and properties; Modifications and indistinguishability, Kolmogorov's continuity theorem, BM [G 2.1-2.2]
- 21.10: Proof of KCT, Wiener measure [G 2.2];

sketch of Lévy-Ciesielski construction [details H 11-13] - 23.10: Simple Markov property, Blumenthal's 0-1 law, path properties of BM [H 29-31]
- 28.10: Filtrations, stopping times; progressive measurability, strong Markov property [H 35-38]
- 30.10: Proof of strong Markov Property [H 38-40]; some remarks on Markov processes [H 33-34]
- 4.11: Feller processes, application of strong Markov property, Markov process which is not strong Markov [H 41-45]
- 6.11: Properties of BM trajectories [H 47-51]
- 11.11.: No lecture
- 13.11.: No lecture
- 18.11.: Continuous-time martingales, examples, Doob's inequalities, upcrossing inequality, convergence theorems [H 55-57], [G 36-40]
- 20.11.: completness of \(\mathcal M^2\), quadratic variation [H 58-64]
- 25.11.: quadratic variation (cont.), stoch. integral of simple processes [H 65-66]
- 27.11.: more about quadratic variation, relation to stopping times
- 2.12.: Properties of stochastic integral
- 4.12.: Stochastic integral with respect to local martingales and semimartingales
- 9.12.: Ito's formula [H 82-87]
- 11.12.: Applications of Ito's formula [H 88-95] (Exponential martingales, Lévy characterisation of BM, Dubins-Schwarz)
- 16.12.: Harmonic functions and BM, transience, recurrence and conformal invariance of BM [H 97-101]
- 8.1.: Change of Measure and Girsanov's transformation [G 81] or [H 102-106]
- 13.1.: Stochastic differential equations - weak and strong solutions, examples [notes] or [G 87-89]
- 15.1.: Stochastic differential equations - Yamada-Watanabe theorem (sketch) existence and uniqueness in Lipschitz case (Picard's iteration) [H 109-112] or [G 90-94]
- 20.1.: Martingale problems - connection to SDE's [H 116-119]
- 22.1.: Relations of PDE's and SDE's, Feynman-Kac formula [H 122-126]
- 27.1.: Tanaka equation and the local time of Brownian motion [H 133+]

**Literature:**

- R. Durrett: Stochastic calculus: A practical introduction
- I. Karatzas, S. Shreve: Brownian motion and stochastic calculus
- J.-F. Le Gall: Mouvement brownien, martingales et calcul stochastique [G]
- D. Revuz, M. Yor: Continuous martingales and Brownian motion
- L.C.G. Rogers, D. Williams: Diffusions, Markov processes and martingales, 1 and 2
- D.W. Stroock, S.R.S. Varadhan: Multidimensional diffusion processes

**Lecture notes:** Handwritten notes [PDF] (low
quality, not intended for distribution)

[H] Notes for a similar lecture given at ETHZ [PDF]

### Seminar in Probability Theory

Thursday 13:15-14:50, Seminarraum 12 (OMP1)

This term the seminar will give an introduction to the percolation theory. Percolation is a simple probabilistic model which exhibits a phase transition (as we will see). The simplest version takes place on \(\mathbb Z^2\), which we view as a graph with edges between neighbouring vertices. All edges of \(\mathbb Z^2\) are, independently of each other, chosen to be open with probability \(p\) and closed with probability \(1 − p\) . A basic questions in this model are

- What is the probability that there exists an open path, i.e., a path all of whose edges are open, from the origin to the infinity?
- Is there (somewhere) an infinite open path?
- What properties have the infinite open connected components?
- ...

**Literature**:

- Geoffrey Grimmett: Percolation, Second Edition, Springer, 1999.
- Geoffrey Grimmett: Probability on Graphs, Cambridge University Press, 2010. Available here.
- Wendelin Werner: Lectures on two-dimensional critical percolation, arXiv:0710.0856.
- Vincent Beffara, Hugo Duminil-Copin: Planar percolation with a glimps of Schramm-Loewner evolution, arXiv:1107.0158.

**Preliminary programm**

- Oct 9: Introduction to the subject, organisation
- Oct 16: Free for preparation of talks
- Oct 23: Overview, phase transition; [Gri99]1.1-1.4, or [Gri10]3.1 (Krasser)
- Oct 30: Uniqueness of the infinite cluster, continuity of \(\theta(p)\); [Gri99]8.2-8.3, [Gri10]5.3 (Sobotnik)
- Nov 6: Basic inequalities (FKG, BK); [Gri99]2.2-2.3, [Gri10]4.2-4.3 (Grocholski)
- Nov 13: Subcritical phase, definition of \(p_T\), exponential decay below \(p_T\); [Gri99]Thm. 6.1, [Gri10]Thm. 5.3 (Freitag, Vokřálová)
- Nov 20: Russo's formula; [Gri99]2.4, [Gri10]4.7; divergence of \(\chi(p)\) below \(p_T\); [Gri99]pp.244-266 (Bazant)
- Nov 27: Sharpness of the percolation threshold, \(p_c=p_T\) [Gri99]5.3, [Gri10]Thm. 5.1 (Außenhofer/Riegelnegg)
- Dec 4: 2nd part of Nov 27
- Dec 11: Percolation in \(d=2\), RSW theory; [Gri10]5.5 (Rößler)
- Dec 18: \(p_c(d=2)=1/2\); [Gri10]5.6 (Oucherif)
- Jan 8: Cardy-Smirnov formula; [Gri10]5.7 (Heider/Wassmer)
- Jan 15: 2nd part of Jan 8
- Jan 22: Inluences and sharp threshold (Belma Klicic)
- Jan 29: ??