### Stochastic Analysis

Tuesday 13:30-15:00, Seminarraum 11 (OMP 1)
Thursday 12:30-14:00, Seminarraum 9 (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

• 01.10.: Introduction, Gaussian r.v. [notes]
• 06.10.: Gaussian vectors, spaces and processes [notes]
• 08.10.: Gaussian measures, pre-BM [notes]
• 13.10.: Kolmogorov continuity theorem, BM, quadratic variation [notes]
• 15.10.: Simple Markov property, 0-1 law, stopping times [notes]
• 20.10.: Strong Markov property, reflection principle [notes]
• 22.10.: LIL and related results, usual conditions [notes]
• 27.10.: Martingales, examples, inequalities, regularity [notes]
• 29.10.: Stopping theorems, local martingales [notes]
• 03.11.: Quadratic variation of local martingale [notes]
• 05.11.: Properties of the quadratic variation, Kunita-Watanabe inequality [notes]
• 10.11.: Stochastic integral w.r.t. martingales [notes]
• 12.11.: Stochastic integral w.r.t. local and semi martingales [notes]
• 17.11.: Ito's formula, exponential martingales [notes]
• 19.11.: Lévy characterisation of BM, Dubins-Schwarz, BDG inequalities [notes]
• 24.11.: Brownian motion and harmonic functions [notes]
• 26.11.: Girsanov transformation [notes]
• 01.12.: Ito's representation theorem, Wiener chaos [notes]
• 03.12.: Stochastic differential equations [notes]
• 10.12.: Picard's Iteration [notes (old)]
• 15.12.: Martingale problems [notes (old)]
• 17.12.: Relation of SDE's and PDE's [notes (old)]
• 07.01.: Semimartingale local time [notes (old)]
• 19.01.: Markov processes: definition and existence
• 21.01.: Markov processes: generator and resolvent
• 26.01.: Markov processes: regularity of trajectories, Markov properties [draft notes]
• 28.01.: Levy processes [see pages 9,10,19,20 in my older notes]

Literature:

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

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

### Seminar in Probability and Dynamical Systems

Tuesday 10:00-11:30, Seminarraum 12 (OMP1)

Topics in preparation

Preliminary programm

• To appear ...