Stochastic Processes (HS 2018)

Lectures: Monday and Tuesday 10:15-12:00, Spiegelgasse 5, Room 05.001

Exercises: Thursday 8:15-10:00, Spiegelgasse 5, Room 05.001 (assistant T. Hayder)

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

• Brownian motion
• Martingales in continuous time
• Markov processes
• Stochastic calculus
• Levy processes
• Stochastic differential equations

Lectures content:

• Sep 17: Introduction, remarks on construction of processes
• Sep 18: Gaussian vectors and processes
• Sep 24: Gaussian spaces (independence,conditioning), Gaussian white noise
• Sep 25: pre-Brownian motion, modifications
• Oct 01: Kolmogorov continuity theorem, Brownian motion, Wiener measure
• Oct 02: Simple Markov property, Blumenthal 0-1 law, consequnces
• Oct 08: Stopping times, strong Markov property
• Oct 09: Reflexion principle, Law of iterated logarithm for BM
• Oct 15: Martingales: inequalities
• Oct 16: Martingales: convergence theorems and regularity
• Oct 22: Martingales: stopping theorems; infinite variation
• Oct 23: Processes of finite variation, local martingales
• Oct 29: Quadratic variation of local martingales
• Oct 30: Properties of quadratic variation, Semimartingales, Kunita-Watanabe inequality
• Nov 05: Stochastic integral for square integrable martingales
• Nov 06: Properties of stochastic integral, extension to semimartingales
• Nov 12: Ito's formular
• Nov 13: Levy's characterisation of BM, Dubins-Schwarz
• Nov 19: Brownian motion and harmonic functions, rekurrence and transience
• Nov 20: Complex Brownian motion; Girsanov transformation
• Nov 26: SDEs: solutions and examples
• Nov 27: SDEs: Existence and uniqueness of strong solutions
• Dec 03: Markov processes: semigroups, resolvent
• Dec 04: Markov processes: generator(notes for the last two lectures)

Questions for exercises:

• Sheet 1 for Sep 27
• Sheet 2 for Oct 4
• Sheet 3 for Oct 11
• Sheet 4 for Oct 18
• Sheet 5 for Oct 25
• Sheet 6 for Nov 1
• Sheet 7 for Nov 8
• Sheet 8 for Nov 15
• Sheet 9 for Nov 22
• Sheet 10 for Nov 29
• Sheet 11 for Dec 6
• Sheet 12 for Dec 13

Literature:

• J.-F. Le Gall: Brownian Motion, Martingales, and Stochastic Calculus
• D. Revuz, M. Yor: Continuous martingales and Brownian motion
• I. Karatzas, S. Shreve: Brownian motion and stochastic calculus
• 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: a script will be written during the semester (in the mean time: handwritten notes for a similar lecture from 2015 [PDF])