401-3642-00L  Brownian Motion and Stochastic Calculus

Semester Spring Semester 2015
Lecturers J. Teichmann
Periodicity yearly course
Language of instruction English


Abstract This is a first course on continuous-time stochastic processes. It covers basic notions of stochastic calculus. The following topics will be discussed:
- Brownian motion
- Markov processes
- Stochastic calculus
- Levy processes
Objective This is a first course on continuous-time stochastic processes. It covers basic notions of stochastic calculus. The following topics will be discussed:
- Brownian motion: definition, construction, some important properties
- Markov processes: basics, strong Markov property, generators and martingale problems
- Stochastic calculus: semimartingales, stochastic integrals, Ito formula, Girsanov transformation, stochastic differential equations
- Levy processes: basic notions, some important properties
Content This is a first course on continuous-time stochastic processes. It covers basic notions of stochastic calculus. The following topics will be discussed:
- Brownian motion: definition, construction, some important properties
- Markov processes: basics, strong Markov property, generators and martingale problems
- Stochastic calculus: semimartingales, stochastic integrals, Ito formula, Girsanov transformation, stochastic differential equations
- Levy processes: basic notions, some important properties
Lecture notes will be available for purchase
Literature - Durrett, R., "Stochastic Calculus. A Practical Introduction", CRC Press, 1996.
- Ikeda, N. and Watanabe, S., "Stochastic Differential Equations and Diffusion Processes", second edition, North Holland, Amsterdam, 1979.
- Karatzas, I. and Shreve, S., "Brownian Motion and Stochastic Calculus", second edition, Springer, Berlin, 1991.
- Revuz, D. and Yor, M., "Continuous Martingales and Brownian Motion", second edition, Springer, Berlin, 1994.
- Rogers, L.C.G. and Williams, D., "Diffusions, Markov Processes, and Martingales", vol. 1 and 2, Wiley, Chichester, 2000, 1994.
- Sato, K., "Levy Processes and Infinitely Divisible Distributions", Cambridge University Press, 1999.
Prerequisites / Notice Familiarity with measure-theoretic probability as in the standard D-MATH course "Wahrscheinlichkeitstheorie" will be assumed. Textbook accounts can be found for example in
- Jacod, J. and Protter, P., "Probability Essentials", second edition, Springer, 2004
or
- Durrett, R., "Probability: Theory and Examples", second edition, Duxbury Press, 1996 (Chapters 1-4 and Appendix)