Stochastic analysis, spring 2013
Stochastic analysys, spring 2013 (Suomeksi)
Teacher
Credits
10 cr.
Type
Advanced course
Prerequisites
Probabllity Theory
Language:
this course will be given in english or finnish depending on the audience.
The teaching material is in english.
Contents
The subject of this course is martingale theory and stochastic integration.
0. Introduction: functions with bounded variations, Riemann-Stieltjes integral. Pathwise quadratic variation, Ito-Föllmer pathwise integral and Ito formula. Brownian motion and its quadratic variation.
I. Kolmogorov extension theorem and construction of stochastic process on its canonical space. Kolmogorov continuity theorem. Paul Levy's construction of Brownian motion.
II. Martingales in discrete time: Conditional expectation, martingale transform, forward and backward martingale convergence theorems, uniformly integrable martingales, square integrable martingales, Doob maximal inequality. Change of measure and Radon-Nikodym derivative.
III. Continuous martingales. Ito isometry. Ito integral and Ito formula. Burkholder Davis Gundy inequality. Local time, Ito-Tanaka formula.
IV Change of measure: Girsanov formula, stochastic exponential, Gronwal lemma. Applications in stochastic filtering.
V. Stochastic differential equations, strong and weak solutions. Applications: Probabilistic solution of partial differential equations. Kakutani's theorem, Feynman-Kac formula.
VI. Ito-Clarck martingale representation. Application: option pricing in Black & Scholes market model.
Study materials
Old webpages
Stokastinen analyysi, kevät 2008
Stokastinen analyysi, syksy 2011
Schedule
During weeks 3-9 and 11-18 tu 12-14 B322, we 10-12 B120, and 2 hours of tutorials. First lecture on tuesday 15.1.
Easter holyday 28.3.-3.4.
Exams
You pass this course by solving exercises in the weekly tutorial sessions and writing and home exam.
Literature
Karatzas and Shreve Brownian motion and stochastic calculus, Second edition, 1998 Springer.
David Williams: Probability with Martingales (Cambridge Mathematical Textbooks).
Mörters and Peres: Brownian motion, Cambridge 2010.
Daniel Revuz ja Marc Yorin kirjaa "Continuous martingales and Brownian motion", 2nd edition Springer 2005
Dieter Sondermann: Introduction to Stochastic Calculus for Finance: A New Didactic Approach, Springer 2007
Timo Seppääinen: Basics of Stochastic Analysis, Lecture Notes, University of Wisconsin-Madison.
Richard Bass: Stochastic Processes, Cambridge 2011.
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Tutorials
Group | Day | Time | Class | Lecturer |
|---|---|---|---|---|
1. | thu | 10-12 | B322 | Dario Gasbarra |