Stochastic analysis (MATS352, 5 cr)
Spring 2014
The course will be given in
Finnish
or in English, depending on the participants.
Information and registration in the Korppi system:
https://korppi.jyu.fi/kotka/r.jsp?course=156335.
Contents
Brownian motion, Poisson random measure,
stochastic integral, Itô formula.
Times and places
Lectures: 24 hours, starting on 17th March, 2014
Mondays at 12.15-14 and Tuesdays at 8.15-10 in MaD380
Exercises: 12 hours, starting on 24th March, 2014
Mondays at 10.15 in MaD380
Notice:
Week 16 no teaching;
moreover, on Monday, 21st April,
no lecture nor exercise session (Easter Monday);
on week 17, exercise session is on Tuesday 22nd April
after the lecture (that is, at 10.15) in MaD381
Examination on Wednesday, 14th May, 2014
Topics for lectures
- 17.3.2014:
Written assignment and other practical information,
continuous time stochastic process
- 18.3.2014:
Filtration, martingale
- 24.3.2014:
Brownian motion: introduction, definition, existence (part 1)
- 25.3.2014:
Gaussian processes, existence
- 31.3.2014:
Gaussian processes, existence (end of the proof);
continuity condition
- 1.4.2014:
Existence of BM (part 2), Hölder continuity of BM
- 7.4.2014:
Non-differentiability of BM, p-variation.
Stochastic integral (part 1)
- 8.4.2014:
Stochastic integral (part 2: simple integrands, Itô isometry)
- 22.4.2014:
Stochastic integral (part 3: space of square-integrable processes,
metric, simple processes dense), more on 2-variation
- 28.4.2014:
Stochastic integral (part 4: extension from simple to
square-integrable processes, some properties)
- 29.4.2014:
Itô's formula
- 5.5.2014:
Lévy processes, Poisson random measure, discussion
Exercises
Session 1
Session 2
Session 3
Session 4
Session 5
Session 6
Literature
- S. Geiss: Stochastic Processes in continuous time,
available
here.
- S. Geiss: Stochastic differential equations,
available
here.
- I. Karatzas & S. E. Shreve:
Brownian Motion and Stochastic Calculus, Springer.
- S. Geiss: Stochastic processes in discrete time,
Lecture notes 62.
- P. Mörters, Y. Peres: Brownian motion, Cambridge University
Press. See also reading seminar.
- D. Appelbaum: Lévy processes and stochastic calculus,
Cambridge University Press.
- P. Billingsley: Convergence of Probability Measures, Wiley.
- A. Gut: Probability: A Graduate Course, Springer.
- O. Kallenberg: Foundations of Modern Probability, Springer.
- P. A. Meyer: Probability and Potentials, Blaisdell.
- D. Revuz & M. Yor:
Continuous Martingales and Brownian Motion, Springer.
- A. N. Shiryaev: Probability, Springer.