planar Brownian motion


The goal is to learn stochastic integration with respect to right continuous semimartingales in continuous time.

Topics : Isonormal Gaussian process and Paul Levy Construction of Brownian motion. Wiener integral. Processes with jumps: Poisson process and counting processes, random measures, Palm measure, Levy processes. General theory of processes in continuous time, filtration, predictable σ-algebra, stopping times and predictable times. Stochastic integrals with respect to processes with locally finite variation, predictable and dual predictable projection, compensator and Doob-Meyer decomposition,Continuous time martingales, existence of right continuous modification, locally square integrable martingales, quadratic and predictable variation. Stochastic integration with respect to continuous martingales, Kunita-Watanabe inequality, Ito isometry and Ito integral, Ito formula and generalizations. Burkholder Davis Gundy inequalities. Ito-Tanaka formula and local times. Change of measure and Girsanov theorem. Stochastic integral representation of local martingales.Stochastic differential equations, weak and strong solutions. Partial differential equations and Feynman-Kac formula. Applications: stochastic filtering, option pricing in mathematical finance.

This course will continue as Stochastic analysis II in the IV period, these contents are for both course periods (10 credits)

11.12.2017 klo 09:00 - 1.3.2018 klo 23:59


Tästä osiosta löydät kurssin opetusaikataulun. Tarkista mahdolliset muut aikataulut kuvauksesta.

Ke 17.1.2018
12:15 - 14:00
To 18.1.2018
12:15 - 14:00
Ke 24.1.2018
12:15 - 14:00
To 25.1.2018
12:15 - 14:00
Ke 31.1.2018
12:15 - 14:00
To 1.2.2018
12:15 - 14:00
Ke 7.2.2018
12:15 - 14:00
To 8.2.2018
12:15 - 14:00
Ke 14.2.2018
12:15 - 14:00
To 15.2.2018
12:15 - 14:00
Ke 21.2.2018
12:15 - 14:00
To 22.2.2018
12:15 - 14:00
Ke 28.2.2018
12:15 - 14:00
To 1.3.2018
12:15 - 14:00

Muu opetus

17.01. - 28.02.2018 Ke 10.15-12.00
Dario Gasbarra
Opetuskieli: englanti


Course book:
Richard Bass, Stochastic processes, Cambridge University Press 2011.
Recommended reading:
Fabrice Baudoin, Diffusion Processes and Stochastic Calculus. European Mathematical Society Ems Textbooks in Mathematics 2014.
Alexander Gushchin, Stochastic calculus for quantitative finance. ISTE Press, Optimization in insurance and finance 2015.
René L Schilling Lothar Partzsch, Brownian motion, an introduction to stochastic processes, De Gruyter 2012.
Sheng-wu He, Jia-gang Wang, Jia-an Yan, Semimartingale Theory and Stochastic Calculus, CRC 1992.
Jean Jacod and Albert Shiryaev, Limit theorems for stochastic processes, 2nd edition Springer 2003.
Hui-Hsiung Kuo, Introduction to stochastic analysis, Springer 2006.
Jean-François Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer 2016.
Mörters and Peres, Brownian motion, Cambridge 2010.
Ashkan Nikeghbali, An essay on the general theory of stochastic processes, Probability Surveys Vol. 3 (2006) 345-412.
Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 2nd edition Springer 2005.



Kurssin suorittaminen

The course is passed by solving the weekly assignments and by writing an home exam.


Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

Probability theory I,II, and their prerequisites

Stochastic calculus for semimartingales

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

The course is focused on stochastic integration theory, with respect to martingales and processes with finite variations, including continuous martingales and processes with jumps.

Lecture notes, Bass "Stochastic Processes" CUP

Exam, other methods will be described later