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 is a continuation of Stochastic analysis I lectured in the III period, these contents are for both course periods (10 credits)

12.2.2018 klo 12:00 - 2.5.2018 klo 23:59


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

Ti 13.3.2018
10:15 - 12:00
Ma 19.3.2018
10:15 - 12:00
Ti 20.3.2018
10:15 - 12:00
Ma 26.3.2018
10:15 - 12:00
Ti 27.3.2018
10:15 - 12:00
Ma 9.4.2018
10:15 - 12:00
Ti 10.4.2018
10:15 - 12:00
Ma 16.4.2018
10:15 - 12:00
Ti 17.4.2018
10:15 - 12:00
Ma 23.4.2018
10:15 - 12:00
Ti 24.4.2018
10:15 - 12:00
Ma 30.4.2018
10:15 - 12:00

Muu opetus

14.03. - 02.05.2018 Ke 10.15-12.00
Dario Gasbarra
Opetuskieli: englanti





Optional course.

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

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

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

Stochastic differential equations and applications

Lecture notes; Bass "Stochastic Processes" CUP

Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later