Course Title

Introduction to Stochastic Differential Equations (in English)

Course number

3341.352

Instructor

Gerald Trutnau

E–Mail

[email protected]

Course homepage http://www.math.snu.ac.kr/∼ trutnau/teachingSDE2013.html Course Objective

The purpose of this course is to introduce the reader to the basic ideas and results of Stochastic Calculus up to the point that he can acquire a sufficient knowledge for the understanding of its role in applications.

References

There will be no main textbook. Here are some references: - Chung, K.L., Williams R.J.: Introduction to stochastic integration, second ed., Probability and its Applications, Birkh¨auser Boston Inc., MA, 1990. - Durrett, Richard: Stochastic calculus. A practical introduction. Probability and Stochastics Series. Boca Raton, FL: CRC Press. - Karatzas, Ioannis; Shreve, Steven E.: Brownian motion and stochastic calculus. 2nd ed. Graduate Texts in Mathematics, Springer-Verlag. - Kuo, Hui-Hsiung: Introduction to Stochastic Integration, Universitext, Springer, 2006. - Revuz, Daniel; Yor, Marc: Continuous martingales and Brownian Motion. 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Berlin: Springer.

Description

Stochastic Calculus is in general considered as a course in the 4th academic year and is usually taught after two preceding probability theory courses (1 year). For example the reader is usually assumed to be familiar with martingale theory and conditional distributions and expectations on the probabilistic side but also with advanced measure theory as well as partial differential equations on the analytic side. However, we will try to present an understandable introduction focusing on concepts as well. We shall provide proofs or outlines of proofs whenever it seems to be possible without too much advanced theoretical machinery or whenever necessary for the illumination of a concept. The student is assumed to have some reasonable background in advanced calculus, measure theory, and basic probability theory, though nonetheless we will always repeat and introduce at least the corresponding statements.

Tentative content

(not necessarily in chronological order + there will also be an updating) Review of Probability, Pathwise Itˆo-Calculus, Kolmogorov’s construction and properties of Brownian motion, continuous time martingale theory and stochastic integration, d-dimensional Itˆo formula and covariation, applications of Itˆo’s formula, Girsanov formula and Novikov’s condition, stochastic differential equations, weak and strong solutions, existence and uniqueness of solutions, one dimensional diffusions, linear stochastic differential equations, connections to partial differential equations, etc.

Teaching Method

Lecture, exercises.

Evaluation

- Attendance (10 % of final score). - Assignment sheets (40 % of final score); Students must solve exercises regularily, and will be given assignment sheets mostly every week. - Midterm (8th week, 75 minutes, 25 % of final score); - Final exam (15th week, 75 minutes, 25 % of final score);