65-461 Stochastic Integration (in English)

Course offering details
Instructors: Sven Karbach
Event type:
Lecture
Displayed in timetable as:
M-VSP-V
Hours per week:
2
Language of instruction:
English
Min. | Max. participants:
- | -
Comments/contents: This course provides an introduction to stochastic calculus. More specifically, we will define and study stochastic integrals, i.e. integrals of stochastic processes with respect to continuous (semi-)martingales and learn how to do calculus with it. Indeed, we proof a stochastic version of the fundamental theorem of calculus called Ito's-formula and discuss several of its far reaching applications. We proceed to introduce and solve stochastic differential equations and present different concepts of solutions. We study the behavior of semi-martingales under absolutely continuous measure changes and have a look at martingale representation theorems. If time permits, we discuss the relation between stochastic differential equations and (deterministic) partial differential equations as given by the Feynman-Kac formula. Along the way we mention some applications in financial mathematics.

Keywords: martingales in discrete and continuous time, the Doob-Meyer decomposition, construction and properties of the stochastic integral, semi-martingales, Itô's formula, (Brownian) martingale representation theorem, Girsanov's theorem, stochastic differential equations, Feynman-Kac formula.
Prerequisites
Some courses along the lines of 'Measure theory' or 'Probability theory'. 

Aim of the course

The students should be able to explain the construction of stochastic integrals,
The students are familiar with optional sampling and stopping for martingales, 
The students can integrate with respect to semimartingales and apply Ito's formula,
The students can explain different concepts of solutions of stochastic differential equations and know how to apply absolutely continuous measure changes for semimartingales by means of Girsanov's theorem.
The students understand the idea of a probabilistic representation of solutions to PDEs,
The students are able to solve problems, where knowledge of the above topics is essential.
Literature: Recommended background reading:

I. Karatzas and S.E. Shreve, Brownian motions and stochastic calculus,
D. Revuz and M. Yor, Continuous martingales and Brownian motion.
Class session overview
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Instructors
Sven Karbach
Appointments
Date From To Room Instructors Appointments
1
Date Tue, 18. Oct. 2022
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
2
Date Tue, 25. Oct. 2022
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
3
Date Tue, 1. Nov. 2022
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
4
Date Tue, 8. Nov. 2022
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
5
Date Tue, 15. Nov. 2022
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
6
Date Tue, 22. Nov. 2022
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
7
Date Tue, 29. Nov. 2022
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
8
Date Tue, 6. Dec. 2022
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
9
Date Tue, 13. Dec. 2022
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
10
Date Tue, 20. Dec. 2022
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
11
Date Tue, 10. Jan. 2023
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
12
Date Tue, 17. Jan. 2023
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
13
Date Tue, 24. Jan. 2023
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
14
Date Tue, 31. Jan. 2023
To 12:15
To 13:45
Room Geom H3
Instructors Sven Karbach
Exams in context of modules
Module(start semester)/ Course Exam Date Instructors Compulsory pass
Ma-M-VSP_1 Advanced Stochastic Processes 1 (WiSe 22/23) / M-VSP-V Stochastic Integration (in English) 1 Oral exam Time tbd Sven Karbach Yes