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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.
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