Real Analysis, Probability, Stochastic Processes, Measure Theory, in particular knowledge about:

-          Sigma algebras and measures

-          Lebesgue integration and Stietjes integrals

-          Product spaces, L_p spaces of random variables

-           Radon-Nikodym theorem

-          Convergences of random variables, uniform integrability

-          Borel-Cantelli lemmas

-          Stochastic processes, Markov processes

-          Discrete time martingales

More background information can be found in the lecture notes about measure theoretic probability, in particular chapters 1-7 and in the notes about stochastic processes chapter 1 (1.1.-1.4), chapter 2 (2.1-2.2) and chapter 3 (3.1-3.3).

Aim of the course

After a brief survey of some basic results from Measure Theoretic Probability Theory, the concept of a martingale is introduced and studied, first in discrete time and then in continuous time. The main example in continuous time is the Brownian motion process. After these preparations we turn to the development of the Itô stochastic calculus. The Itô isometry and the Itô formula are derived. We will also introduce general semi-martingale theory. The theory is applied to obtain solutions of certain classes of stochastic differential equations. We will treat Girsanovs change of measure theorem.

After successfully finishing this course, the student is able to:

  • work with random walks, Brownian motion, local martingales and semimartingales
  • derive and use the properties of (local) martingales to analyze paths of stochastic processes
  • derive and use the properties of stochastic integral
  • derive Itô's formula and use it in solving simple stochastic differential equations

Rules about Homework/Exam

Grade is determined on the basis of the written exam only. No homework.

Lecture Notes/ Literature

For the lectures on SDEs we use the most material coming from D. Revuz and Y. Yor "Continuous martingales and Brownian motion".

The following lecture notes are also an excellent introduction to SDEs:

For an introduction to measure theoretic probability see: or the book by Probability with martingales by David Williams. A very advanced book is: Foundations of modern probability by Kallenberg

The required measure theory can be found in many books. The following lecture notes could be a helpful start if you did not do a course on measure theory: Lecture notes on measure theory by Mark Veraar Summary of measure theory. A more complete book on measure theory is for instance: Measure and Integration Theory by Bauer.


W. Ruszel (TUD) &  P. Mandal (UT)