Stochastic Differential Equations - 6EC - M1

Prerequisites

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 https://staff.fnwi.uva.nl/s.g.cox/mtp_2016.pdf, in particular chapters 1-7 and in the notes about stochastic processes http://www.math.leidenuniv.nl/~spieksma/colleges/sp-master/sp-hvz1.pdf chapter 1 (1.1.-1.4), chapter 2 (2.1-2.2) and chapter 3 (3.1-3.3).

Aim of the course

This course is a theoretical course on stochastic analysis and in particular stochastic differential equations. Connections to partial differential equations will be discussed as well.

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 Girsanov’s change of measure theorem, connections to PDE’s and Feynman-Kac formula.

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      integrals
  • derive Itô's formula and use it in solving simple      stochastic differential equations
  • apply Girsanov’s change of measure theorem
  • derive the connection to PDE’s
  • derive and use the Feynman-Kac formula

Lecturer

W. Ruszel (TUD)

Minitests

At the beginning of each lecture there will be a minitest. If the tests are passed with 60% successfully the student can earn 0.5 bonuspoints which add to the endgrade for the course.