This is a
BMS course that will be taught in English to facilitate
participation of international students.
Content:
Gaussian processes; white noise; Brownian motion and its path
properties; filtrations and stopping times;
continuous time martingales; continuous semimartingales;
quadratic variation; stochastic integration; Itô’s
formula; Burkholder’s inequality; change of measure; martingale
representation; stochastic differential
equations: existence and uniqueness, Markov property, link with
partial differential equations.
Prerequisites:
Analysis I
and II, Stochastics I and II. Recommended: Analysis III and
basic knowledge of Functional Analysis.
Current information: - An updated version of the lecture notes is
available: Lecture notes (Without proofs. Version of 01-08-2016) . - Solution for Sheet 13 is available: Solution . - Solution for Sheet 10, exercise 10.2 is available: Solution . - Solution for Sheet 9, exercise 9.2 is available: Solution . - Solution for Sheet 8, questions 8.1.b and 8.1.c is available: Solution . - A new reference has been added: Lecture notes of P. Priouret .
- I. Karatzas and S. Shreve. Brownian motion and
stochastic calculus. 2nd ed. Graduate Texts in
Mathematics, 113. New York etc.: Springer-Verlag (1991)
- D.
Revuz and M. Yor. Continuous martingales and Brownian
motion. 3rd ed.
Graduate Texts in Mathematics, 293. Berlin: Springer
(1999)
- P. Mörters and Y.
Peres. Brownian Motion. 1st ed. Cambridge Series in
Statistical and Probabilistic Mathematics. Cambridge
University Press (2010)