The aim of the course is to provide a sound knowledge of Brownian motion, the stochastic integral, Ito's calculus, and stochastic differential equations. During the course, some applications and connections with the analysis of partial differential equations will be illustrated.
Program:
Motivation. Stochastic processes (basics).
Recap on probability: notions of convergence, multivariate Gaussian distributions, conditional expectation.
Brownian motion: construction and fundamental properties.
Discrete and continuous time martingales.
Stochastic integral: construction and properties.
Itô calculus: Itô's formula, first applications (e.g. Dirichlet problem), Girsanov's theorem, martingale representation.
Stochastic differential equations: notions of existence and uniqueness, fundamental theorem of existence and uniqueness, examples, Markov property and diffusions, Feynman-Kac formula.
Program:
Motivation. Stochastic processes (basics).
Recap on probability: notions of convergence, multivariate Gaussian distributions, conditional expectation.
Brownian motion: construction and fundamental properties.
Discrete and continuous time martingales.
Stochastic integral: construction and properties.
Itô calculus: Itô's formula, first applications (e.g. Dirichlet problem), Girsanov's theorem, martingale representation.
Stochastic differential equations: notions of existence and uniqueness, fundamental theorem of existence and uniqueness, examples, Markov property and diffusions, Feynman-Kac formula.
- Teacher: Markus Fischer