Foundations of Stochastic Analysis (2020-21)

Course Term: Michaelmas
Course Overview:

The course surveys the basics of stochastic calculus as a preparation for developments in the Michaelmas term courses
Prerequisites
It will be assumed that students have a good understanding of probability and measure and at least a first course in stochastic processes

Course Syllabus:

1. Continuous martingales and Brownian motion
Basic theorems and properties of continuous martingales and continuous local martingales. Properties of Brownian motion.
2. Stochastic integration
Construction of the stochastic integral,L2theory and extension to local martingales and semi-martingales. Ito’s formula, all in the setting of continuous martingales.
3. Useful theorems
Dambanis-Dubins-Schwarz, Girsanov, Martingale Representation4: SDEs classical existence and uniqueness results
Strong and weak solutions. Pathwise uniqueness and uniqueness in law. Existence and uniqueness of strong solutions via picard
5. Martingale problems
Connections between SDEs and generators; weak solutions to SDEs through martingale problems

Lecturer(s):

Prof. Ben Hambly
Learning Outcomes:

This is an 8 hour course held in the first two weeks of the CDT in Random Systems