Lecturer: Xue-Mei Li
Term(s): Term 1
Status for Mathematics students: List C
Commitment: 30 lectures
Assessment: 3-hour examination
Prerequisites: A willingness, even an enthusiasm, to work with random variables is the key prerequisite. No single module is a prerequisite. Earlier probability modules will be some use. The framework is measure theory, so it is a nice illustration of the ideas from MA359 Measure Theory, or ST342 Maths of Random Events, or ST318 Probability Theory. The content will also link with some content from modules on ODE's and PDEs. A student without any of the above would have to work hard.
The module complements the module MA4F7/ ST403 Brownian Motion.
We will introduce continuous time martingales, stochastic integration, and basic tools in stochastic analysis including Ito’s formula, various inequalities for local martingales and for stochastic integrals. We will also introduce stochastic differential equations and study their basic properties. Time permitting, we will also discuss completeness, strong completeness of SDEs and differentiation of probabilistic semi-groups.
Revuz and Yor, Continuous Martingales and Brownian Motion
Ikeda and Watanabe: Stochastic Differential Equations and Diffusion Processes