Measure Theoretic Probability (Mastermath course)
Aim of the course
The aim of this course is to provide an introduction to stochastic calculus. More concretely, we will define stochastic integrals: i.e., we define the integral of a stochastic process with respect to a continuous (semi-)martingale. We proceed to introduce and solve stochastic differential equations (which are actually stochastic integral equations). Finally, we discuss the relation between stochastic differential equations and (deterministic) partial differential equations as given by the Feynman-Kac formula. Along the way we mention some applications in financial mathematics: indeed, stochastic differential equations are widely used as models for financial products.
For the experts, some keywords are: martingales in discrete and continuous time, the Doob-Meyer decomposition, construction and properties of the stochastic integral, semi-martingales, Itô's formula, (Brownian) martingale representation theorem, Girsanov's theorem, stochastic differential equations, Feynman-Kac formula.
Asma Khedher (first half) and Sonja Cox (second half)