Part A integration, B8.1 Martingales Through Measure Theory and B8.2 Continuous Martingales and Stochastic Calculus, is expected.
Stochastic differential equations have been used extensively in many areas of application, including finance and social science as well as in physics, chemistry. This course develops the theory of Itô's calculus and stochastic differential equations.
Prof. Harald Oberhauser
The student will have developed an appreciation of stochastic calculus as a tool that can be used for defining and understanding diffusive systems.
Recap on Brownian motion, quadratic variation, Ito's calculus: stochastic integrals with respect to local martingales, Ito's formula.
Lévy's characterisation of Brownian motion, exponent and Cameron-Martin martingales, exponential inequality, Burkholder-Davis-Gundy inequalities, Girsanov's Theorem, the Martingale Representation Theorem, Dambis-Dubins-Schwarz.
Local time, motion and Tanaka's formula.
Stochastic differential equations: strong and weak solutions, questions of existence and uniqueness, diffusion processes. Discussion of the one-dimensional case, a comparison theorem. Numerical schemes.
Conformal invariance of Brownian motion.