Ok, I will do that. I sure my specialization modules that I took during my master will be useful for me when doin my PhD in Mathematical Finance. Those specialization modules are:
Stochastic Processes
The module covers the following topics: Weak convergence, compactness criteria, Markov processes: recurrence and transience, harmonic functions, stationary processes, ergodic theorem for Markov chains, stochastic processes in continuous time: renewal processes, Poisson process, Levy processes, Brownian motion, Donsker's invariance principle, Martingales and stopping times in continuous time, stochastic integral with Brownian motion as integrator, Ito formula. Students learn to construct mathematical models and to analyze complex random phenomena, especially with time dependencies.
Probability Theory
The module discusses supplements to measure theory, Borel-Cantelli, 0-1-law, additions to the laws of large numbers and to the central limit theorem, large deviations, law of iterated logarithm, conditional expectations and stochastic kernels, martingales, limit theorems. The module conveys familiarity with the measure theoretic construction of probability theory and the fundamental limit theorems as well as the ability to understand advanced topics in stochastics.
Stochastic Integration and Stochastic Differential Equations
The module covers the following topics: Doob inequalities, Doob-Meyer decomposition in continuous time, quadratic variation and covariation, Ito isometry and stochastic integral with semimartingales as integrator, Ito formula in the general case, Stratonovich integral, Ito calculus, stochastic treatment of parabolic and elliptic PDEs, Levy's theorem, random time changes in stochastic integrals. Further topics are change of measure: Girsanov's theorem, white noise, stochastic differential equations: existence and uniqueness of strong solutions, weak solutions. The module conveys familiarity with the methods of modern stochastic analysis, especially in the analysis of stochastic phenomena in continuous time.
Stochastic Processes in Physics and Biology
The module covers the following topics: Markov chains and population genetics, branching processes, continuous time Markov processes and molecular motors, gene regulation, rate equations, Master equation and Fokker-Planck equation, Kramers-Moyal expansion, Smoluchowski equation, phase separation kinetics, Langevin equations and non-equilibrium growth processes, diffusion limited aggregation, directed percolation, diffusion-reaction models, linear response theory, Onsager relations, mode-coupling theory and glass transition. The module aims to convey fundamental abilities in modeling and analyzing complex biological systems, using the methods of physics.
Soft Condensed Matter Physics and Critical Phenomena
The module covers the following topics: Mean-field theory, field theories, critical phenomena and renormalization group, generalized elasticity (XY model, liquid crystals, gels), hydrodynamics, topological defects, walls, kinks and solitions, response theory and nonequilibrium thermodynamics. The module aims to convey a fundamental understanding the collective phenomena occurring in macroscopic particle systems in condensed matter.
Mathematical Quantum Mechanics
This course introduces the basic elements of mathematical quantum mechanics. First the fundamentals of quantum mechanics and the measurement process (EPR-paradox and Bell inequality) and the mathematical basics of unbounded and self-adjoint operators (domain of definition, graphs, adjoints, spectrum, criteria for self-adjointness, spectral theorem, quadratic forms) will be discussed. then Coulomb-Schrödinger operators, the essential spectrum, invariance under compact perturbations and the minimax principle will be presented. This is followed by elements of the theory of many-particle systems (density functional theory, second quantization, fundamentals of quantum field theory) and its applications (e.g. Hartree-Fock approximation, superconductivity and superfluidity). At the end the basics of scattering theory (one-particle problems, the existence of wave operators) will be discussed.
Mathematical Statistical Mechanics
The module covers the following topics: Gibbs measures: DLR conditions, existence, uniqueness (Dobrushin's theorem), phase transitions, absence of spontaneous symmetry breaking in two dimensions. Ising model: high temperature phase, Peierls argument, cluster expansion, Fortuin-Kasteleyn representation, FKG inequality, spontaneous symmetry breaking in continuous models. Non-equilibrium model systems: Exclusion processes, matrix product ansatz, interacting particle systems. The main goal of this module is to acquire a deeper mathematical and physical understanding of phase transitions and collective phenomena that occur in macroscopic interacting particle systems.
