Advanced Probability Theory · Mathematics
Basic topics: Conditional expectations; Total variation distance; Relative entropy; Martingale theory (definition, properties, decomposition, convergence, inequalities); Optional sampling; Harmonic function; Brownian motion (definition, properties, quadratic variation; characterization, Dambias-Dubins-Schwarz); Stochastic integration and stochastic differential equations (weak and strong solutions; existence and uniqueness); Diffusion; Semimartingale; Girsanov theorem. Advanced topics: Robbins-Monroe stochastic approximation; Gittins Whittle dynamic allocation problem; Snell optimal stopping; Stochastic control; The martingale problem of Stroock and Varadhan; Stock prices; Kalyan filter; Gradient flow and Langevin dynamics.
First half is advanced probability starting from product measure. Second half are seminars related to all aspects of stochastic analysis.
Discrete-time martingale, continuous-time martingale, stochastic calculus
Advanced course in probability theory, stochastic calculus, martingale theory, stochastic control etc.
Martingales, Stochastic Calculus and its applications (Stochastic Control, Optimal Stopping, a little Portfolio Theory). We used Øksendal as the primary reference.
Prof Karatzas is a very good instructor and very very knowledgeable in this field. Definitely take this course if you get the chance, but be warned that it can be fairly theoretical. 1 Columbia University: Arts & Sciences Fall 2023 Course: MATHGU4156_001_2023_3-ADVANCEDPROBABILITYTHEORY : MATHGU4156_001_2023_3 - ADVANCED PROBABILITY THEORY Instructor: Ioannis Karatzas
Discrete and continuous time martingales, stochastic processes, Brownian motion, Ito integration
A rigorous and beautiful introduction to the study of Martingales, Optimal Stopping, Brownian motion, Stochastic calculus, and Stochastic Differential Equations.
Stochastic Analysis with its applicatons
Stochastic processes - Conditional expectation and martingales, stopping time, convergence, Brownian motion, stochastic differential equations Applications - dynamic allocation problems, stochastic approximation, portfolio theory, optimal filtering, statistical mechanics (relative entropy, Wasserstein distance, Fischer information), Diffusion, etc.