Ioannis Karatzas

Karatzas is truly passionate about teaching. In addition to his beautiful chalkboard calligraphy and funny, unexpected metaphors, his love for the material is apparent in every lecture. I highly recommend taking a course with him during your time here if you are interested in pure math and wish to have a rigorous foundation in probability and analysis. The course is also a great preparation to studying martingales and Brownian motion in Advanced Probability, which he also teaches.

WARNING: he does not give standard Columbia grades. Pathological exams. Decent, but overrated, lectures. He introduced new material for the final at the "optional problem solving session." On the friday before a monday exam, we were forbidden from emailing him. His TAs (who are also his PhD students) regularly skipped their office hours. He refused to even discuss the content type or range of material on the exams, and then complained that students kept asking him about the exam. Even by upper level math course standards the class was full of very smart people, but he apparently set the curve somewhere between a C and B-. I know someone who did all the work but failed.

Unless you're a genius, don't take a lower level class with this guy: you will suffer and NOT do well(unless you think a B or B+ is "superb"). He is the only professor I have ever emailed that told me not to email him. I think that should speak for itself.

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.

Stochastic process;markov chains martingale; Ito's calculus

Requires a massive time-commitment, and very very hard. Valuable class but not to be undertaken lightly 1 Columbia University: Arts & Sciences Fall 2021 Course: MATHGR6151_001_2021_3-ANALYSIS&PROBABILITYI : MATHGR6151_001_2021_3 - ANALYSIS & PROBABILITY I Instructor: Ioannis Karatzas

This class is super challenging. With that said, I think Karatzas is a good instructor and I think I learned a lot. I had taken real analysis I but not II. The pace is fast so you may feel like you don't fully understand some topics, but I also think that's just a product of the difficulty of the course. One nice thing about the course is that the beginning is much more typical proofy prob theory (e.g. DCT and Borel-Cantelli) but the later part switches to Markov chains which is a nice change of pace.

Karatzas is a brilliant man -- one of the only few math teachers who knows his stuff well enough to teach it well. The downside is that this is a very difficult class for people who are not used to rigorous theorem proving. Analysis was traditionally taught by Gallagher, who I've heard was so boring that nobody attended class. I guarantee that Karatzas is just the opposite. His teaching style is extremely engaging, and though keeping with mathematical rigor, he knows how to use intuition to get the message across. As for the competition in this class, be prepared to meet some of the smartest people on campus, since this is a required class for math majors. But that shouldn't deter you from taking an absolutely enlightening class. Just forget about pumping up your GPA -- take this class for your enjoyment of learning.

First half is advanced probability starting from product measure. Second half are seminars related to all aspects of stochastic analysis.

Markov Chains, OST, Martingales, Brownian Motion, Stochastic Calculus with application in Finance and Physics