I'm currently finishing up my PhD in mathematics (as of Fall 2021), specifically in probability theory. Unsurprisingly, I have found that teaching helps my learning process. So, I create some lecture content every now and then. I intend to produce a full 2 year sequence worth of material following this tentative progression:
Course 1 - Probability space, random variables, expectation, variance, independence, laws of large numbers, convergence in distribution, characteristic functions, central limit theorems,
Course 2- discrete time stochastic processes including: optional times, stopping times, martingales, markov chains
Course 3-continuus time stochastic processes (same topics in discrete time, extended to continuous-time case), brownian motion, stochastic integration
Course 4- SDEs and further topics