Advanced Probability Theory

This course will cover advance topics in probability, including: the theory of martingales in discrete and in continuous time; Brownian motion and its properties, stochastic integration, ordinary and partial stochastic differential equations; Applications to optimal filtering, stopping, control, and finance; Continuous-time Markov chains, systems of interacting particles, relative entropy dissipation, notions of information theory; Electrical networks, random walks on graphs and groups, percolation