MTH-210.4 Advanced Probability Theory
Syllabus
1. Martingale Theory (Conditional Expectation, Optional Stopping Theorem, Doob's inequality, Convergence Theorems)
2. Markov Chains (Construction, stationary measure, convergence to equilibrium, ergodic theorem, recurrence and transience, application to random walks )
3. Continuous time stochastic processes (Brownian motion, Kolmogorov consistency and continuity theorem, sample path properties, first passage times, reflection principle, Markov property)
4. Introduction to stochastic calculus (Stochastic integrals, Ito formula, SDEs, connection to PDEs).