This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.
Prerequisites
Undergraduate background in real analysis (Math 431) and probability (Math 230 or 340)Syllabus
- Brownian Motion and Stochastic Processes, Martingales
- Construction and properties of Brownian motion, Kolmogorov extension theorem, continuity theorem
- Ito Integrals with respect to Brownian motions and Ito processes
- Ito’s Formula and Quadratic Variation
- Stochastic Differential Equations: Existence and Uniqueness, Weak and strong solutions, Markov Property
- PDEs and SDE: Infinitesimal generators, optional stopping, stoping times, localization
- Connection with PDEs: Forward and Backward Kolmogorov equation, Dirichlet Problems and hitting probabilities, Poisson Equations and Feynman-Kac formula
- Levy-Doob theorem, martingales representation theory
- Kolmogorv-doob like inequalities for martingales
- Girsonov’s Theorem, Time changes
- One Dimensional Feller processes: reachable and unreachable boundary points, nature scale, speed measures
- Bessel process
- Invariant measures
- Tanaka formula
- Examples of SDE models
References
- Fima C. Klebaner, Introduction to stochastic calculus with applications
- Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus
- Richard Durrett, Stochastic calculus
- Bernt Øksendal, Stochastic differential equations
- H. P. McKean, Stochastic integrals
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion