Math 545 Syllabus

This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.


Undergraduate background in real analysis (Math 431) and probability (Math 230 or 340)


  • 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


  • 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