Time: Mondays and Wednesdays 2:00-3:15 pm
Room: Courant Institute / Warren Weaver Hall 1302
Instructor: Professor Nina Holden, firstname.lastname@nyu.edu
Office hours: Mondays 3:15-5 pm, WWH 813
Course description: The course is targeted at Mathematics PhD students. Stochastic processes in continuous time. Brownian motion. Poisson process. Martingales and semimartingales. Stochastic integral. Stochastic differential equations. Markov processes. Connections with PDEs. Convergence of stochastic processes.
Text:
There will be no official textbook. The following are some useful books (those with hyperlink available through NYU):
Stochastic Processes by Bass
Brownian Motion, Martingales, and Stochastic Calculus by Le Gall
Stochastic Processes by Varadhan (Courant Lecture Series in Mathematics, volume 16)
Theory of Probability and Random Processes by Koralov and Sinai
Brownian motion and stochastic calculus by Karatzas and Shreve
Continuous Martingales and Brownian Motion by Revuz and Yor
Grading and problem sets: Weekly problem sets (due Wednesdays before class), a midterm exam and a final exam. The problem sets will be posted on Brightspace and should also be submitted there. No late submissions will be accepted. Homework counts for 50% of the final score, with the homework having the lowest score dropped from the computation. The final exam and the midterm exam each count 25% of the final score.
Prerequisites: Fluency with the material of the graduate course Probability Theory I is required. Some but not all of those topics are: measures, Lebesgue integration, various modes of convergence of random variables, characteristic functions, law of large numbers, central limit theorem, conditional expectations with respect to sigma-algebras, filtrations, sequences of random variables forming martingales or Markov chains, stopping times, inequalitites for martingales.