Time: Tuesdays 1:25-3:15 pm
Room: Courant Institute / Warren Weaver Hall 1302
Instructor: Professor Nina Holden, firstname.lastname@nyu.edu
Office hours: Thursdays 2-4 pm, WWH 813
Course description: The course is targeted at Mathematics PhD students. Stochastic processes in continuous time. Brownian motion. Poisson process. Processes with independent increments. Stationary processes. Semi-martingales. Markov processes and the associated semi-groups. Connections with PDEs. Stochastic differential equations. Convergence of processes.
Text:
There will be no official textbook. Some useful books are (those with hyperlink available through NYU):
Stochastic Processes by Bass
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
Grading and problem sets: Weekly problem sets in addition to a take-home 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 70% of the final score, with the homework having the lowest score dropped from the computation. The final exam will count for 30% 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. We will also need elements of functional analysis and real analysis.