Detailed course outline
The course is a sequence of modules that take several weeks each. Here is a tentative schedule.
- Brownian motion
- interpretation as homogeneous noise
- Markov property, transition probabilities
- digression on multi-variate Gaussians and linear algebra
- relation to the heat/diffusion partial differential equation
- dynamics of the probability density, conditional expectations
- the direction of time, well-posed and ill-posed PDE problems
- boundary conditions corresponding to hitting times
- solution using the method of images
- boundary conditions corresponding to hitting times
- probability flux
- simulation of Brownian motion, analysis of simulation data
- a finite difference method for the heat equation
- Ito calculus for Brownian motion
- adapted functions of Brownian motion, filtration (informal)
- the Ito integral, causal approximation
- almost sure convergence, Borel Cantelli lemma
- convergence of Ito integral approximation (informal)
- Ito's lemma
- martingales
- quadratic variation
- Ito isometry formula
- the Feynman Kac correspondence
- Diffusion processes
- the Ornstein Uhlenbeck process
- geometric Brownian motion
- backward equation and the generator
- forward equation, adjoint, probability dynamics
- Ito integral and Ito's lemma for general diffusions (informal)
- quadratic variation
- stochastic differential equations
- coupling to Brownian motion
- simulating general diffusions
- random walk and finite difference approximations
- Change of measure
- probability space and measure (informal)
- expectation and the probability integral (informal)
- probability density and likelihood ratio
- hypothesis testing, the Neyman Pearson lemma
- importance sampling
- change of measure/drift for Brownian motion
- Girsanov theorem for general diffusions
- rare event simulation
Academic integrity and behavior
Please feel free to ask questions and make relevant comments during class. If something is unclear to you, it probably is unclear to many other students.
Class time in graduate math courses is intense and limited. Students should arrive to class on time. Please do not use laptops or other devices during class except to take notes (this is rare) or for an emergency (more rare). Please refrain from talking that would distract others.
Please review the NYU academic integrity policy. The following specific policies apply throughout this class.
- Students may consult with each other while learning how to do exercises, but all written answers must be prepared individually. Students may not hand in work copied from outside sources or other students.
- Code may not be shared or copied from outside sources.
- If code or written answers show that one student has copied from another, both students will be penalized.
I (the instructor) will try to create an environment that does not encourage cheating. I will ensure that the work load is managable by an individual student working independently. I will work with the grader and TA to identify cheating. I will listen to anyone's thoughts or complaints on this issue. Please let me know if the work load is unmanageable or if you suspect others of cheating.