# Category Archives: Uncategorized

## HW5. MA3160 Fall 2017

Exercise 1. Three balls are randomly chosen with replacement from an urn containing 5 blue, 4 red, and 2 yellow balls. Let X denote the number of red balls chosen. (a) What are the possible values of X? (b) What … Continue reading

## MA3160. Fall 2017. Midterm 1 sample

Practice midterm 1   We will do the correction in class on 09/28.

## Lecture 6. Rough paths Fall 2017

In the previous lecture we defined the Young’s integral when and with . The integral path has then a bounded -variation. Now, if is a Lipschitz map, then the integral, is only defined when , that is for . With … Continue reading

## MA5311. Take home exam

Exercise 1. Solve Exercise 44 in Chapter 1 of the book. Exercise 2.  Solve Exercise 3 in Chapter 1 of the book. Exercise 3.  Solve Exercise 39 in Chapter 1 of the book. Exercise 4. The heat kernel on is given by . By … Continue reading

## MA5161. Take home exam

Exercise 1. The Hermite polynomial of order is defined as Compute . Show that if is a Brownian motion, then the process is a martingale. Show that   Exercise 2. (Probabilistic proof of Liouville theorem) By using martingale methods, prove that if … Continue reading