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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

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MA3160. Fall 2017. Midterm 1 sample

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

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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

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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

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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

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MA5311. Take home exam due 03/20

Solve Problems 1,2,8,9,10,11 in Milnor’s book. (Extra credit for problem 6)

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MA5161. Take home exam. Due 03/20

Exercise 1. Let . Let be a continuous Gaussian process such that for , Show that for every , there is a positive random variable such that , for every and such that for every , \textbf{Hint:} You may use … Continue reading

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