Problems for Wednesday, August 31st. From the textbook:
Page 4, exercise 1.2 (added August 23rd)
Page 8, problems 1.3, 1.5, 1.6
Pages 13-14, problems 1.8, 1.9, 1.16, 1.18, 1.21, 1.22, 1.23
Problems for Wednesday, September 7th.
From the textbook: Pages 20-22, problems 1.26, 1.32, 1.33, 1.34, 1. 37, 1.38, 1.42, 1.43.
Problem 1 and 2 on this sheet.
Problems for Wednesday, September 14th.
Pages 46-48, problems 2.4, 2.7, 2.10, 2.16, 2.17
Page 49, exercise 2.20
Page 53, problems 2.22, 2.30
Page 56, exercise 2.37
Problems for Wednesday, September 21st.
Pages 64-65, problems 2.40, 2.41, 2,42, 2.43
Page 84, problems 3.2, 3.3, 3.4, 3.5, 3.6
Problems for Wednesday, September 28th.
Pages 95-96: 3.7, 3.13, 3.18, 3.19
Page 120, Exercise 4.1. There are misprints in part (a) and (b). The statements should be: \(\liminf_n\Lambda_n=\cup_{m=0}^{\infty}\cap_{j=m}^{\infty}\Lambda_j\) and \(\limsup_n\Lambda_n=\cap_{m=0}^{\infty}\cup_{j=m}^{\infty}\Lambda_j\)
Problem 1 and 2 from this problem set.
Problems for Wednesday, October 5th:
Page 128, problems 4.7, 4.9, 4.11, 4.13, 4.23
Problem 1 from the Mandatory Assignment 2020
Problem 1 on the Exam 2019
Problem 5 on Trial Exam 3, 2019
Problems for Wednesday, October 12th: Fewer problems this week as we have an assignment going on (but not as little as it may look, and some of the problems are quite hard!)
Page 147-148, Problems 5.9 (assume p>0), 5.10, 5.11, 5.12
Notes: There is a number of confusing misprints in problem 5.11: In a), \(S_n\) should be \(S_k\), in d), \(X_k^2\) should be \(S_k^2\), and in e), the first occurrence of \(S_n\) should be \(S_k\).
Here's a hint for Exercise 5.12: Before you attempt the problem, it is a good idea to show that if \(\lim_{n\to\infty}\lim_{m\to\infty}P[\sup_{n<k\leq m}|S_n-S_k|\geq\delta]=0\) for all \(\delta>0\), then the sequence \(S_n\) converges a.s. (think Cauchy sequences).
Problems for Wednesday, October 19th:
Page 143, Exercise 5.6
Page 145, Exercise 5.8
Page 161-162: Problems 6.2, 6.3, 6.4, 6.5
There is a misprint in 6.4; the right hand side of the formula should be \(e^{\frac{z^2\sigma^2}{2}+z\mu}\) - plus instead of minus in the last term (and don't make this problem into a competition in contour integration if you don't want to; a hand waiving argument suffices).
Problems for Wednesday, October 26th:
Page 161-162, Problems 6.6, 6.8, 6.9, 6.12
Page 175: Problems 6.21, 6.22
There is a condition lacking in 6.6a): The \(X_n\)'s should be assumed to be independent (Extra problem: Show by example that the statement is false without this extra condition).
Problems for Wednesday, November 2nd:
Page 188: Problems 6.26, 6.29
Problems for Wednesday, November 9th:
Page 192: Exercise 7.1 (There is a misprint in the hint: The reference should be to Corollary 1.8)
Page 211: Problems 7.14, 7.15, 7.16.
Exam 2019, Problem 4 (this is a different take on Problem 6.12)
Problems for Wednesday, November 16th:
Page 274: Problems 8.2, 8.3, 8.5, 8.6, 8.7
Pages 288-290: Problems 9.2, 9.5
Comment: Problems 8.2, 8.3, 8.5, and 8.6 can be done quite quickly if you have the right approach. In 8.7 the way of attack should be clear, but you have to juggle the limits the right way.
Problems for Wednesday, November 23rd: Optional problems: 9.7, 9.8, 9.9, 9.10 (for the notation U(0,1) in problem 9.10, see page 89).
I shall not do the problems above in class as they are not particularly relevant for the exam, but they show an important application of martingales and stopping times to stochastic optimization theory. Note that there is a misprint two lines above the start of Problem 9.7: \(E[X_T]\)should have been \(E[Z_T]\).
"Real" problems: