A selection of the exercises below will be solved in class on the date indicated:
Problems for Thursday, August 27th. From the textbook:
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 Thursday, September 3rd.
From the textbook: Pages 20-22, problems 1.26, 1.28, 1.32, 1.33, 1.34, 1. 37, 1.38, 1.42, 1.43.
Problem 1 and 2 on this sheet.
From the textbook: Page 42: Exercise 2.2
Problems for Thursday, September 10th. From the textbook:
Extra problem (replaces 2.11 in the textbook)
From the textbook:
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
Problems for Thursday, September 17th. Fewer problems this week, but also harder ones:
From the textbook:
Page 56, Exercise 2.37
Page 84, problems 3.2, 3.3, 3.4, 3.5, 3.6
Problems for Thursday, September 24th.
Pages 95-96: 3.7, 3.13, 3.18, 3.19
As there are few problems for the first sections of Chapter 3, we go back to Chapter 2 to pick up some more:
Pages 64-65, problems 2.40, 2.41, 2,42, 2.43
Problems for Thursday, October 1st:
Page 120, Exercise 4.1. There are misprints in part (a) and (b). The statements should be: \(\liminf_n=\bigcup_{m=1}^{\infty}\bigcap_{j=m}^{\infty}\Lambda_j\) og \(\limsup_n=\bigcap_{m=1}^{\infty}\bigcup_{j=m}^{\infty}\Lambda_j\)
Page 126, Exercise 4.3
Page 128, problems 4.5, 4.7, 4.9
Problem 1 on the Exam 2019
Problem 5 on Trial Exam 3, 2019
Problems for Thursday, October 8th: Trying to give fewer problems this week as we have an assignment going on:
Page 126-128, problems 4.11, 4.13, 4.23, 4.28
Mandatory assignment 2019, Problem 1.
Problems for Thursday, October 15th:
Trial Exam 1, 2019), Problem 4
Page 143, Exercise 5.6
Page 145, Exercise 5.8
Page 147-148, Problems 5.9 (assume p>0), 5.10, 5.11, 5.12
There is a number of confusing misprint 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 \(S_k\).
Otherwise, the degree of difficulty varies a lot this week, but much should be manageable. I do think, however, that Exercise 5.12 should have had a hint or a part a): Before you attempt the problem, it is a good idea to show that if \(\lim_{n\to\infty}\lim_{n\to\infty}P[\sup_{n< k\leq m}|S_n-S_k|\geq\delta]=0\) for all δ>0, then the sequence \(\{S_n\}\)converges a.s. (think Cauchy sequences).
Problems for Thursday, October 22nd:
Page 161-162: Problems 6.2, 6.3, 6.4, 6.5, 6.6, 6.8
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). There is also 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)
Otherwise the problems this week do not seem too unreasonable; just remember that if two distributions have the same characteristic function, then they are equal.
Problems for Thursday, October 29th:
Page 162: Problems 6.9, 6.11, 6.12
Page 175: Problems 6.21, 6.22, 6.24
There are some misprints in Problem 6.11: In lines 5 and 6, \(f\) should have been \(\phi\), and in line 7, \(\phi(t)0)\) should have been \(\phi(t_0)\).
Problems for Thursday, November 5th:
Page 188: Problems 6.26, 6.29, 6.31
Page 199: Exercise 7.1
Page 211: Problems 7.14, 7.15, 7.16.
Problems for Thursday, November 12th:
Page 274: Problems 8.2, 8.3, 8.5, 8.6, 8.7
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 Thursday, November 19th, and the rest of the semester:
Pages 288-290: Problems 9.3, 9.5, 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 sequence 9.7-9.10 in class as I did in detail last year. The sequence is not particularly relevant for the exam, but it shows 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 \(E[Z_T]\).