HOMEWORK
Homework is from the textbook by Durrett.
Homework is due in class on Wednesdays.
Please hand in the starred problems.
Homework 1, due Wed Jan 14, 2015:
2.1.18, 3.1.2, 3.2.9, 3.2.11, *3.2.12, 3.2.13, *3.2.14, *3.2.15 (hint:
use strong law of large numbers).
For solutions to the starred problems on homework assignment 1, click here.
(You will need the username and
password given out in class to access these.)
Homework 2, due Wed Jan 28, 2015:
2.1.8, *3.3.2(i),
3.3.4, 3.3.5, *3.3.7, 3.3.8, 3.3.9, 3.3.10, *3.3.14.
(A hint for 3.3.14 will be given in class.)
For solutions to the starred problems on homework assignment 2, click here.
Homework 3, due Wed Feb 4, 2015:
3.3.23, *3.3.28, 3.4.1, *3.4.2, 3.4.5, 3.4.6, *3.4.7.
(There appears to be a typo in 3.3.28. In the displayed equation (with two
equalities), C lambda in two places should be C/lambda. This does not affect the main conclusion of the problem.)
For solutions to the starred problems on homework assignment 3, click here.
Homework 4, due Wed, Feb 11, 2015:
5.1.1, 5.1.6, 5.1.8, *5.1.9, *5.1.10, *5.1.11.
For solutions to the starred problems on homework assignment 4, click here.
Homework 5, due Wed Feb 25, 2015:
5.2.2, *5.2.5, *5.2.6, 5.2.8, *5.2.9, 5.3.13 (read the Galton-Watson example as well).
Extra problem: Suppose {G(n), n=0,1,2,...} and {F(n),n=0,1,2,...}
are filtrations wiht G(n) contained in F(n) for each n.
Supppose that {M(n), n=0,1,2,...} is a martingale with respect to {F(n),n=0,1,2, ...} and suppose M(n) is measurable with respect to G(n) for each n.
Prove that {M(n),n=0,1,2,...} is a martingale with respect to {G(n),n=0,1,2,...}. Prove that the same is true if "martingale" is replaced with "submartingale"
(or "supermartingale") in the above.
Use this to show that if
{M(n), F(n), n=0,1,2,...} is a (sub/super)martingale and T is a stopping time, then
{M(min(n,T)), F(min(n,T)), n=0,1,2,...} is a (sub/super)martingale.
For solutions to the starred problems and the extra problem on homework assignment 5,
click here.
Homework 6, due March 11, 2015.
5.4.4, 5.4.6, *5.4.8, *5.5.2, *5.5.8
For solutions to the starred problems,
click here.