练习
Show that if then and sum over to get a second proof of .
证明
两面取期望得到 然后再归纳即可。
练习
Generalize the proof of Theorem 4.4.1 to show that if is a submartingale and are stopping times with then .
证明
令
为可预测策略，那么
是下鞅，从而
练习
Suppose are stopping times. If then
is a stopping time.
证明
有 只要证明
和
都在
中。 对于
，由于
故
. 对于
，在上面一定有
，故 其中
，从而等号右侧
.
练习
Use the stopping times from the previous exercise to strengthen the conclusion of Exercise 4.4.2 to .
证明
由 是停时知道 是下鞅。
练习
Prove the following variant of the conditional variance formula. If then
练习
Suppose in addition to the conditions introduced above that and let . Exercise 4.2.2 implies that is a martingale. Use this and Theorem 4.4.1 to conclude
练习
The next result gives an extension of Theorem 4.4.2 to . Let be a martingale with and . Show that
Hint: Use the fact that is a submartingale and optimize over .
练习
Let be a submartingale and .
Prove this by carrying out the following steps:
(i) Imitate the proof of 4.4.2 but use the trivial bound
for
to show
(ii) Use calculus to show .
练习
Let and be martingales with and .
练习
Let , be a martingale and let for . If then a.s. and in .
练习
Continuing with the notation from the previous problem. If and then a.s. In particular, if and then a.s.