In this post, we prove some fundamental martingale inequalities that, once again, are due to Joe Doob

**Theorem (Doob’s maximal inequalities)** *Let be a filtration on probability space and let be a continuous martingale with respect to the filtration .*

- Let and . If , then we have
- Let and . If , then we have

**Proof:**

Let and . If then, from Jensen’s inequality the process is a submartingale. Let and

with the usual convention that . It is seen that is an almost surely bounded stopping time. Therefore, from the Doob’s stopping theorem

But from the very definition of ,

.

which implies,

This concludes the proof of the first part of our statement.

Let now and .

Let us first assume that:

,

The previous proof shows that for ,

We deduce,

From Fubini’s theorem,

Similarly, we obtain

Hence,

By using now Hölder’s inequality we obtain,

which implies

As a conclusion if , we have:

Now, if , we consider for , the stopping time . By using the above result to the martingale , we obtain

from which we may conclude by using the monotone convergence theorem.

Hi, Professor Baudoin. I think in the first line of the proof of the part 2, it should be p>1.

Thanks. Corrected.