Talk:Doob's martingale inequality

Useless?

The what??? How is this useful??? ~user:orngjce223how am I typing? 15:46, 22 December 2006 (UTC)

How is it useful? For one example, see the application to Brownian motion that is included in the article. For a more general viewpoint, consider the following problem: I know how much a stochastic process X "tends to vary" over small time steps, but I wish to know the probability that it "only varies a little bit" over a long interval of time. An estimate of this probability is provided by Doob's martingale inequality. For further discussion, I would suggest reading Øksendal (see the article), [http://www.amazon.com/Basic-Stochastic-Processes-Zdzislaw-Brzezniak/dp/3540761756/sr=1-1/qid=1166809090/ref=pd_bbs_sr_1/102-0052142-5680906?ie=UTF8&s=books Brzezniak & Zastawniak], or [http://www.amazon.com/Brownian-Stochastic-Calculus-Graduate-Mathematics/dp/0387976558/sr=8-1/qid=1166809189/ref=pd_bbs_sr_1/102-0052142-5680906?ie=UTF8&s=books Karatzas & Shreve]. Sullivan.t.j 17:45, 22 December 2006 (UTC)

Landsburg's brainteaser as an example

Bennett Haselton posted Using a Brainteaser to Discover a Theorem which mentions this article:

[A] commenter pointed out I had rediscovered a result called “Doob’s martingale inequality”, although I think my example is vastly easier to understand than the Wikipedia article.

where the last two words link to this article. The brainteaser which the example is addressing comes from Steven Landsburg.

Perhaps someone will find this useful for expansion or explanation?

CRGreathouse (t | c) 20:01, 31 July 2023 (UTC)

Two sources

I recommend

Grimmett-Stirzacker, Probability and Random Processes, section 12.6, The Maximal Inequality.

Here we see two related inequalities, for submartingales and supermartingales respectively. (They are not dual to each other, but are two different inequalities.) They give a proof that uses stopping times, which is shorter that the one in the Wikipedia article, but less elementary. They present several consequences, and give names to some of the inequalities.

Also

Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Appendix 2.

shows one way to combine the two inequalities. 2A02:1210:2642:4A00:5037:21D1:4158:2342 (talk) 14:42, 1 September 2023 (UTC)