Martingale problems

Can anyone tell me how and why they're used?

The solution to a martingale problem under broad conditions is a Markov process, apparently. Okay. Is that important?

Why do we care about solving martingale problems, anyway?

I can find some sources out there on the Net, of course, but all the ones I've found are extremely thin on motivation.

And by extremely thin on motivation, I mean they have absolutely none whatsoever.

In what sense?

I know very little of Martingale problems (I'm assuming that you're talking about the basic betting "strategy" that used to be advocated for gambling and games of chance), but can you be a bit more specific about what you're looking for?

Googling "Martingale problems applications" comes up with a use in evolution, at least.

This one appears to be systems control.

As for the first line of your post, that would be a negative from me.

Asset Pricing.

Hi Jesse. The context is dynamics.

In some circumstances, a certain kind of discrete-time equation can be asymptotically approximated by a continuous-time equation that's easier to analyze.

Lately martingale problems have been popping up in this branch of the dynamics literature as a new way to prove that a certain continuous-time equation is the right one for a given discrete-time equation.

The proofs that I've encountered toss martingale problems about in a rather offhand way, as if it should be obvious that the solution to martingale problem x is markov process y. Maybe it is obvious to a mathematician.

A martingale is any process with a mean change of zero. It’s a sexy generalization of a random walk, in which naughty things are allowed to happen in higher moments. (E.g., skewness, memory in the variance.) IIRC, a martingale in this sense does have a connection to the martinglae betting strategy you mentioned, but I forget what the connection is.

Limiting the search to "Martingale Applications" seems to indicate that the theories are used very heavily in trading and finance.

IIRC, a martingale in this sense does have a connection to the martinglae betting strategy you mentioned, but I forget what the connection is.

Well, the basis behind the betting strategy was ostensibly that by doubling down after any loss, a bettor with an unlimited bankroll and infinite time would come out ahead. Assuming a situation where the bettor bets $100 at even odds on a coin toss, the ev on the bet is still $0, regardless.

If you look at something like stock pricing, using a random walk model (haven't they moved away from that in the years since I've traded), one could argue that stock pricing follows a similar pattern. The stock's ev is it's current value. Although that's kind of an odd example of a discrete process...

Wikipedia offers quite a few Examples of Martingales.

And if this makes sense to you -

Martingale problems were used to construct and study properties of multidimensional diusions (Stroock and Varadhan (1979)), Infinite particle systems and Icing models (Holley and Stroock (1976)), processes associated with Boltzman equation (Tanaka (1978), Horowitz and Karandikar (1990)).

you're welcome to it.

You might look at Probability with Martingales...

Edited - David Williams, the author of that, is an LT member. Mayhap he'll jump in.

>9 Jesse_wiedinmyer:
Yes, I remember that now. You'd eventually win if you didn't go broke first. Lol.

random walk model (haven't they moved away from that in the years since I've traded)
I believe the finance kids have moved on to geometric differential diffusions with time-varying variance and drift. But the context for me isn't finance.

>11 Jesse_wiedinmyer:
The multidimensional diffusions thing makes sense, in context, but the rest is Greek.

The clearest thing I can find is http://www.ima.umn.edu/postdoc-seminar/2003-2004/kurtz/imamp.pdf

A martingale problem is defined on page 8. This guy defines most of his notation, which is more than I can say for anything else I’ve found. The other sources begin like this: “Assume a hyper GLAVIN system that is HOYVEN. Obviously this implies that Poisedon is at war with Mercury, GLAVIN.” Then it gets worse.

>12 Jesse_wiedinmyer:
I'm going to tap him on the shoulder. Can't hurt.

Actually, there's another LT member who also wrote the book on martingales - you might ask HIM...

http://www.librarything.com/profile/VKrishnan

That's two of them...

Gotta love this site.

43 of them, alas, as there are 42 David Williams's.

Go to the work via the touchstone and it will give you the proper profile. I already left a message on his profile and sent an email to Mr. Krishnan (he only has 4 books listed, so I'm guessing he's not as active on-site).

But the context for me isn't finance.


Mind me asking what your context is?

>18 Jesse_wiedinmyer: Huh? But how...? There are so many listed.

Follow the touchstone link in #12, then go to the top right corner of the work page.

That takes me here. http://www.librarything.com/librarything_author.php

Then there's a David Williams murder mystery guy or something.

The block with the LTAuthor icon has this -

LibraryThing Author

David Williams is a LibraryThing Author, an author who lists their personal library on LibraryThing.

profile page | author page

Click on "profile page."

Mind me asking what your context is?

People are trying to learn about a Markov process in discrete time. The continuous-time (CT) equation that asymptotically describes the discrete-time (DT) learning process is also a certain Markov process. Mathematicians working on this proceed more-or-less clearly, until they pull an UTTERLY RANDOM martingale problem out of (a hat, thin air, their ass, take your pick) and say "The solution to this mart prob is obviously the asymptotic CT equation."

Said martingale problem has absolutely no apparent connection with anything that has come before it. GAAAAAH!

#23: (No, I think the problem is that ALL FORTY-TWO "David Williams" get linked to the "LT Author" badge that only David Williams #I has applied for.)

Bob, Yes. I think it's a combining thing.

#24

So you're basically wondering how they know that the two are interchangable?

>27 Jesse_wiedinmyer:
No. it's much worse than that. There's no interchangeable thing; it's like this:

"How can we make sure that the basement of this house we're building won't flood? The answer, of course, is to acquire the One Ring of Sauron and dance a jig around it thrice at midnight. If you can solve that problem, you - as any idiot can see - will have solved the problem of basement floods."

This idiot, though, fails to see the connection between the One Ring of Sauron and flooding basements. None of the authors of these papers ever establish the connection. They just posit a random problem... To me, it is seriously like reading a paper that accidentally got mixed with another paper in the typesettig process.

"The proof is trivial, and is left as an exercise for the student."

I hate it when that happens. You know, just not enough room in the margins to write the fucker out.

>29 AsYouKnow_Bob:
Yeah. "We won't bother to state the crucial step, because it's so trivial."

In ways that are largely incomprehensible to me, these guys discuss the well-posedness of martingale solutions.

Mayhap here?

>32 Jesse_wiedinmyer:, 33
I'll bookmark those, thanks. I'm going to bed. Back to the math wars tomorrow.

Mr. Krishnan has expressed being otherwise occupied at the moment, but did choose to add this -

I will briefly raise the point that a martingale need not be a Markov process and vice versa. For example a Poisson process N(t) is a Markov process but it is not a martingale because it has a trend lambda*t. However, of one subtracts the mean {N(t)-lambda*t) then it is also martingale. I have given a control system example in my book Nonlinear Filtering and Smoothing republished by Dover (2005) in Chapter 10 under fault
detection problems.

Got it. I drfited off thinking about this last night after the discussion here, so this thread helped.

I then checked some stuff in the papers this morning. The martingale problem seems arbitrary because in at least one of the papers they neglect to mention why they choose it. In others they mention it (in the appendix, aargh), but it’s easy to miss.

The proofs go like this:

1. Here’s a discrete-time (DT) process. We can prove that it has some well-defined continuous-time (CT) limit, without knowing what that limit is.

2. Here’s a martingale problem. We can show, arguing abstractly, that the CT limit mentioned in 1 solves that problem (even though we don’t know what that CT limit is yet).

3. Here is a very specific, well-defined CT process. We DO know what this one is. It is chosen, presumably, by trial and error, luck, or long experience working with this kind of problem. We can show that this also solves the same martingale problem.

4. We can also show that this martingale problem has a unique solution. Therefore, the unknown CT limit in 2 must be the known CT process in 3.

In the main paper I’ve been reading this is mostly unsaid. The crucial point of uniqueness of the solution is mentioned in the middle of a paragraph and given no emphasis. In other papers some of the proof is in the main text and other parts are put in the Appendices. Damn it. The main proof idea is pretty elegant. I wish they’d lay it out clearly so people could understand it more easily.

I think that what the mathematicians actually do when they’re writing the paper initially is to start with step 3, go to 4, then back up to 1, then 2.

Well, that's kind of the gist of proof, no? You set up equivalence between various statements to arrive at a tautologous yet non-obvious statement.

I think some proofs are like that, but not all. E.g., I guess when I think of a proof that all X's are Y, but not vice-versa, I think there's a set/subset relation, not an equivalence relation.

I can accept that.

By the way, earlier today this thread was #8 in Google's results for "martingale problems." It since has dropped off the map, though. (How fleeting is fame.)

The classic treatment of this question is in Markov Processes: Characterization and Convergence by Stewart Ethier and Thomas G. Kurtz. The point of the exercise is usually a way of proving results about limits of sequences of processes. E.g., one has a sequence of finite-state, finite-time Markov chains, which are converging on some limiting continuous process, and one wants to prove that the limiting process is also Markov. It is then often easier to show that the limiting process must also solve a martingale problem, and hence must also be Markov.

Not to be too self-promotional, but I tried to lay out some of the motivation in ch. 12 of my online notes on stochastic processes, with an illustration in ch. 18, and possibly elsewhere that I've forgotten. But really the best source on this is Ethier and Kurtz.

Thanks very much for responding, Dr. Shalizi.