Hi there,

I'm working on a PhD project that involves parameter estimation for diffusion processes. I'm based in a machine learning research group, and the emphasis here is strongly on "practical" research. From the point of view of many people in this field, it's "nice" to have theoretical results, but at the end of the day if the model doesn't describe real-world data well, the results are deemed to be uninteresting (I'm not saying this is true in general, this is just the consensus among my group).

We've worked out some theory, and now we're starting to look for real-world models to apply it to. I was wondering if it would be appropriate to start a big-list, soft-question type thread to ask for instances where diffusion processes have been used in applied mathematics. I would ask for some justification of why the continuous-time setting is more appropriate than, say a model based on Markov chains. Ideally, I'd get a collection of responses along the lines of John Baez's Azimuth post:
http://www.azimuthproject.org/azimuth/show/Hopf+bifurcation

I'm not convinced the question is appropriate for mathoverflow. Perhaps it's too statistical. I'd appreciate any opinions you have on this.

Many thanks
Simon

How well any part of what you say will fly well on this site will depend on the experience of the audience. I do not know the number of people on MathOverflow with enough experience in your field who can give you an expert answer, but I am not optimistic about it. For the array of things you want to know, I suggest emailing colleagues.

In spite of the above, there are things you can do to improve your chances of success with a MathOverflow question. Make sure to include motivation and a couple of examples either supporting the kind of answer you are or are not seeking. I think what you have posted is suitable for three MathOverflow questions, so I recommend picking the most important and most specific question first. You can ask a series of related questions, but I recommend waiting a day or two in between questions for feedback..

Instead of a soft question, you might start out with the impression that continuous time models seem more prevalent in modeling diffusion (I assume that is the case), and give a couple of examples. Then propose why a Markov process might be a more suitable model for this and other types of real world behaviour, and show examples if you can supporting this statement. Then ask if there are any more examples using Markov, and indicate what you have done to find them. This becomes a type of reference request which is specific and quickly answered.

Based on feedback to a very specific question, the followup question to be posted later might be softer and ask for a big list. A third might ask the more general and possibly contentious question (continuous or Discrete?). Of course, use your own sequence of more specific to more general, as I am just guessing above what is important and what is a good sequence to ask.

In all of the above, try to clue the reader as to what you have done to answer your own question, make it specific so that the reader does not have to do much outside reading to answer (e.g. linking to Baez is fine, but better is a synopsis of what Baez does that you want plus a link), provide motivation so that some readers can answer the motivation and/or the question as well, and read the how to ask for points that I have missed.

Good Luck.

Gerhard "Ask Me About System Design" Paseman, 2011.08.19