Brownian Motion and Stochastic Integrals: Worked Problems and Solutions
The book would be structured like The Cauchy Schwarz Master Class. That is, each chapter would be organized around a small set of Challenge Problems which would provide coaching about some particularly useful idea --- or brazen trick. The intention would be to provide friendly advice about problem solving while engaging questions about Brownian motion that call for increasing sophistication. Each chapter would conclude with a big selection of interesting problems, and all of the problems would be given complete solutions.
The seed for the book would be the 200 or so problems and solutions that I have built up from 12 years of mid-term and final take-home exams from my course Stat 955: Stochastic Calculus and Financial Application.
Unfortunately --- or fortunately --- perhaps as many of half these problems will not be interesting enough to "make the cut," so there is still a lot of work left to be done. I would also want to avoid major poaching from SCFA, though some minor poaching is both inevitable and appropriate.
Connections with Other Problem Books?
There are not many "problem books" in probability theory or stochastic processes. The market is clearly underdeveloped --- yet it does exist to a modest extent.
Dynkin and Yuskevich have an elegant problem book on Markov processes, and it should be more widely known. Unfortunately, the English version has always been so outrageously priced that it has probably sold very few copies.
Grimmett and Stirzaker also have their One Thousand Exercises in Probability, which can be quite useful as a classroom supplement. Still, their book is decidedly more elementary than the one I have in mind.
More to the point, neither of these books offers coaching in problem solving. For such coaching one usually has to call on the classic "problem solving" literature of Engle, etc. Nevertheless, coaching can be stirred into the dissuasion of almost any problem --- and the discussion is almost always enriched by its inclusion.
Another Example: I just saw Martingales and Markov Chains: Solved Exercises and Elements of Theory by Paolo Baldi, Laurent Mazliak, and Pierre Priouret. I have ordered this and will take a serious look. From the Amazon preview, it seems that there would not be much conflict with this book. I would focus more on continuous time, have many more pictures, and lean on the "physical rather than the formal."
The Intended Audience?
I think that a book like the one suggested here would be useful to a wide range of students, scientists, economists, and engineers.
Still, I am concerned that it may diverge a bit too far from the conventional scheme of things. Conceivably, Brownian motion may not appeal to the people who are naturally drawn to problem books, and people who like problem books may not have a taste for Brownian motion. This would be a pity, but it is possible.
I had no such worries when starting out with The Cauchy Schwarz Master Class. Inequalities have always been a mainstay within the problem solver community.
There is a principle about the definition of an audience that I find credible: if an audience is genuinely well defined then that particular audience is probably saturated. The implication is that the spots where the audience is ill defined are precisely the spots where new books should be directed. Naturally, this may be wishful thinking. After all, I did come up with this "principle" just as I was squirming in my seat worrying about the potential audience.
This project seems like fun and it deserves an honest try, even if some marketing problems remain to be solved. By late spring 2007, I promise to post at least a couple of sample chapters for feedback on their tone and content.
I would love to get any kind of advanced feedback that you can provide.
Are there books in this area that I have not seen?
Is there perhaps a simple variation on this theme that would be more appealing to you?
Finally, I need you advice about scope. Should keep a broad focus --- science, engineering, economics --- or should I lean on the financial modeling side things? There is plenty to do even with such a restricted focus, but my current sense is that it is best to stay broad --- and stay mathematical.
Still, it always pays to be flexible. I am open to any and all suggestions. Let me know what you think!
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