# Teaching stochastic calculus to students who know no measure theory (or PDE, or…)

I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)).

I'm to teach the (elementary) stochastic integration and differential equations to the students with practically zero background (the question "in what?" is meaningless here: the answer is "in everything"). I do not want to take the ancient Egyptian approach however ("Do this like we have shown you and your answer will be right!", or whatever else was written on that papyrus a few thousand years ago).

So, my current plan (after covering some basic probability and discrete time Markov chains, which, thanks God, is a part of the course) is to introduce countably many independent standard Gaussians, construct the Brownian motion on dyadic rationals by successive splitting of each step $\xi$ into $\frac 12(\xi+\eta)$ and $\frac 12(\xi-\eta)$ drawing independent copies from my countable independent list as I need them, show the relevant continuity (or some crude version of it), extend to reals by continuity, assume that all functions are piecewise continuous (or even constant: after all it is still a dense linear space in $L^2$), integrate with respect to $dW$ using dyadic Riemann procedure and solve the stochastic ODE by the Euler method using the same sequence of discrete dyadic approximations again and again. This seems possible in principle with the full level of rigor if you restrict the classes of random functions you consider and think in advance of how to pass to the limits without ever invoking any non-baby versions of Fubini or Dominated Convergence, but some details promise to be a lot of nuisance to figure out from scratch. So I wonder

1) Is there any decent book (or, even better, lecture notes) that follow this or similar approach?

2) Do you have any better idea of how to carry out this task?

The final goal is not very lofty: just to make sure that the student eyebrows won't reach the stratosphere when someone will later talk to them about option pricing, etc. in the financial math. courses, but I still want to do the job decently by common mathematical standards.

Thanks in advice for any ideas, references, etc.

• Are you aware of the new StackExchange site for math educators? It's at matheducators.stackexchange.com – Joonas Ilmavirta Nov 28 '14 at 15:50
• @ Joonas Ilmavirta I haven't been before you posted. Do you suggest to move the question there? I'm a bit concerned that it will be slightly out of alignment with "How to teach addition, subtraction, multiplication and division of binary numbers? Are there any activities that can be recommended?" and "How do I make my student understand concepts such as “x divided by x”?" but if other folks think it belongs there rather than here, I do not mind trying. :-) – fedja Nov 28 '14 at 16:40
• I would welcome more higher education questions at MESE, so I would prefer to have this question there. The site is young and it can still be steered in new directions. Also, this question is not about research mathematics (in a strict interpretation). Some of the MO folks are also there, so you might get a good answer there as well. There is a danger that those who could give the best answer(s) are not active there, so it's your call, but I would suggest migration. – Joonas Ilmavirta Nov 28 '14 at 16:54
• Joonas Ilmavirta As I said, I do not mind trying but I do have my reservations about it. Whichever way the folks will finally decide is fine with me. I would just appreciate if people don't get carried away with resurrecting the old controversies about "What exactly is appropriate for what SE subfora" and the game of close-open tug of war in this thread. There is no need for any of us to go online to get that kind of entertainment. :-) – fedja Nov 28 '14 at 17:06
• Your question is appropriate in both places (I don't really want to follow a strict interpretation), so I see no reason for any kind of tug-of-war with it. And such battles are far from entertainment for me. :o) You could always try this one at MESE, but I don't want to persuade you into anything against your own judgement. Maybe a third opinion would be welcome here... (We can't vote about migration in the usual way since MESE is in beta.) – Joonas Ilmavirta Nov 28 '14 at 17:36

I think the situation in which you are is not so uncommon with the proliferation of Financial Mathematics Master's programs...

One widely adapted solution is Steve Shreve's "Stochastic Calculus for Finance II: Continuous time Models". Whereas the book develops much of the more advanced techniques in close relationship with option prices, the chapter's 1-4 are an excellent introduction into Brownian motion and Ito calculus.

A bit on the lighter side I like Thomas Mikosch's "Elementary Stochastic Calculus with Finance in View", whereas on the more rigorous side I recommend Kuo's "Introduction to Stochastic Integration". In my classes I use the letter one, but only after an introduction into measure-theoretic probability. Its main shortcoming for financial applications is imho the treatment of the Feynman-Kac formula, which is best substitute from other sources (as Shreve).

• Thanks a lot. Steve's book certainly has a good order of exposition, which I may well adopt. Still, it looks like he sweeps too many things under the rug. The most amazing blunder is that he never bothers to prove that Brownian motions do, indeed, exist while (unless you invoke very heavy tools that are way beyond the scope of this course) passing directly from the existence of the limiting distribution for a family of stochastic processes to the existence of a stochastic process with the limiting distribution is not for the weak of heart, unless I miss something. – fedja Nov 28 '14 at 19:14
• I agree that he sweeps some things under the rug, one could add his treatment of quadratic variation or the non-treatment of local martingales (a quite important topic even for option pricing, e.g. in local and stochastic volatility models). But the balance between ease of exposition and rigor is hard to strike (do you really want to discuss tightness of probability measures in such a class?) and I think he is doing a very good job. I feel amending a textbook by other sources is now easier than ever, as many textbooks are in electronic form available for students for free... – Stephan Sturm Nov 28 '14 at 19:59
• Exactly. I do not say that we disagree much on anything, rather wonder what exactly those "other sources to amend the book" are. :-) Of course, "alternatives" are of interest too. – fedja Nov 28 '14 at 20:23
• As to "do you really want to discuss tightness of probability measures in such a class?", the answer is "certainly not!", but, as I showed above, all I need to construct a Brownian motion is an a.s. convergence (uniform on bounded subsets of dyadic rationals, in my case) and that's precisely the level at which I want to keep the whole story. – fedja Nov 28 '14 at 20:42
• This is a very good answer. Also consider Lin Introductory Stochastic Analysis for Finance and Insurance. – mdg Feb 3 '15 at 9:44

In addition to Steve Shreve's "Stochastic Calculus for Finance II: Continuous time Models" you should definitely use Steve Shreve's "Stochastic Calculus for Finance I" about the binomial discrete time model.

Then, rather than worry about existence of Brownian motion you can spend time on the Girsanov theorem and the mathematical proof of the fundamental theorems of asset pricing in volume II. The interplay between the math and the financial interpretation in terms of complete markets etc. is very interesting.

• It certainly is, but I can get any interesting result you want by introducing a non-existing object axiomatically and developing a fascinating theory of it to include the desired statement, so the proof of existence in this case is a must because I have never yet seen a student who was born with intuition and natural feelings about things like Borel sigma-algebra and general continuous time processes that wouldn't be way off from how it really is (unlike real number, when everybody is born with a decent idea of what that might be and just cannot express it in words, so definition is enough). – fedja Nov 29 '14 at 12:19
• Thanks for your comment nevertheless. I'll definitely think of it :-). – fedja Nov 29 '14 at 12:21
• Proving existence is unnecessary. – mdg Feb 3 '15 at 9:53