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.