Could anyone give me some references where I could find  

(a) discrete version(s) of [Ito's lemma](http://en.wikipedia.org/wiki/Ito%27s_lemma)  

(b) a proof how it converges to the continuous form in the limit  

(c) its usage within stochastic difference equations  

(d) a deduction of a discrete version of the Black Scholes model.


Every little bit of information would help.



Discrete version of Ito's lemma
1. [*Stochastic Calculus for Finance II: Continuous-Time Models*](http://rads.stackoverflow.com/amzn/click/0387401016) by Shreve or
2. Shreve or Øksendal's [*Stochastic Differential Equations*](http://rads.stackoverflow.com/amzn/click/3540047581)
3. Øksendal
4. Williams' [*Probability with Martingales*](http://books.google.com/books?id=e9saZ0YSi-AC) or Shreve


