Discrete version of Ito's lemma Could anyone give me some references where I could find
(a) discrete version(s) of Ito's 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.
 A: Terry Tao wrote a nicely motivated discussion of the discrete Black-Scholes equation.
A: Szabados and Székely. Stochastic integration based on simple, symmetric random walks. 2009. MRNumber 2472013
Szabados. A discrete Itô's formula. 1990. MRNumber 1116806
Csörgö and Revész. On strong invariance for local time of partial sums. 1985. (Note: the authors actually had two papers in that year, and Szabados cites the wrong one.) MRNumber 805116
Kudzma. Itô's formula for a random walk. 1982. MRNumber 684465.
A: If the goal is a discrete version of Black-Scholes, this seems backwards to me. The standard discrete description of dynamic hedging using discrete steps for time and price on a binomial tree (you can easily extend it to a recombining multinomial tree, too) is for me the best discrete version of Black-Scholes. In particular, this description makes it quite clear why you want to use pseudo-probabilities and not real probabilities.
The continuous version of Black-Scholes is easily derived from this by taking a limit.
I guess I'm a Luddite, but I've never understood the need to know Ito's lemma and stochastic calculus when doing mathematical finance. The real world is discrete. Simple continuous limits such as Black-Scholes are extremely useful, but I've never fully understood why anyone would want to use overly sophisticated continuous stochastic models. The real world is far too noisy.
A: Eric Forgy's work is rather abstract but also very interesting:


*

*http://phorgyphynance.wordpress.com/my-papers/

*http://phorgyphynance.files.wordpress.com/2008/06/discretesc.pdf
A: *

*Stochastic Calculus for Finance II: Continuous-Time Models by Shreve or 

*Shreve or Øksendal's Stochastic Differential Equations

*Øksendal

*Williams' Probability with Martingales or Shreve

A: Proposition 1.13.1 of this book.  The result is based on the Clarke-Ocone formula and is a discrete-time (as opposed to a simple process in continuous time approach, as posted by most of the others).  Subtle difference, if you're interested...
Cheers!
