Discrete version of Ito's lemma
- 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
Terry Tao wrote a nicely motivated discussion of the discrete Black-Scholes equation.
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.