# Shreve I.2.5 Discrete versions of Ito’s lemma

Øksendal (sixth edition) example 3.1.9: almost surely,
$B_t^2 - t = \int_0^t 2B_s dB_s$

This has a discrete version which holds everywhere: let $X_n=\pm 1$ and $S_n=\sum_{i=1}^n X_i$, then
$S^2_n-n = 2\sum_{i=0}^{n-1} S_i X_{i+1}$
To verify just note that both sides increase by $2S_{n-1}X_n$ when going from $n-1$ to $n$.

Øksendal’s exercise 4.2:
$B_t^3 = \int_0^t 3B_s ds + \int_0^t 3B_s^2 dB_s$

Here the discrete version is not a perfect analogue:
$S_n^3 - S_n = 3\sum_{i=0}^{n-1} (S_i + S_i^2 X_{i+1})$
The extra term $S_n$ seems related to the fact that $(dB_t)^3 = 0$.