Let $W_t$ be a $d$-dimensional random-walk (here $t=0,1,\dots)$ and suppose that $f:\mathbb{R}^d\rightarrow \mathbb{R}$ is a twice continuously-differentiable function. Does a "discrete" Ito Lemma of the form hold $$ f(W_{t+1})-f(W_{t})= -\sum_{i=1}^d\frac1{2}\partial^2_{i,j} f(W_t) + \sum_{i=1}^d\partial_{i}f(W_t)W_t? $$
If not...what would it look like?