Let $f:\mathbb{R} \to \mathbb{R}$ be in $C^{2}_c(\mathbb{R})$, that is, twice differentiable with continuous second derivative and compact support.
I would like to show that $$ f(W_t) - f(W_0) = \int_{0}^{t} f'(W_s)\circ dW_s $$ where here $\circ dW_s$ denotes the Stratonovich integral, given by: $$ \int_0^t A_s \circ dW_s = \lim_{n \to \infty} \sum_{ k= 1}^{nt} \frac{A_{j/n} + A_{(j-1)/n}}{2} (W_{j/n} - W_{{j-1}/n}) $$ with the limit taken in the appropriate topology. One proves Ito's lemma by taylor expanding the espression $f(W_t) - f(W_0)$ and writing it as a sort of telescoping series. I would like to the same for the Stratonovich integral, however I am running into some issues as when we Taylor expand for the Ito integral, we expand around the left endpoint, whereas I am not sure how to Taylor expand for the Stratonovich integral. I understand of course that the expansion should be around the midpoint, but the algebra is giving me some issues.