Question

Does it make sense to define the Stratonovich integral like this ? $I_{s}(t)=\int^{t}_{s}F(f_s(u),\circ du):= \lim_{\Delta \to 0} in\ pr. \Sigma^{n-1}_{k=0}\frac{1}{2} \Big( F(f_s(t_{k+1}),t_{k+1}) $ $+F(f_s(t_{k}),t_{k+1})-F(f_s(t_{k+1}),t_{k}) -F(f_s(t_{k}),t_{k}) \Big)$ where $F(x,t,\omega)$ is a semimartingale helix and $f_s (t)$ is a semimartingale and $\Delta :=\max(t_{k+1}-t_k)$. The common version is $\Sigma \big( X_{t_{k+1}}+X_{t_{k}} \big)/2 \big( W_{t_{k+1}}-W_{t_{k}} \big).$ How do they become equivalent to each other?