Given a function $h = h(t,x)$ and ad diffusion process $X_t$, we define the Stratonovitch integral $$ \int_0^T h(s, X_s)\circ dW_s $$
to be the mean-square limit of the sums $$ S_n = \sum_{j=1}^n h(t_j^{(n)}, \frac{1}{2}(X_{t^{(n)}_j}+X_{t_{j+1}^{(n)}}))(W_{t_{j+1}^{(n)}}-W_{t_j}^{(n)}) $$ for partitions $\pi_n = (0 = t_1^{(n)} <t_{2}^{(n)}<\ldots<t_{n+1}^{(n)} = T)$ with $|\pi_n|\to 0$ as $n\to \infty$.
In the papier An optimal polynomial approximation of Brownian motion, James Foster, Terry Lyons, Harald Oberhauser (https://arxiv.org/abs/1904.06998) at page 14, there are in the high order Stratonovitch-Taylor expansion terms that contain integrals such as:
$$\int_s^t \int_s^u \circ dW_vdu, \int_s^t\int_s^u \int_s^v \circ dW_r\circ dW_v du$$
I do not understand to what differential terms or $L^2$-limits those terms correspond to. What bugs me is that there is no term at the left of $\circ dW$ in the integrands, no $h$.