The transformation from Ito integral to Stratnonvich integral In the book Introduction to SDE by Evans, it says that if $\mathbf{X}$ solves the Ito sde
$$
\left\{\begin{aligned} d \mathbf{X} &=\mathbf{b}(\mathbf{X}, t) d t+\mathbf{B}(\mathbf{X}, t) d \mathbf{W} \\ \mathbf{X}(0) &=\mathbf{X}_{0} \end{aligned}\right.$$
if and only if $\mathbf{X}$ solves the Stratonovich sde
$$
\left\{\begin{aligned} d \mathbf{X} &=\left[\mathbf{b}(\mathbf{X}, t)-\frac{1}{2} \mathbf{c}(\mathbf{X}, t)\right] d t+\mathbf{B}(\mathbf{X}, t) \circ d \mathbf{W} \\ \mathbf{X}(0) &=\mathbf{X}_{0} \end{aligned}\right.
$$
where
$$ 
c^{i}(x, t) :=\sum_{k=1}^{m} \sum_{j=1}^{n} b_{x_{j}}^{i k}(x, t) b^{j k}(x, t).
$$
However, if I use the conversion formula that is 
$$
\begin{aligned}&\left[\int_{0}^{T} \mathbf{B}(\mathbf{W}, t) \circ d \mathbf{W} \right]^{i} \\ &=\left[\int_{0}^{T} \mathbf{B}(\mathbf{W}, t) d \mathbf{W}\right]^{i}+\frac{1}{2} \int_{0}^{T} \sum_{j=1}^{n} b_{x_{j}}^{i j}(\mathbf{W}, t) d t \end{aligned}$$
to the ito sde then we have
$$
\begin{aligned}
d \mathbf{X}&=\mathbf{b}(\mathbf{X}, t) d t+\mathbf{B}(\mathbf{X}, t) d \mathbf{W}\\
&=\mathbf{b}(\mathbf{X}, t) d t+\mathbf{B}(\mathbf{X}, t) \circ d \mathbf{W}-\frac{1}{2} \sum_{j=1}^{n} b_{x_{j}}^{i j}(\mathbf{W}, t) d t
\end{aligned}
$$
which is not the same one in the Stratonovich sde.
 A: I think there is an issue in the conversion formula. 1. The function $\textbf{B}(.,t)$ depends on the process $\textbf{X}$ and not on the brownians $\textbf{W}$ and 2. The finite variation process should depend on the function $\textbf{B}$ and not $\textbf{b}$.
Let us write the conversion formula for a general Ito process $\textbf{X}$ which verify the following SDE
\begin{equation*}
d\textbf{X}_t = \textbf{b}(\textbf{X}_t,t)dt + \textbf{B}(\textbf{X}_t, t)d\textbf{W}_t
\end{equation*}
Note that the usual approach for such proof is to start with elementary processes and use a density argument to have the general case.
By definition of the Stratonovich integral, we have :
\begin{align}
\left[\int_0^T \textbf{B}(\textbf{X}_t, t)\circ d\textbf{W}_t\right]^{i,\bullet}  = \lim_{n\to\infty} \frac12\sum_{i=0}^{p(n)-1}\left[\textbf{B}^{i,\bullet}(\textbf{X}_{t^n_{i+1}},t^n_i) + \textbf{B}^{i,\bullet}(\textbf{X}_{t^n_{i}},t^n_i)\right](\textbf{W}_{t^n_{i+1}}-\textbf{W}_{t^n_{i}})\quad (1)
\end{align} 
where $0 = t_0 < t^n_1 < ... < t^n_p(n) = t$ is a subdivision of $[0,T]$ such that $\sup_i(t^n_{i+1}-t^n_{i}) \to_{n\to \infty} 0$. 
We know that by Taylor approximation:
\begin{align*}
\textbf{B}^{i,\bullet}(x + h ,t) =  \textbf{B}^{i,\bullet}(x,t) + \sum_{j=1}^n \frac{\partial\textbf{B}^{i,\bullet}(x,t)}{\partial x_j}h + o(h)
\end{align*}
Therefore, replacing the previous equation in (1). We have:
\begin{align*}
\left[\int_0^T \textbf{B}(\textbf{X}_t, t)\circ d\textbf{W}_t\right]^{i,\bullet} 
= \lim_{n\to\infty} \frac12\sum_{i=0}^{p(n)-1}\left[\textbf{B}^{i,\bullet}(\textbf{X}_{t^n_{i+1}},t^n_i) + \textbf{B}^{i,\bullet}(\textbf{X}_{t^n_{i}},t^n_i)\right](\textbf{W}_{t^n_{i+1}}-\textbf{W}_{t^n_{i}}) \\
= \lim_{n\to\infty} \frac12\sum_{i=0}^{p(n)-1}\left[\sum_{j=1}^n \frac{\partial\textbf{B}^{i,\bullet}(\textbf{X}_{t^n_{i}},t^n_{i})}{\partial x_j}(\textbf{X}_{t^n_{i+1}}-\textbf{X}_{t^n_{i}}) + 2\textbf{B}^{i,\bullet}(\textbf{X}_{t^n_{i}},t^n_i)\right](\textbf{W}_{t^n_{i+1}}-\textbf{W}_{t^n_{i}}) \\
= \lim_{n\to\infty} \frac12\sum_0^{p(n)-1}\left[\sum_{j=1}^n \frac{\partial\textbf{B}^{i,\bullet}(\textbf{X}_{t^n_{i}},t^n_{i})}{\partial x_j}(\textbf{X}_{t^n_{i+1}}-\textbf{X}_{t^n_{i}}) \right](\textbf{W}_{t^n_{i+1}}-\textbf{W}_{t^n_{i}}) + \left[\int_0^T \textbf{B}(\textbf{X}_t, t)d\textbf{W}_t\right]^{i,\bullet} \\
= \lim_{n\to\infty} \frac12\sum_{i=0}^{p(n)-1}\left[\sum_{k=1}^m\sum_{j=1}^n \frac{\partial{B}^{i,k}(\textbf{X}_{t^n_{i}},t^n_{i})}{\partial x_j}{B}^{j,k}(\textbf{X}_{t^n_{i}},t^n_{i})  \right](\textbf{W}_{t^n_{i+1}}-\textbf{W}_{t^n_{i}})^2 + \left[\sum_{j=1}^m\frac{\partial{B}^{i,j}(\textbf{X}_{t^n_{i}},t^n_{i})}{\partial x_j}{b}^{j}(\textbf{X}_{t^n_{i}},t^n_{i})  \right]\underbrace{(t_{i+1}^n-t_{i}^n)(\textbf{W}_{t^n_{i+1}}-\textbf{W}_{t^n_{i}})}_{(a)} + o([\textbf{W}_{t^n_{i+1}}-\textbf{W}_{t^n_{i}}]) + \\
\left[\int_0^T \textbf{B}(\textbf{X}_t, t)d\textbf{W}_t\right]^{i,\bullet}  \\
= \frac12\left[\sum_{k=1}^m\sum_{j=1}^n \frac{\partial{B}^{i,k}(\textbf{X}_{t},t)}{\partial x_j}{B}^{j,k}(\textbf{X}_{t},t)\right]dt + 
\left[\int_0^T \textbf{B}(\textbf{X}_t, t)d\textbf{W}_t\right]^{i,\bullet} 
\end{align*}
Where the equalities hold in $L^2$ (to be precise the third equality holds in probability). 
Note that there (a) has bounded variation. Then it is well known that the co-variation $<t,W_t>$ is null. 
