In a paper I'm reading appears the following SDE:
$$\mathrm{d}u=\omega u^\perp\mathrm{d}t+(\sqrt{2d}\circ\mathrm{d}B_t)\cdot u^\perp u^\perp $$
here $u:\mathbb{R}\rightarrow \mathbb{R}^2$, $\omega:\mathbb{R}\rightarrow\mathbb{R}$, $d$ is a real number, $B$ a two-dimensional standard Brownian motion, "$\cdot$" is the standard scalar product and for $x=\begin{pmatrix}x_1\\x_2\end{pmatrix}$ the vector $x^\perp$ is defined by $x^\perp=\begin{pmatrix}x_2\\-x_1\end{pmatrix}$. I don't understand what this SDE looks like in integral notation. It should start like
$$ \begin{pmatrix}u_1(t)\\u_2(t)\end{pmatrix}=\begin{pmatrix}u_1(0)\\u_2(0)\end{pmatrix}+\int_0^t\omega(s)\begin{pmatrix}u_2(s)\\-u_1(s)\end{pmatrix}\mathrm{d}s+.....$$
How is the stochastic part defined here? Any help is appreciated.
In the latest version of https://arxiv.org/pdf/1307.1953.pdf, the typo in 2.15 was fixed to
$$\mathrm{d}u=\omega u^\perp\mathrm{d}t+(\sqrt{2d}\circ\mathrm{d}B_t)\cdot u^\perp $$
for $u^{\perp}=(u_{2},-u_{1})$ and $B=(B_{1},B_{2})$
$$=\omega \binom{u_{2}}{-u_1}\mathrm{d}t+2d\binom{u_{2}\circ dB_{1}}{-u_1\circ dB_{1}} $$
and so as noted here Conversion between solution to Stratonovich SDE and Itô SDE
$$=(\omega+\frac{1}{2}) \binom{u_{2}}{-u_1}\mathrm{d}t+2d\binom{u_{2}dB_{1}}{-u_1 dB_{1}}. $$