As mentioned here https://math.stackexchange.com/questions/2296945/conversion-between-solution-to-stratonovich-sde-and-it%C3%B4-sde

  the conversion to Stratonovich sde is the following

You are considering the Stratonovich SDE $$ dX = h(X) \, dt + \gamma (X) \circ dW, $$ where I suppose the following hold:

1. $h \colon \mathbb{R}^n \to \mathbb{R}^n$,
2. $\gamma = (\gamma_{ij})_{i,j = 1}^{n,m} \colon \mathbb{R}^n \to \mathbb{R}^{n\times m} $,
3. $W$ is an $m$-dimensional Brownian motion.

Such a SDE is equivalent to the Itô SDE $$ dX = \left( h(X) + \frac{1}{2} c(X) \right) dt + \gamma (X) \, dW, \tag{1} \label{1}$$ where $c = (c_i)_{i=1}^n \colon \mathbb{R}^n \to \mathbb{R}^n$ is defined by $$ c_i (x) = \sum_{k=1}^{m} \sum_{j=1}^{n} \frac{\partial \gamma_{ik}}{\partial x_j} (x) \, \gamma_{jk} (x).$$ In order to obtain \eqref{1}, apply exactly the conversion formula that you can find, e.g., at page 123 of the book An introduction to stochastic differential equations by Lawrence Evans.I hope that this can help to clarify what the notes you are looking at do, but perhaps you have to give some more details about the functions appearing into the SDE.

So here we can use the existence conditions for the Ito sde: Weak solutions of linear parabolic PDEs and corresponding SDEs

As mentioned in this answer, the conversion to Stratonovich SDE is the following:

You are considering the Stratonovich SDE $$ dX = h(X) \, dt + \gamma (X) \circ dW, $$ where I suppose the following hold:

1. $h \colon \mathbb{R}^n \to \mathbb{R}^n$,
2. $\gamma = (\gamma_{ij})_{i,j = 1}^{n,m} \colon \mathbb{R}^n \to \mathbb{R}^{n\times m} $,
3. $W$ is an $m$-dimensional Brownian motion.

Such a SDE is equivalent to the Itô SDE $$ dX = \left( h(X) + \frac{1}{2} c(X) \right) dt + \gamma (X) \, dW, \tag{1} \label{1}$$ where $c = (c_i)_{i=1}^n \colon \mathbb{R}^n \to \mathbb{R}^n$ is defined by $$ c_i (x) = \sum_{k=1}^{m} \sum_{j=1}^{n} \frac{\partial \gamma_{ik}}{\partial x_j} (x) \, \gamma_{jk} (x).$$ In order to obtain \eqref{1}, apply exactly the conversion formula that you can find, e.g., at page 123 of the book An introduction to stochastic differential equations by Lawrence Evans.I hope that this can help to clarify what the notes you are looking at do, but perhaps you have to give some more details about the functions appearing into the SDE.

So here we can use the existence conditions for the Ito sde: Weak solutions of linear parabolic PDEs and corresponding SDEs