I should point out that I have not found myself in the subject of stochastic processes from a Mathematical stand-point. Rather my aim is to appreciate the origin of certain terms in equations describing fluid flow, in particular, simulating turbulent combustion using PDF-based methods. Ultimately to appreciate which terms are "physical" in origin and which are not.
As a starting point I am trying to convert from the Stratonovich form of this equation: $$dx(t) =\Bigl(U-\frac{1}{2}\frac{\partial \Gamma_T}{\partial x}-\frac{1}{\rho}\frac{\partial \rho}{\partial x} \Gamma_T\Bigr)dt +{\sqrt{2\Gamma_T}}\circ dW(t)$$ To the corresponding Ito version, given by: $$dx(t) =\Bigl(U-\frac{1}{\rho}\frac{\partial \rho}{\partial x} \Gamma_T\Bigr)dt +{\sqrt{2\Gamma_T}} dW(t)$$ I can see that the Ito interpretation of the stochastic integral includes the additional "drift" term: $$+\frac{1}{2}\frac{\partial \Gamma_T}{\partial x}$$
I have tried a number of different strategies such as Wong-Zakai correction, the "simpler" version: $$X\circ dW=XdW+\frac{1}{2}dXdW$$ to no avail. Applying the Wong-Zakai correction to the stochastic integral, I can go no further than this: $$\int_a^b{\sqrt{2\Gamma_T}}\circ dW(t)=\int_a^b{\sqrt{2\Gamma_T}}dW(t)+\frac{1}{2}\int_a^b\frac{\partial }{\partial x}{\sqrt{2\Gamma_T}}dt$$
Whilst the original equation does not show, I believe that $\Gamma_T$ is a function of both $x$ and $t$. As such the second integral on the RHS above is stochastic itself?
Could someone please provide some guidance in going from this correction equation to the final drift term?