I'm considering the set of SDEs (in the sense of Ito)
$\begin{align*} \mathrm d x &= -yx \mathrm d t+ x^2 \mathrm d B_t \\ \mathrm d y &= -y^2 \mathrm d t + xy \mathrm d B_t\end{align*}$
where the two Brownian motions are identical (i.e there is just one noise driving the dynamics).
Now as can be seen easily,
$y\cdot \mathrm d x - x\cdot \mathrm d y = 0$ and then by Ito's formula, after defining $w = \frac{x}{ y}$,
$\mathrm d w = \frac{\mathrm d x}{y} - \frac{x\mathrm d y}{y^2} + \frac{1}{2} \cdot (-\frac{1}{y^2})(2\mathrm d x\cdot \mathrm d y) + \frac{1}{2} \frac{2x}{y^3}(\mathrm d y) = 0$
This means that the ratio of the processes $x$ and $y$ is a constant.
Now the problem I actually have is modified by a factor of 2:
$\begin{align*} \mathrm d x &= -\tilde yx \mathrm d t+ x^2 \mathrm d B_t \\ \mathrm d \tilde y &= - 2 \tilde y^2 \mathrm d t + 2x\tilde y \mathrm d B_t\end{align*}$
Structurally, this new set of SDEs looks very similar to the one above, but I can't find a process like in the case above which has a similar "good" property (like the ratio being constant). Is there a trick to this I'm missing?
Actually, I got it: The quantity $\frac{x}{\sqrt{y}}$ is not constant, but at least
$$\mathrm d (x y^{-1/2}) = \frac{1}{2}x^3y^{-1/2} \mathrm d t$$ which is a nice smooth quantity.