it is possible to obtain a variance process $V$ under the putative risk neutral measure that may not satisfy the Feller condition, implying that the putative risk neutral measure may fail to be equivalent to the real world probability measure.

Consider a two-currency model, where the dynamics of the exchange rate $S^{1,2}$ is of the Heston type , such that the variance process $V$ fulfills the Feller condition, i.e. $2\kappa\theta-\sigma^2\geq0$. Let $Z^{{\mathbb{Q}}^{1}},Z^{{\mathbb{Q}}^{2}}$, as in , be the candidate Brownian motions under the putative risk neutral measures ${\mathbb{Q}}^1$ and ${\mathbb{Q}}^2$, respectively. The following holds:

1.  For the putative risk neutral measure ${\mathbb{Q}}^1$ we get:

    1.  \[lemma2:point11\] If $\lambda(t)=a\sqrt{V(t)}$, $a\in{\mathbb{R}}$, then the drift of the variance process $V$ under ${\mathbb{Q}}^1$ is $$\begin{aligned}
        \kappa\left(\theta-V(t)\right)-\sigma\rho aV(t)\end{aligned}$$ and the Feller condition is always satisfied under ${\mathbb{Q}}^1$, which is then a true equivalent martingale measure.

    2.  \[lemma2:point12\] If $\lambda(t) = a\sqrt{V(t)}+ \frac{b}{\sqrt{V(t)}}$, $a,b\in{\mathbb{R}}$, then the drift of the variance process $V$ under ${\mathbb{Q}}^1$ equals $$\begin{aligned}
        \kappa\left(\theta-V(t)\right)-\sigma\sqrt{V(t)}\rho\left(a\sqrt{V(t)}+ \frac{b}{\sqrt{V(t)}}\right)\end{aligned}$$ and the Feller condition may be violated, implying that ${\mathbb{Q}}^1$ may not be equivalent to ${\mathbb{P}}$.

    3.  \[lemma2:point13\] If $\lambda(t) = \frac{b}{\sqrt{V(t)}}$, $b\in{\mathbb{R}}$, then the drift of the variance process $V$ under ${\mathbb{Q}}^1$ is $$\begin{aligned}
        \kappa\left(\theta-V(t)\right)-\sigma\rho b\end{aligned}$$ and the 