relations $$\begin{aligned}
\boldsymbol{\pi}^i(t)=Diag^{1/2}(\boldsymbol{V}(t))\boldsymbol{a}^i +Diag^{-1/2}(\boldsymbol{V}(t))\boldsymbol{b}^i ,\ \ i=1,...,N,\end{aligned}$$ where $Diag^{1/2}(\boldsymbol{u})$ denotes the diagonal matrix whose diagonal entries are the respective square roots of the components of the vector $\boldsymbol{u}\in \mathbb R^d$. The family of short-rate processes $r^i, i=1,..., N$ is assumed to be given in the form $$\begin{aligned}
\label{eq:shortRateDynamics}
r^i(t)=h^i+\langle \boldsymbol{H}^i,{\mathbf{V}}(t)  \rangle+\langle \boldsymbol{G}^i,{\mathbf{V}}^{-1} (t) \rangle,\end{aligned}$$ where ${\mathbf{V}}^{-1} $ is a vector whose components are the inverses of those of ${\mathbf{V}}$.

Under Assumption \[piandr\], we can express the dynamics of the GOP as $$\begin{aligned}
\frac{{\mathrm{d}}D^i(t)}{D^i(t)}=& \left( r^i(t)+(\boldsymbol{a}^i)^\top Diag(\boldsymbol{V}(t))\boldsymbol{a}^i+(\boldsymbol{b}^i)^\top Diag^{-1}(\boldsymbol{V}(t))\boldsymbol{b}^i
+2(\boldsymbol{a}^i)^\top \boldsymbol{b}^i\right){\mathrm{d}}t\\
&+ (\boldsymbol{a}^i)^\top Diag^{1/2}(\boldsymbol{V}(t))d\boldsymbol{Z}(t)+(\boldsymbol{b}^i)^\top Diag^{-1/2}(\boldsymbol{V}(t))d\boldsymbol{Z}(t).\end{aligned}$$ Here we suppress the explicit formulation of the dependence of $r^i$ on $\boldsymbol{V}$. Consequently, the dynamics of the exchange rate $S^{i,j}$ is given by the SDE $$\begin{aligned}
\label{exchangerates}
\begin{aligned}
\frac{{\mathrm{d}}S^{i,j}(t)}{S^{i,j}(t)}=& \Big((r^i(t)-r^j(t))+2(\boldsymbol{a}^i)^\top \boldsymbol{b}^i -(\boldsymbol{a}^i)^\top \boldsymbol{b}^j-(\boldsymbol{a}^j)^\top \boldsymbol{b}^i\Big.\\ 
&\Big.+(\boldsymbol{a}^i)^\top Diag(\boldsymbol{V}(t))(\boldsymbol{a}^i-\boldsymbol{a}^j)
+(\boldsymbol{b}^i)^\top Diag^{-1}(\boldsymbol{V}(t))(\boldsymbol{b}^i-\boldsymbol{b}^j)\Big){\mathrm{d}}t\\ 
&+ ((\boldsymbol{a}^i-\boldsymbol{a}^j)^\top Diag^{1/2}(\boldsymbol{V}(t))+(\boldsymbol{b}^i-\boldsymbol{b}^j)^\top Diag^{-1/2}(\boldsymbol{V}(t))){\mathrm{d}}\boldsymbol{Z}(t).
\end{aligned}\end{aligned}$$ Notice that the dynamics of the exchange rates are fully functionally symmetric w.r.t. the construction of product/ratios thereof.

Strict Local Martingality {#subsecstrictlm}
-------------------------

In Section \[sec:unifying\], we observed that, for a sufficiently general specification of the market price of risk for a given currency, we can have a situation where an equivalent risk neutral probability measure may fail to exist, due to the 