behavior of the variance process near zero under the putative risk neutral measure.

In this subsection, we investigate the conditions under which the $i$-th benchmarked savings account, $\hat{B}^i(t)=\frac{B^i(t)}{D^i(t)}$, is a strict $\mathbb P$-local martingale, $i=1,...,N$. As observed in Section \[sec:benchmarkApproach\], $\hat{B}^i(t)$, after normalization to one at the initial time, corresponds to the Radon-Nikodym derivative for the putative risk neutral measure of the $i$-th currency denomination. Should $\hat{B}_i(t)$ be a strict $\mathbb P$-local martingale, we note that classical risk neutral pricing is not applicable. However, real world pricing in line with is still applicable, see [@bookplaten10], and provides the minimal possible price.

Given and , the dynamics of $\hat{B}^i(t)$ are given by the SDE $$\begin{aligned}
\begin{split}
{\mathrm{d}}\hat{B}^i (t) &= - \hat{B}^i(t)((\boldsymbol{a}^i)^\top (Diag(\boldsymbol{V}(t)))^{1/2}{\mathrm{d}}\boldsymbol{Z}(t)\\
&\quad+(\boldsymbol{b}^i)^\top (Diag(\boldsymbol{V}(t)))^{-1/2}{\mathrm{d}}\boldsymbol{Z}(t)).
\end{split}\label{putativeRadonNikodym}\end{aligned}$$ Upon integration of the above SDE we obtain $$\begin{aligned}
\mathbb{E} \left[ \hat{B}^i (t) \right]&
=\hat{B}^i_0 \prod^d_{k=1} \mathbb{E} \left[ \xi_k^i (t)\right], \end{aligned}$$ where we define the exponential local martingale process $\xi_k^i = \left\{ \xi_k^i(t) \, , \, t \geq 0 \right\}$ via $$\begin{aligned}
 \label{eqRNprocess}
\begin{split}
\xi_k^i (t)&:= \exp \left\{ -  \rho_k \int^t_0 (a^i_kV_k(s)^{1/2}+b^i_kV_k(s)^{-1/2}) {\mathrm{d}}W_k(s) \right.\\
&\quad\quad\quad\left.- \frac{1}{2}  \rho_k^2 \int^t_0
(a^i_kV_k(s)^{1/2}+b^i_kV_k(s)^{-1/2})^2 {\mathrm{d}}s \right\} \, .
\end{split}\end{aligned}$$ The putative change of measure with respect to the $i$-th currency denomination 