T]$ and $i = 1,\ldots N$, where for $N,d\in {\mathbb{N}}$, the $N$-dimensional family of predictable, ${\mathbb{R}}^d$-valued stochastic processes $\boldsymbol{\pi}=\left\{\boldsymbol{\pi}^i(t)=(\pi^i_1(t),\ldots ,\pi^i_d(t)), \ 0 \leq t\leq \bar T \right\}$ represent the market prices of risk with respect to the $i$-th currency denomination. The processes $\boldsymbol{\pi}^i$ are assumed to be integrable with respect to the $d$-dimensional standard Brownian motion ${\mathbf{Z}}$.

The GOP can be shown to be in many ways the *best* performing portfolio. In particular, in the long run its value outperforms almost surely those of any other strictly positive portfolio. Here we assume that it remains finite in finite time in all currency denominations. If we were to consider a model where the GOP explodes in any of the currency denominations, then the model would allow an obvious form of economically meaningful arbitrage, since one could generate in that currency denomination in finite time unbounded wealth from finite initial capital. Given the uniqueness of the GOP, all exchange rates $S^{i,j}(t)$ can be uniquely determined as ratios of different denominations of the GOP in the respective currencies.

\[AnsatzFX\] The family of exchange rate processes $S^{i,j}=\left\{ S^{i,j}(t), \ 0 \leq t\leq \bar T\right\}$ is determined by the ratios $$\begin{aligned}
S^{i,j}(t)=\frac{D^i(t)}{D^j(t)}\end{aligned}$$ for $0 \leq 