denomination corresponding to the portfolio strategy $\boldsymbol\delta$, i.e. $$\begin{aligned}
\label{eq:portfolio}
\mathtt{V}^{i,\delta}(t)=\sum_{j=1}^N\delta_j(t)B^{i,j}(t).\end{aligned}$$ The strategy $\delta$ is said to be self-financing if $$\begin{aligned}
\label{eq:portfolioSefFinancing}
{\mathrm{d}}\mathtt{V}^{i,\delta}(t)=\sum_{j=1}^N\delta_j(t){\mathrm{d}}B^{i,j}(t).\end{aligned}$$ In line with [@bookplaten10], we assume limited liability for all investors. For this purpose, we introduce ${\mathcal{V}}^+$ as the set of all self-financing strategies forming strictly positive portfolios. For our purposes, we will be interested in a particular strategy $\delta^\star\in{\mathcal{V}}^+$, which yields the **growth optimal portfolio** (GOP), which can be shown to be equivalent to the numéraire portfolio (NP), and is defined as follows:

\[def:gop\] A solution $\delta^{\star}$ of the maximization problem $$\begin{aligned}
\sup_{\delta\in {\mathcal{V}}^+}\mathbb{E}\left[\log\left(\frac{\mathtt{V}^{i,\delta}(T)}{\mathtt{V}^{i,\delta}(0)}\right)\right],\end{aligned}$$ for all $i = 1,\ldots ,N$ and $0\leq T\leq {\bar T}$ is called a growth optimal portfolio strategy.

It has been shown in [@bookplaten10] that the GOP value process is unique in an incomplete jump-diffusion market setting. We summarize the discussion above in the following assumptions.

\[assumption:gop\] We assume the existence of the **growth optimal portfolio** (GOP), and denote by $D^i=\left\{D^i(t)\in (0,+\infty), \ 0 \leq t\leq \bar T \right\}, \ i=1,...,N$ the value of the GOP denominated in the $i$-th currency. The dynamics of the GOP are given by $$\begin{aligned}
\begin{aligned}
\frac{{\mathrm{d}}D^i(t)}{D^i(t)}=  r^i(t){\mathrm{d}}t+\langle \boldsymbol{\pi}^i(t),\boldsymbol{\pi}^i(t){\mathrm{d}}t+{\mathrm{d}}\boldsymbol{Z}(t)\rangle, \ D^i(0)>0 
\label{eq:sdegop}
\end{aligned}\end{aligned}$$ for $t\in [0, \bar 