Let’s consider a market in finite discrete time with trading dates $0,1,\dots,T$ probability space $(\Omega, \mathcal{F}, \mathbb{P})$, filtration $\{\mathcal{F_t}\}_{t \in \{0,1, \dots, T\}}$, $N$ traded assets, whose prices are described by a $\mathbb{R}^N$-valued stochastic-process $S=\{S_t\}_{t \in \{0,1, \dots, T\}}$ such that $\forall t: \ S_t^0>0$ a.s. (i.e. $S^0$ is a numéraire).
Now a strategy $\Psi=\{\Psi_t\}_{t \in \{1, \dots, T\}}$ is a $\mathbb{R}^N$-valued predictable process. It’s corresponding value process is $\tilde{V}(\Psi)=\{\tilde{V}_t(\Psi)\}_{t \in \{0,1, \dots, T\}}$, where $\tilde{V}_0(\Psi):=\Psi_1 \cdot S_0$ and $\tilde{V}_t(\Psi):=\Psi_t \cdot S_t$. We call the strategy it „self-financing“ if $\forall t<T: (\Psi_{t+1}-\Psi_t)\cdot S_t=0$. We call it „$a$-admissible“ if $\forall t: \tilde{V}_t(\Psi) >a$ a.s. and just „admissible“ is it is $a$-admissible for some $a$.
Now an arbitrage opportunity is a self-financing strategy $\Psi$ such that $\tilde{V}_0(\Psi)\leq 0$ a.s. and $\tilde{V}_T(\Psi)\geq 0$ a.s. and additionally either $\mathbb{P}(\tilde{V}_T(\Psi)> 0)>0$ or $\mathbb{P}(\tilde{V}_0(\Psi)< 0)>0$.
I am now supposed to show that in a market with numéraire there exist arbitrage opportunities if and only if there exist admissible arbitrage opportunities.
The strategy of the proof is probably to take an arbitrage opportunity $\Psi$ and from it construct another arbitrage opportunity $\Phi$ that is admissible. I tried various things, but didn’t succeed. I presume that we somehow must put more into the first component of $\Phi$, corresponding to the numéraire in order to make it admissible, while simultaneously making sure that it stays self-financing. I‘d be very happy if someone could give me some hints.
