# Different definitions of arbitrage

Consider the following setup: Let $S=\left(S_1,\ldots,S_n\right)$ be a $n$-dimensional price process and denote by $V$ its value process defined by $V_t=\phi_t\dot\ S_t$ for $t=0,\ldots,T$. In "Stochastic Finance" by Föllmer and Schweizer, we have the following definition for arbitrage in chapter 5:

Definition: A self-financing trading strategy $\phi$ is called an arbitrage opportunity if its value process V satisfies $V_0\leq 0$ a.s., $V_T\geq 0$ a.s. and $P(V_T>0)>0$. If there is no arbitrage opportunity, then the financial market is said to satisfy NA.

In the book "The mathematics of arbitrage" by Schachermayer and Delbaen, we have the following definition of arbitrage in chapter 2:

Definition: The set $K=\{(\phi\dot\ S)_T|\phi\in\mathcal{H}\}$, where $\mathcal{H}$ is the set of trading strategies. Define $C=\{g\in L^\infty(\Omega,\mathcal{F},P)|\exists f\in K\ f\geq g \}$ as the set of contingent claims super-replicable at time T. A financial market satisfy the (NA) (no arbitrage), if $$C\cap L^\infty_{\geq 0}=\{0\}$$

Is it possible to prove the equivalence of these definitions?

• @Gordon Fixed it! – peer Jan 18 '17 at 17:09
• If you look at the questions I asked you can see some useful stuff, otherwise I may be able to show the equivalence. – Wolfy Jan 18 '17 at 18:23
• @BillytheKid: You are welcome to provide an answer. – Gordon Jan 18 '17 at 18:42
• @BillytheKid Can you provide me with a link? – peer Jan 18 '17 at 19:11
• Yes will do shortly just not near a computer at the moment – Wolfy Jan 18 '17 at 19:25