# General version of the fundamental theorem of asset pricing in continuous time

I know the fundamental theorem of asset pricing in discrete time which says that there are no arbitrage opportunities if and only if there exists a equivalent martingale measure. As far as I understand this equivalence is not true for the general setting; i.e. existence of equivalent martingale measure implies no arbitrage but not the other way around.

Can anyone please state the general fundamental theorem of asset pricing?

Definition No free lunch with vanishing risk (NFLVR). We say that a process $$S$$ satisfies NFLVR if there does not exist a sequence of admissible integrands $$H^n$$ such that, if $$f_n = (H^n \cdot S)_\infty$$, then $$f_n^- \to 0$$ uniformly and $$f_n \to f_0$$ where $$f_0$$ is a $$[0,\infty]$$-valued random variable satisfying $$P(f_0 > 0 ) > 0$$.
Theorem Let $$S$$ be a bounded $$\mathbb{R}$$-valued semi-martingale. There is an equivalent martingale measure for $$S$$ if and only if $$S$$ satisfies NFLVR.
For a proof of this theorem see the book The Mathematics of Arbitrage by Delbaen and Schachermayer, or their original paper A general version of the fundamental theorem of asset pricing. For a counterexample to "no arbitrage $$\implies$$ existence of an equivalent martingale measure", see On the fundamental theorem of asset pricing with an infinite state space by Back and Pliska.