Fundamental theorem of asset pricing

In more general markets

When stock price returns follow a single Brownian motion, there is a unique risk neutral measure. When the stock price process is assumed to follow a more general sigma-martingale or semimartingale, then the concept of arbitrage is too narrow, and a stronger concept such as no free lunch with vanishing risk (NFLVR) must be used to describe these opportunities in an infinite dimensional setting.

In continuous time, a version of the fundamental theorems of asset pricing reads:

Let ${\displaystyle S=(S_t{)}_{t\ge 0}}$ be a d-dimensional semimartingale market (a collection of stocks), ${\displaystyle B}$ the risk-free bond and ${\displaystyle (\mathrm{\Omega },\mathcal{F},P)}$ the underlying probability space. Furthermore, we call a measure ${\displaystyle Q}$ an equivalent local martingale measure if ${\displaystyle Q\approx P}$ and if the processes ${\displaystyle {\left(\frac{S_t^i}{B_t}\right)}_t}$ are local martingales under the measure ${\displaystyle Q}$.

1. The First Fundamental Theorem of Asset Pricing: Assume ${\displaystyle S}$ is locally bounded. Then the market ${\displaystyle S}$ satisfies NFLVR if and only if there exists an equivalent local martingale measure.
2. The Second Fundamental Theorem of Asset Pricing: Assume that there exists an equivalent local martingale measure ${\displaystyle Q}$. Then ${\displaystyle S}$ is a complete market if and only if ${\displaystyle Q}$ is the unique local martingale measure.