I was reading the papers co-authored by Harrison, Kreps and Pliska, that initiated the formal research on the connection between pricing, martingale measures, arbitrage and completeness. I have some issues relating some of the ideas, though, and would be happy if somebody could help me.
In a more or less abstract setting, one considers an ordered topological space X of cash flows, a subspace M⊆X of traded instruments and a positive linear functional π:M→R. The 1st FTAP that relates absence of arbitrage to the existence of EMM concerns continuous extension of π to X which is still positive. As an example, one can consider M={m=m0+∫T0HtdSt}
I guess, the existence of a unique positive extension of π to the whole X implies the completeness of the market: that is, we can price everything in a consistent way and there is a unique way to do this. Apparently, this automatically leads to the representability of any x∈X as x0+∫T0HtdSt.
At the same time, I do not understand, how the existence of at least one continuous positive extension may be related to the absence of arbitrage. That is, I would imagine an arbitrage being rather existence of two strategies H′ and H″ such that m′0+∫T0H′tdSt=m″0+∫T0H″tdSt
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