On the Second Fundamental Theorem of Asset Pricing

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Let X¹,, Xᵈ be sigma-martingales on (Ω, F, P). We show that every bounded martingale (with respect to the underlying filtration) admits an integral representation w.r.t. X¹,, Xᵈ if and only if there is no equivalent probability measure (other than P) under which X¹,,Xᵈ are sigma-martingales. From this we deduce the second fundamental theorem of asset pricing- that completeness of a market is equivalent to uniqueness of Equivalent Sigma-Martingale Measure (ESMM).