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There are two subparts to Fundamental Asset Pricing theorem.

  1. The Law Of One Price (LOOP thereafter) holds if and only if there exists a state price vector.
  2. In a market in which the LOOP holds, the state price vector is unique if and only if the market is complete.

How can we prove (1) and (2) mathematically?

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I just comment your second point, because in the definition i now of the LOP the state price vector (martingale measure) is involved.

  1. assuming the LOP holds then: state price vector is unique <=> the market is complete.

to proof "=>" look at the trinomial model, show that the model is not complete by trying to find a hedge for a call, afterwards calculate the set of risk-neutral measures.

"<=" market complete => for each derivate $D$ (including indicator variables for each measurable subset) a hedge $H$ exists such that $$D_t = \mathbb{E}_t^Q[D_T e^{r_{T-t}}] ~~~~~~~~~~~~~~(1)$$ for any risk-neutral measure $Q$ and an $t\geq0$ where $r$ is the discount-rate. Now observe $D_0$ is deterministic (not random) and the same whatever $Q$ we use for pricing. Now assume there a two different risk neutral measure $Q_1,Q_2$ such that $Q_1(A)\neq Q_2(A)$ for some measurable subset. If we "hedge" the indicator variable $1_A$ we get $$\mathbb{E}_0^{Q_1}[1_A e^{r_{T-t}}]=Q_1(A)$$ and $$\mathbb{E}_0^{Q_2}[1_A e^{r_{T-t}}]=Q_2(A).$$ But, this conflicts $(1)$ since the above would induce $\mathbb{E}_0^{Q_1}[1_A e^{r_{T-t}}]\neq\mathbb{E}_0^{Q_2}[1_A e^{r_{T-t}}]$ , hence there exists only one risk neutral measure $Q$ and therefore only one state price vector.

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