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Let $H$ replicate ${\xi }_1$, so ${\xi }_1=H^T{P}_1$. The positive-definite Gram matrix $Q=\mathbb{E}[P_1{P}_1^T]$ is invertible. Multiplication by $P_1$ and taking expected values giveAll quantities are integrable: the finite-atom representation from completeness gives finite terminal values, and the matrix $Q$ has finite entries. Positive definiteness also guarantees uniqueness of holdings, since $h^T{P}_1=0$ would imply $h^{T}Qh=0$ and hence $h=0$.

Thus the replication cost isConsequently the one-period Gram-matrix replication formula isThe symmetry of $Q^{-1}$ makes the displayed orientations consistent. Positivity of this $Z$ would require an additional no-arbitrage condition; positive definiteness of the payoff Gram matrix alone proves the representation and uniqueness, not positivity of prices across states.