One-period Gram-matrix replication formula ($H=Q^{-1}\mathbb{E}[P_1{\xi }_1],Q=\mathbb{E}[P_1{P}_1^T]$)

In a one-period complete market, an invertible payoff Gram matrix identifies the unique holdings replicating each claim. The pricing weight is $Z=P_0^T{Q}^{-1}{P}_1$ and the random portfolio kernel is $W=Q^{-1}{P}_1$. Positive definiteness gives uniqueness of holdings; positivity of $Z$ requires an additional no-arbitrage condition.