= Positive regression deflator in a complete finite market
{title2=$Z_t=P_t^\top V_t^{-1}P_{t-1}$}
Let $V_t=\mathbb E[P_tP_t^\top\mid\mathcal F_{t-1}]$ be positive definite. <Market completeness> restricts each parent atom to at most $n$ successors, while positive definiteness forces exactly $n$ independent successor price vectors. The positive <martingale deflator> supplied by the <fundamental theorem of asset pricing> then has the unique one-step ratio $Z_t=P_t^\top V_t^{-1}P_{t-1}>0$. Its product deflates the prices into <martingales>.
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