Positive regression deflator in a complete finite market ($Z_t=P_t^{\top }{V}_t^{-1}{P}_{t-1}$)

Let $V_t=\mathbb{E}[P_t{P}_t^{\top }\mid {\mathcal{F}}_{t-1}]$ 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.