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A martingale deflator is a strictly positive adapted process $Y$ such that every deflated cum-dividend asset gain has zero conditional drift:Using the definitions of ${\pi }^H$ and ${\xi }^H$,The holdings $H_t$ are ${\mathcal{F}}_{t-1}$-measurable, so the right side is a martingale transform of the deflated asset-gain local martingale. Hence $Z$ is a local martingale.

The fundamental theorem of asset pricing says, in this discrete-time formulation, that the market has no arbitrage if and only if it admits a strictly positive martingale deflator. Under a chosen positive numeraire this is equivalent to the existence of an equivalent martingale measure for numeraire-discounted gains.