= Martingale deflator
A <martingale deflator> rescales asset values so that their deflated gains have a stated martingale property. For a price $S$ with cumulative cash dividends $D$, the correctly deflated gain process is
$$
Y_tS_t+\int_0^tY_u\,dD_u,
$$
with a strictly positive adapted deflator $Y$. Dividends are deflated at their payment times; multiplying $Y_t$ by the undiscounted total $S_t+D_t$ is generally different. A <local martingale deflator> requires the local-martingale property, while true expectation pricing requires the corresponding true-martingale and <integrability> conditions.
For a positive <bank account> $B$, put $Z_t=Y_tB_t/(Y_0B_0)$. If $Z$ is a true mean-one <martingale>, it defines an equivalent <probability measure> by its terminal density. Deflated-price <local martingales> then yield an <equivalent local martingale measure>. Obtaining an <equivalent martingale measure>, and ordinary conditional-expectation pricing, additionally requires true martingales for the asset-price ratios. Positivity and a local property alone do not justify that promotion.
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