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A refined version of the ‘fundamental theorem of asset pricing’ is Delbaen-Schachermayer 1994 ftap-delbaen-schachermayer.  It says that no arbitrage in continuous time requires semimartingale models.  Therefore my various efforts to look at the massive evidence for non-Markovian behavior of actual data would seem misguided from the theory perspective, but measurements on actual data has to be accounted for.  Non-Markovian nature of the daily returns of 1900 stocks cannot simply be theorized away.  The simplest solution is to consider fractional diffusion models, i.e. model probability density function of the return process as $D^\gamma_t p(x,t) + L_x p(x,t) = 0$ whose corresponding stochastic process will not be Markovian and then approximate it by a diffusion scaled by $t^{\gamma-1}$ as is done by Norros et. al. (norros-valkeila-virtamo-fbm see the last comments of the paper) where they approximate fractional Brownian motion by a martingale.  This is not satisfactory, but it does produce equivalent martingale measures without arbitrage.