  We discuss martingales, detrending data, and the efficient market hypothesis
for stochastic processes x(t) with arbitrary diffusion coefficients D(x,t).
Beginning with x-independent drift coefficients R(t) we show that Martingale
stochastic processes generate uncorrelated, generally nonstationary increments.
Generally, a test for a martingale is therefore a test for uncorrelated
increments. A detrended process with an x- dependent drift coefficient is
generally not a martingale, and so we extend our analysis to include the class
of (x,t)-dependent drift coefficients of interest in finance. We explain why
martingales look Markovian at the level of both simple averages and 2-point
correlations. And while a Markovian market has no memory to exploit and
presumably cannot be beaten systematically, it has never been shown that
martingale memory cannot be exploited in 3-point or higher correlations to beat
the market. We generalize our Markov scaling solutions presented earlier, and
also generalize the martingale formulation of the efficient market hypothesis
(EMH) to include (x,t)-dependent drift in log returns. We also use the analysis
of this paper to correct a misstatement of the fair game condition in terms of
serial correlations in Fama's paper on the EMH.
