McCauley, Joseph L. and Bassler, Kevin E. and Gunaratne, Gemunu H. (2007): Martingales, Detrending Data, and the Efficient Market Hypothesis.

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Abstract
We discuss martingales, detrending data, and the efficient market hypothesis for stochastic processes x(t) with arbitrary diffusion coefficients D(x,t). Beginning with xindependent 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 2point 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 3point 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. We end with a discussion of Levy’scharacterization of Brownian motion and prove that an arbitrary martingale is topologically inequivalent to a Wiener process.
Item Type:  MPRA Paper 

Institution:  University of Houston, NUI Galway 
Original Title:  Martingales, Detrending Data, and the Efficient Market Hypothesis 
Language:  English 
Keywords:  Martingales; Markov processes; detrending; memory; stationary and nonstationary increments; correlations; efficient market hypothesis 
Subjects:  C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C53  Forecasting and Prediction Methods; Simulation Methods G  Financial Economics > G0  General C  Mathematical and Quantitative Methods > C2  Single Equation Models; Single Variables 
Item ID:  2256 
Depositing User:  Joseph L. McCauley 
Date Deposited:  15. Mar 2007 
Last Modified:  13. Feb 2014 19:21 
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URI:  http://mpra.ub.unimuenchen.de/id/eprint/2256 