Martingales, detrending data, and the efficient market hypothesis

Joseph L. McCauley, Kevin E. Bassler and Gemunu H. Gunaratne

Physica A: Statistical Mechanics and its Applications, 2008, vol. 387, issue 1, 202-216

Abstract: We discuss martingales, detrending data, and the efficient market hypothesis (EMH) 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 non-stationary 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 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's characterization of Brownian motion and prove that an arbitrary martingale is topologically inequivalent to a Wiener process.

Keywords: Martingales; Markov processes; Detrending; Memory; Stationary and non-stationary increments; Correlations; Efficient market hypothesis (search for similar items in EconPapers)
Date: 2008
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Citations: View citations in EconPapers (14)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:387:y:2008:i:1:p:202-216

DOI: 10.1016/j.physa.2007.08.019

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