Ole Peters, William Klein
Geometric Brownian motion is non-stationary. It is non-ergodic in the sense that the time-average growth rate observed in a single realization differs from the growth rate of the ensemble average. We prove that the time-average growth rate of averages over a finite number, N, of realizations is independent of N. A stability analysis shows that the time at which such averages begin to deviate from ensemble-average behavior increases logarithmically with N.
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http://arxiv.org/abs/1209.4517
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