Excel econometrics efficiency fractal fractal geometry linear optimization mathematics normal distribution. Limitations of efficiency and of martingale models. A case against the lognormal distribution. Nonlinear forecasts, rational bubbles, and martingales. Buy options. Proportional growth with or without diffusion, and other explanations of scaling. Mandelbrot on price variation. Hu, now at the University of California, Santa Barbara.

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  • Benoit Mandelbrot (), a Polish-born French mathematical physicist.

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    Hu, now at the University of California, Santa Barbara. The variation of certain speculative prices.

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