Discontinuity and scaling: their scope and likely limitations

  • Benoit B. Mandelbrot
Chapter

Abstract

Chapter E1 stated emphatically my view that Gaussianity, random walks and martingales are attractive hypotheses, but disagree with the evidence concerning price variation. This chapter presents, in largely non-mathematical style, the processes I propose as replacements for Brownian motion. Their foundations include an evidence-based theme and a conceptual tool.

The theme is discussed in Section 1: it is discontinuity and the related notions of concentration and cyclicity. The tool, scaling, is discussed in Section 2. The possible limitations of scaling expressed by cutoffs and crossovers are discussed in Section 3. Section 4 comments on alternative approaches that contradict scaling, and instead replace Brownian motion by a “patchwork” of step-by-step “fixes.” Section 5 describes some paradoxes of scaling.

Stationarity and scaling express invariances with respect to translation in time and change in the unit of time. Diverse principles of invariance are essential to my work, in economics as well as in physics.

Keywords

Brownian Motion Price Change Fractional Brownian Motion Future Career Serial Dependence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Benoit B. Mandelbrot
    • 1
  1. 1.Mathematics DepartmentYale UniversityNew HavenUSA

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