Scaling, correlations, and cascades in finance and turbulence

The question of information cascades in finance appears in the literature. We use the dynamics of Kolmogorov's 1962 (K62) turbulence model, an example of multiaffine scaling, to illustrate how evidence for diffusion from large to small length scales, or correspondingly an information cascade from large to small times in finance, could be inferred from a certain multiaffine scaling exponent. As an alternative to the derivations given by Kolmogorov, Onsager, and Heisenberg, we also show how to derive the K41 model from ‘time’ reversible dynamics. We then discuss and compare five different analyses of finance data by econophysicists, including one where the information cascade was suggested. We explain why there is as yet no compelling evidence for an information cascade in finance. We point out the different finance data analyses are largely in disagreement with each other, and suggest the use of a stronger condition in data analysis in order to resolve the differences. We observe that errors are incurred for large returns by using price differences instead of the logarithmic return, which is a dimensionless, additive variable.