Renormalization and fixed points in finance, since 1962

In diverse sciences that lack Hamiltonians, the analysis of complex systems is helped by the powerful tools provided by renormalization, fixed points and scaling. As one example, an intrinsic form of exact renormalizability was long used by the author in economics and related fields, most notably in finance. In 1962–3, its use led to a model of price variation founded on the (Cauchy-Polyà-Lévy) stable distribution, with striking data collapse that accounted for observed large deviations from Gaussianity. In 1965, a different form of exact renormalization led to fractional Brownian motion, which neglected large deviations but accounted for long dependence and the resulting non periodic cyclic behavior. Finally, from a seed planted in 1972, exact renormalizability and scaling led to a model of price variation of which the M1963 and M1965 models are special examples. This broader model, fractional Brownian motion in multifractal time, accounts simultaneously for both large deviations and long dependence. These three steps are in loose parallelism with space, time and joint renormalization in statistical physics. This presentation surveys the old works and many new developments described in the author's 1997 books on fractals and scaling in finance.