Fat tails and black swans: Exact results for multiplicative processes with resets

Abstract

We consider a class of multiplicative processes which, added with stochastic reset events, give origin to stationary distributions with power-law tails—ubiquitous in the statistics of social, economic, and ecological systems. Our main goal is to provide a series of exact results on the dynamics and asymptotic behavior of increasingly complex versions of a basic multiplicative process with resets, including discrete and continuous-time variants and several degrees of randomness in the parameters that control the process. In particular, we show how the power-law distributions are built up as time elapses, how their moments behave with time, and how their stationary profiles become quantitatively determined by those parameters. Our discussion emphasizes the connection with financial systems, but these stochastic processes are also expected to be fruitful in modeling a wide variety of social and biological phenomena. City sizes, word usage, surname abundance, personal income, or stock market returns are examples of power-law distributed quantities. Such a kind of distribution, ubiquitous in the natural and social sciences, holds “atypical” properties that have awoken the interest of researchers for over a century.1–5 The dynamics of these kinds of data exhibit extreme, catastrophic, “unexpected” black-swan-like events.6 Distribution moments, such as the average or the variance, are highly volatile and poorly predict future properties of the process. In that context of uncertainty, knowledge of the dominant mechanisms underlying power-law distributions is relevant to directly compare the short-time properties of data series to actual asymptotic properties, and to eventually evaluate the reliability of forecast algorithms. Stochastic multiplicative processes (SMPs) with reset events, introduced two decades ago7 as a generic mechanism to generate power laws, have multiple applications in a variety of situations.8,9 In this contribution, we derive several finite-time properties of SMPs with reset events with the aim of improving our understanding of the poor predictability of the dynamical process. The discussion of our results in a financial context clarifies the relationship between gain and risk in investing strategies, and provides clues to control the frequency and magnitude of extreme events.