Fat Tails and Black Swans: Exact Results for Multiplicative Processes with Resets

TL;DR

This paper derives exact results for multiplicative stochastic processes with resets, revealing how power-law distributions emerge and evolve over time, with implications for social, economic, and ecological systems.

Contribution

It provides a comprehensive set of exact analytical results for complex multiplicative processes with resets, including their dynamics and stationary distributions.

Findings

Power-law tails develop over time in the processes.

Stationary distributions are quantitatively determined by process parameters.

Moments of the distributions exhibit specific time-dependent behaviors.

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 behaviour 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.