Double stochastic opinion dynamics with fractional inflow of new opinions

TL;DR

This paper introduces a novel stochastic model for opinion dynamics and financial order flow that incorporates fractional inflow and power-law cancellation times, capturing long-range dependence and opinion lifespan effects.

Contribution

It combines fractional Lévy stable motion with power-law distributed cancellation times, extending to social opinion dynamics with finite opinion lifespan.

Findings

Model captures long-range dependence in order flow and opinion dynamics.

Applicable to financial markets and social systems with finite opinion lifespan.

Provides a framework for testing long-range dependence in complex time series.

Abstract

A recent analysis of empirical limit order flow data highlights the necessity for a more refined order flow model that integrates the power-law distribution of limit order cancellation times. These cancellation times follow a discrete probability mass function derived from the Tsallis $q$-exponential distribution, or equivalently, the second form of the Pareto distribution. By combining fractional L'{e}vy stable motion as the model for limit order inflow with the power-law distribution for cancellation times, we propose an innovative approach to modeling order imbalance in financial markets. We extend this model to a broader context, illustrating its applicability to opinion dynamics in social systems where opinions have a finite lifespan. This proposed model exemplifies a stochastic time series characterized by stationary increments and broken self-similarity. Consequently, it offers a novel framework for testing methods to evaluate long-range dependence in such time series.