Two-dimensional Brownian motion with dependent components: turning angle analysis

TL;DR

This paper analyzes a two-dimensional correlated Brownian motion model focusing on the distribution of turning angles, supported by theoretical, simulation, and real-world data analysis, including financial and physical systems.

Contribution

It introduces a model for dependent components in 2D Brownian motion and studies its turning angle distribution, extending traditional independent-component models.

Findings

Dependent components significantly affect turning angle distribution.

The model aligns well with real-world financial and physical data.

Extensions to time-varying correlations are feasible.

Abstract

Brownian motion in one or more dimensions is extensively used as a stochastic process to model natural and engineering signals, as well as financial data. Most works dealing with multidimensional Brownian motion consider the different dimensions as independent components. In this article, we investigate a model of correlated Brownian motion in $\mathbb{R}^2$, where the individual components are not necessarily independent. We explore various statistical properties of the process under consideration, going beyond the conventional analysis of the second moment. Our particular focus lies on investigating the distribution of turning angles. This distribution reveals particularly interesting characteristics for processes with dependent components that are relevant to applications in diverse physical systems. Theoretical considerations are supported by numerical simulations and analysis of two real-world datasets: the financial data of the Dow Jones Industrial Average and the Standard and Poor's 500, and trajectories of polystyrene beads in water. Finally, we show that the model can be readily extended to trajectories with correlations that change over time.