Diffusion on the circle and a stochastic correlation model

TL;DR

This paper introduces new diffusion models on the circle for time-varying correlation, deriving an approximation for the von Mises process's transition density to enable likelihood-based inference and applications in finance.

Contribution

It develops a practical likelihood-based inference method for circular diffusion models, especially the von Mises process, with applications to financial correlation modeling.

Findings

Derived an analytical approximation for the von Mises diffusion transition density

Established statistical properties of the estimators for circular diffusions

Demonstrated the models' effectiveness through simulations and market data applications

Abstract

We develop diffusion models for time-varying correlation using stochastic processes defined on the unit circle. Specifically, we study Brownian motion on the circle and the von Mises diffusion, and propose their use as continuous-time models for correlation dynamics. The von Mises process, introduced by Kent (1975) as a characterization of the von Mises distribution in circular statistics, does not have a known closed-form transition density, which has limited its use in likelihood-based inference. We derive an accurate analytical approximation to the transition density of the von Mises diffusion, enabling practical likelihood-based estimation. We study inference for discretely observed circular diffusions, establish consistency and asymptotic normality of the resulting estimators, and propose a stochastic correlation model for financial applications. The methodology is illustrated through simulation studies and empirical applications to equity-foreign exchange market data.