Cutoff stability of multivariate geometric Brownian motion

TL;DR

This paper investigates the cutoff convergence phenomena of multivariate geometric Brownian motion, providing detailed conditions and representations for stability and convergence rates of key statistical quantities.

Contribution

It introduces a comprehensive analysis of cutoff stability for multivariate geometric Brownian motion, including explicit stability conditions and spectral characterizations.

Findings

Established cutoff convergence for autocorrelation, Wasserstein distance, and anti-concentration probabilities.

Derived necessary and sufficient mean square stability conditions involving spectral parts.

Provided a complete spectral representation under diagonalizable drift and volatility matrices.

Abstract

This article establishes cutoff convergence or abrupt convergence of three statistical quantities for multivariate (Hurwitz) stable geometric Brownian motion: the autocorrelation function, the Wasserstein distance between the current state and its degenerate limiting measure, and, finally, anti-concentration probabilities, which yield a fine-tuned trade-off between almost sure rates and the respective integrability of the random modulus of convergence using a quantitative Borel--Cantelli Lemma. We obtain in case of simultaneous diagonalizable drift and volatility matrices a complete representation of the mean square and derive nontrivial, sufficient and necessary mean square stability conditions, which include all real and imaginary parts of the volatility matrices' spectra.