Non-commutative geometric Brownian motion exhibits nonlinear cutoff stability

Abstract

This article quantifies the asymptotic ε-mixing times, as ε tends to 0, of a multivariate geometric Brownian motion with respect to the Wasserstein-2-distance. We study the cases of commutative, and first order non-commutative drift and diffusion coefficient matrices, respectively, in terms of the nilpotence of the respective iterated Lie commutators.

Geometry-preserving Lie Group Integrators For Differential Equations On The Manifold Of Symmetric Positive Definite Matrices

“…We are interested here in a multivariate generalization of a Geometric Brownian Motion (GBM), given by the (Itô) SDE [17]:…”

“…We can derive ODEs followed by the mean (taking expectations in (17)) and covariance matrix (using Itô's Lemma [18] on XX T , taking expectations, and a few algebraic manipulations). They provide a broad summary of the statistics of the process (much simpler than e.g.…”

“…The multivariate GBM SDE is given by Eq. (17). When A and B commute, and A + 1 2 B 2 has no eigenvalue with a negative real part, [17] provides a closed form solution for an initial value of x 0 under the form:…”

“…With our choice of parameters (normal commuting matrices for A and B, see Sec. 6.5), we can obtain [17] a closed form solution on the variance of the process at each time step, and for any initial (deterministic)…”