Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Model Reduction of Multivariate Geometric Brownian Motions and Localization in a Two-State Quantum System

C. Chen, M. Colangeli, M. H. Duong, M. Serva

Abstract

We develop a systematic framework for the model reduction of multivariate geometric Brownian motions (GBMs), a fundamental class of stochastic processes with broad applications in mathematical finance, population biology, and statistical physics. Our approach leverages the interplay between the method of invariant manifolds and adiabatic elimination to derive closed-form reduced equations for the deterministic drift. An extended formulation of the fluctuation-dissipation theorem is subsequently employed to characterize the stochastic component of the reduced description. As a concrete application, we apply our reduction scheme to a GBM arising from a two-state quantum system, showing that the reduced dynamics accurately capture the localization properties of the original model while significantly simplifying the analysis.

Model Reduction of Multivariate Geometric Brownian Motions and Localization in a Two-State Quantum System

Abstract

We develop a systematic framework for the model reduction of multivariate geometric Brownian motions (GBMs), a fundamental class of stochastic processes with broad applications in mathematical finance, population biology, and statistical physics. Our approach leverages the interplay between the method of invariant manifolds and adiabatic elimination to derive closed-form reduced equations for the deterministic drift. An extended formulation of the fluctuation-dissipation theorem is subsequently employed to characterize the stochastic component of the reduced description. As a concrete application, we apply our reduction scheme to a GBM arising from a two-state quantum system, showing that the reduced dynamics accurately capture the localization properties of the original model while significantly simplifying the analysis.

Paper Structure

This paper contains 31 sections, 6 theorems, 150 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model reduction of multivariate geometric Brownian motions
  3. Multivariate geometric Brownian motions
  4. Dynamics of the mean and covariance of GBMs
  5. Model reduction of the deterministic part and covariance dynamics
  6. Model reduction of GBMs
  7. A GBM from a two-state quantum system
  8. A two-state quantum system
  9. The process: cartesian coordinates
  10. Averages and interpretation
  11. Covariance matrix
  12. Stationary covariance matrix
  13. Model reduction via higher-order ODEs
  14. The deterministic part
  15. Reduced deterministic system: large $\beta$ regime
  16. ...and 16 more sections

Key Result

Proposition 1

Suppose that $F$ is an $n\times n$ matrix and $u=u(t)\in\mathbb{R}^n$ solves the following system of linear differential equations Let $p_n(\lambda):=\sum_{i=0}^n a_i \lambda ^i$ be the characteristic polynomial of $F$. Then $u(t)$ satisfies the $n$-th order differential equations

Figures (3)

  • Figure 1: Time evolution of $p_{zz}(t)$, computed from the the third-order ODE \ref{['eq: exact equation for variances']} (black solid line) and from the adiabatic elimination \ref{['eq: high temperature regime first order']} (black dashed line), with initial conditions $p_{zz}(0) = 0, p_{zz}'(0) = 0, p_{zz}"(0) = 0$. The dotted horizontal line marks the asymptotic value $1/3$. The parameters are set to the values $\alpha = 0.5$ and $\beta^2 = 2$ (left panel) and $\beta^2 = 100$ (right panel).
  • Figure 2: Behavior of the eigenvalues of the matrix $M_{\epsilon}$ in Eq. \ref{['Meps']} (black dashed lines) and $4\alpha \epsilon a_3^*(\epsilon)$ (solid gray line) as functions of $\epsilon$, using $\alpha=0.5$ and $\beta=1$. The critical point $\epsilon_c" \simeq 0.59185$ marks the onset of a pair of complex conjugate eigenvalues.
  • Figure 3: Time evolution of $p_{zz}(t)$ computed from the original system \ref{['eqcov']} (black solid line), and time evolution of the reduced dynamics $\tilde{p}_{zz}(t)$ computed from the invariant manifold reduction \ref{['exred']} (gray short-dashed line) and from the adiabatic elimination \ref{['adiab']} (black dashed line), with initial condition $p_{zz}(0) = 0$. Results are shown for $\epsilon = 0.5 < \epsilon_c"$ (left panel) and $\epsilon = 0.01$ (right panel). The inset in the right panel offers a magnified view of the early-time dynamics. The dotted horizontal line marks the asymptotic value $1/3$. In both panels, parameters are set to $\alpha = 0.5$ and $\beta = 1$.

Theorems & Definitions (13)

  • Proposition 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 2
  • Remark 3
  • Lemma 3: hamm2023wasserstein
  • ...and 3 more