Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The stability of the multivariate geometric Brownian motion as a bilinear matrix inequality problem

Gerardo Barrera, Eyleifur Bjarkason, Sigurdur Hafstein

TL;DR

The paper studies the origin stability of the linear SDE with multiplicative noise, i.e., the multivariate geometric Brownian motion, and shows that global asymptotic stability in probability (GASiP) can be certified by a Lyapunov function $V(x)=\|x\|_Q^p$ whose verification reduces to a Bilinear Matrix Inequality (BMI) feasibility problem in the decision variables $Q$ and $c$. When $p=2$, the BMI collapses to a Linear Matrix Inequality (LMI), linking GASiP to exponential mean-square stability, but the authors emphasize that GASiP can be established even when 2-ES fails. The main contributions include a general BMI formulation for GASiP without commutativity assumptions, a detailed construction in the $n=2,\ell=1$ case, and extensive numerical demonstrations across 2D and 3D models (random oscillators, satellite dynamics, inertia systems, diagonal/off-diagonal noise, cancer remission, and smoking models). The results highlight the practical value of the BMI approach, enabling stability certification beyond the conservative confines of 2-ES and suggesting heuristics to solve BMIs via sequences of LMIs, with potential extensions to time-varying coefficients.

Abstract

In this manuscript, we study the stability of the origin for the multivariate geometric Brownian motion. More precisely, under suitable sufficient conditions, we construct a Lyapunov function such that the origin of the multivariate geometric Brownian motion is globally asymptotically stable in probability. Moreover, we show that such conditions can be rewritten as a Bilinear Matrix Inequality (BMI) feasibility problem. We stress that no commutativity relations between the drift matrix and the noise dispersion matrices are assumed and therefore the so-called Magnus representation of the solution of the multivariate geometric Brownian motion is complicated. In addition, we exemplify our method in numerous specific models from the literature such as random linear oscillators, satellite dynamics, inertia systems, diagonal and non-diagonal noise systems, cancer self-remission and smoking.

The stability of the multivariate geometric Brownian motion as a bilinear matrix inequality problem

TL;DR

The paper studies the origin stability of the linear SDE with multiplicative noise, i.e., the multivariate geometric Brownian motion, and shows that global asymptotic stability in probability (GASiP) can be certified by a Lyapunov function whose verification reduces to a Bilinear Matrix Inequality (BMI) feasibility problem in the decision variables and . When , the BMI collapses to a Linear Matrix Inequality (LMI), linking GASiP to exponential mean-square stability, but the authors emphasize that GASiP can be established even when 2-ES fails. The main contributions include a general BMI formulation for GASiP without commutativity assumptions, a detailed construction in the case, and extensive numerical demonstrations across 2D and 3D models (random oscillators, satellite dynamics, inertia systems, diagonal/off-diagonal noise, cancer remission, and smoking models). The results highlight the practical value of the BMI approach, enabling stability certification beyond the conservative confines of 2-ES and suggesting heuristics to solve BMIs via sequences of LMIs, with potential extensions to time-varying coefficients.

Abstract

In this manuscript, we study the stability of the origin for the multivariate geometric Brownian motion. More precisely, under suitable sufficient conditions, we construct a Lyapunov function such that the origin of the multivariate geometric Brownian motion is globally asymptotically stable in probability. Moreover, we show that such conditions can be rewritten as a Bilinear Matrix Inequality (BMI) feasibility problem. We stress that no commutativity relations between the drift matrix and the noise dispersion matrices are assumed and therefore the so-called Magnus representation of the solution of the multivariate geometric Brownian motion is complicated. In addition, we exemplify our method in numerous specific models from the literature such as random linear oscillators, satellite dynamics, inertia systems, diagonal and non-diagonal noise systems, cancer self-remission and smoking.
Paper Structure (19 sections, 4 theorems, 145 equations)

This paper contains 19 sections, 4 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Main results and consequences
  4. Construction of the BMI problem for $n=2$ and $\ell=1$
  5. Examples and simulations
  6. Dimension two
  7. Random linear oscillator
  8. Satellite dynamics
  9. Two-inertia systems
  10. Diagonal noise system
  11. Off-diagonal noise systems
  12. Dimension three
  13. Cancer self-remission and tumor stability
  14. Stochastic stability of a mathematical model of smoking
  15. Conclusions
  16. ...and 4 more sections

Key Result

Theorem 2.5

The null solution $(X(t;0_\textsf{n}))_{t\geq 0}$ of eq:modelito is GASiP, if and only if for a $p>0$ there exist positive constants $c_1,c_2,c_3,c_4,c_5$, a scalar function $V\in \mathcal{C}(\mathbb{R}^n, \mathbb{R})\cap \mathcal{C}^2(\mathbb{R}^n\setminus \{0_\textsf{n}\},\mathbb{R})$ and $Q\succ where $\|x\|_Q:=(x^*Qx)^{1/2}$ for all $x\in \mathbb{R}^n$.

Theorems & Definitions (19)

  • Definition 2.1: Stability in Probability (SiP)
  • Definition 2.2: Asymptotically stable in probability (ASiP)
  • Definition 2.3: Global Asymptotic Stability in Probability (GASiP)
  • Definition 2.4: Exponential $p$-Stability (p-ES)
  • Theorem 2.5: Lyapunov's criterium
  • Lemma 2.6: Lyapunov function
  • Remark 2.7: LMI and BMI problems
  • Definition 2.8: Bilinear Matrix Inequality (BMI)
  • Theorem 2.9: GASiP as a BMI problem
  • Remark 2.10: One-dimensional Geometric Brownian motion
  • ...and 9 more