Core Concepts
Study stability of multivariate geometric Brownian motion using Lyapunov functions and BMI problems.
Abstract
The content discusses the stability analysis of multivariate geometric Brownian motion through the use of Lyapunov functions and Bilinear Matrix Inequality (BMI) problems. It explores the conditions for global asymptotic stability in probability and exponential p-stability, providing insights into random dynamical systems modeled by stochastic differential equations. The manuscript delves into specific models from various fields such as physics, biology, and finance to exemplify the proposed method.
Introduction:
Discusses random dynamical systems and stochastic differential equations.
Introduces the concept of strong solutions for linear SDEs.
Preliminaries and Main Results:
Defines stability concepts like Stability in Probability (SiP), Global Asymptotic Stability in Probability (GASiP), and Exponential p-Stability (p-ES).
Presents Lyapunov's criterion for verifying GASiP.
Construction of BMI Problem for n = 2 and ℓ= 1:
Derives a Bilinear Matrix Inequality problem for n = 2 case with detailed computations.
Illustrates how to formulate BMI feasibility problems to ensure system stability.
Stats
Since no key metrics or figures were provided in the content, there are no stats to extract.
Quotes
"The solution of (1.1) is called a multivariate geometric Brownian motion."
"The null solution matrix On×n ∈Mn of (1.6) is asymptotically stable."