Stability of Multivariate Geometric Brownian Motion

Основные понятия

Study stability of multivariate geometric Brownian motion using Lyapunov functions and BMI problems.

Аннотация

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.

Статистика

Since no key metrics or figures were provided in the content, there are no stats to extract.

Цитаты

"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."

Ключевые выводы из

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

by Gerardo Barr... в arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.16765.pdf

Дополнительные вопросы

How does the Magnus expansion impact the representation of solutions?

The Magnus expansion plays a crucial role in representing solutions for non-commutative stochastic systems like the multivariate geometric Brownian motion. When no commutativity assumptions are made between the drift matrix and noise dispersion matrices, the solution to the system becomes complex. The Magnus expansion provides an exponential shape to the solution, expressed as exp(Y(t))x, where Y(t) is given in terms of iterated commutators. This results in a more intricate representation of the solution involving nested commutators, making it challenging to analyze and interpret.

What are the implications of non-commutativity assumptions on system stability?

Non-commutativity assumptions have significant implications on system stability. In stochastic systems like SDEs with non-commuting matrices, determining stability becomes more complex. The lack of commutativity leads to nonlinear terms in equations and complicates finding closed-form expressions for solutions. It also affects how Lyapunov functions can be constructed and applied to assess stability since traditional methods may not directly apply due to non-commuting elements.

How can Lyapunov functions be applied to other types of stochastic systems?

Lyapunov functions play a vital role in analyzing and determining stability properties for various types of stochastic systems beyond linear SDEs. These functions can be utilized in modeling random dynamical systems driven by both deterministic forces and external random fluctuations. By constructing suitable Lyapunov functions that satisfy specific criteria such as positivity or homogeneity, one can assess stability properties like global asymptotic stability or exponential p-stability for different types of stochastic differential equations (SDEs). Additionally, Lyapunov functions provide a robust framework for studying mean-square stability and assessing system behavior under uncertainties inherent in stochastic processes.