Sufficient conditions for non-stability of stochastic differential systems

Zhenhai Yan

doi:10.1186/s13660-015-0894-y

Abstract

In this paper, we use the local $C^{2}$ -conjugate transformation to transform nonlinear stochastic differential systems to ordinary differential systems and give some sufficient conditions for the exponential stability and instability of stochastic differential systems. Our conditions just depend on the derivatives of drift terms and diffusion terms at equilibrium points.

Highlights

1 Introduction The exponential stability of nonlinear stochastic differential systems is always a point which has attracted a significant amount of concern in the last two decades and produced a lot of results and methods
Because of the complexities of differential systems, one used to employ Lyapunov function to discuss these problems in most of the literature, the methods have the advantage of discussing the stability of the systems if one only knows the existence of these solutions
Remark In the proof of Theorem, the constant NW depends on W (t), which means that NW may be different for different trajectories W (t) of Brownian motion

Summary

Introduction

The exponential stability of nonlinear stochastic differential systems is always a point which has attracted a significant amount of concern in the last two decades and produced a lot of results and methods (see [ – ]). It is an ordinary differential equation and the Brownian motion W (t) is a parameter in ) is an ordinary differential equation with the parameter W (t), we can discuss the stability of (A ) b(x) and σ (x) are continuous differentiable; (A ) there exists a positive constant M such that |b(x)| + |σ (x)| < M( + |x|). Let A = (aij)i,j≤m be a m × m matrix and suppose that every m dimensional vector is a m × matrix. Let A = ∂σ ( ), σ (x) and Ax be smooth C -equivalent, H be the C -conjugate mapping between σ (x), and Ax. Suppose that G(y) is the inverse mapping of H(x), ∂G( ) = (∂H( ))–.

One dimensional situation

High dimensional situation I