Sufficient conditions for non-stability of stochastic differential systems
- Zhenhai Yan^{1, 2}Email author
https://doi.org/10.1186/s13660-015-0894-y
© Yan 2015
Received: 12 May 2015
Accepted: 12 November 2015
Published: 2 December 2015
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.
Keywords
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 (see [1–4]). 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.
However, there are many inconveniences in the application on account of the difficulties in the construction of the Lyapunov function. So we try to transform nonlinear stochastic differential systems to ordinary differential systems in which the Brownian motion is just a parameter of them, and then discuss the stability in different situations.
We remark that equation (1.2) is different from equation (1.1). It is an ordinary differential equation and the Brownian motion \(W(t)\) is a parameter in (1.2). We can get the solution of (1.2) for many \(b(x)\), such as \(b(x)=kx\) or \(b(x)=kx+hx^{n}\).
By the law of the iterated logarithm for Brownian motion (see [6], p.112), the exponential stability of \(Y(t)\) is equivalent to that of \(X(t)\). Because (1.2) is an ordinary differential equation with the parameter \(W(t)\), we can discuss the stability of (1.2) with a lot of methods.
The main purpose of this paper is to extend this result to a wider range. This paper will be organized as follows: We shall first give preliminaries: some definitions and fundamental lemmas which will be used in the following and together with the transformations in different situations. Next we shall give the main results of this paper. In this section we not only discuss the exponential stability of nonlinear stochastic differential systems in a one dimensional situation but also consider the high dimensional situation in two cases.
2 Preliminaries: transformation
- (i)
\(\mathscr{F}_{0}\) includes all the events whose probability is 0;
- (ii)
\(\mathscr{F}_{t}\) is right continuous.
Let \(\{W(t)\}_{t\geq 0}\) be the standard d dimensional Brownian motion which is defined on \((\Omega, \mathscr{F}, \{\mathscr{F}_{t}\}_{t\geq0}, P)\) and is adapted to \(\{\mathscr{F}_{t}\}_{t \geq0}\).
- (A1)
\(b(x)\) and \(\sigma(x)\) are continuous differentiable;
- (A2)
there exists a positive constant M such that \(|b(x)|+|\sigma (x)|< M(1+|x|)\).
- (A3)
0 is the only 0 point of \(b(x)\) and \(\sigma(x)\), and \(\partial \sigma(0)\) is not 0.
If \(\sigma(x)\) and \(b(x)\) satisfy (A1), (A2), and (A3), then (2.1) has the unique solution \(X(t)\equiv0\) corresponding to the initial value. We call the solution a trivial solution or equilibrium position.
From Lemma 2.1 of [8], we have the following result.
Lemma 1
Definition 1
Definition 2
- (i)
H is \(C^{2}\)-differentiable homeomorphism;
- (ii)
\(\partial H(x)U(x)=V(H(x))\), \(\forall x\in O_{\rho}(0)\).
2.1 One dimensional situation
Let \(m=d=1\). By (A3), we can suppose that \(\sigma(x)>0\) when \(x\in(0, \infty)\). If \(x\in(-\infty, 0)\), then we have \(\sigma(x)<0\) and \(\sigma'(0)>0\), where \(\sigma'(x) \) is the derivative of \(\sigma(x)\).
We notice that equation (2.8) is an ordinary differential equation, where \(W(t)\) is a parameter of it. Thus (2.8) has a unique solution for every trajectory \(W(t)\) of Brownian motion.
2.2 High dimensional situation I
Let \(A=\partial\sigma(0)\), \(\sigma(x)\) and Ax be smooth \(C^{2}\)-equivalent, H be the \(C^{2}\)-conjugate mapping between \(\sigma (x)\), and Ax. Suppose that \(G(y)\) is the inverse mapping of \(H(x)\), then \(\partial G(0)=(\partial H(0))^{-1}\). By Definition 2, we have \(\partial H(x)\sigma(x)=A H(x)\), \(\forall x\in O_{\rho}(0)\). Taking derivative of both sides above, we can get \(\partial H(0)A=A\partial H(0)\), which means that \(\partial H(0)\) and A are commutative.
2.3 High dimensional situation II
Next we suppose \(d>1\). Let \(W(t)=(W_{1}(t),\ldots,W_{d}(t))\) and \(\sigma(x)=(\sigma_{1}(x),\ldots,\sigma_{d}(x))\). Here \(W_{1}(t), \ldots, W_{d}(t)\) are d independent Brownian motions, \(\sigma_{i}(x)\) is a function defined on \(\mathbb{R}^{m}\), where \(i=1, 2, \ldots, d\).
Let \(A_{1}=\partial\sigma_{1}(0),\ldots, A_{d}=\partial\sigma_{d}(0)\). Now suppose that \(\sigma_{k}(x)\) and \(A_{k} x\) are \(C^{2}-\)equivalent, where \(k=1, \ldots, d\). For \(m=1\), the \(C^{2}-\)equivalence will be automatically established.
3 Main results: exponential stability and instability
3.1 One dimensional situation
Let \(\gamma=b'(0)-\frac{\sigma'(0)}{2}\) in equation (2.8), then we have the following results.
Theorem 1
If \(\gamma<0\), then the trivial solution of (2.1) is exponentially stable.
Proof
We only prove Theorem 1 in the case \(x_{0}>0\), because the proof of it is similar in the case \(x_{0} \leq0\).
The proof is similar in the case \(x_{0}<0\). So we omit it here. Then we complete the proof. □
Remark 1
Theorem 2
If \(\gamma>0\), then the trivial solution of (2.1) is unstable.
Proof
By the law of iterated logarithm of Brownian motion, almost surely, we know that \(\limsup_{t\rightarrow\infty}X(t;x_{0})=\infty\) when \(T=\infty\), which is a contradiction.
So \(P\{T<\infty\}=1\), which is the same as \(\sup_{t>0}|X(t;x_{0})|\geq \epsilon\), a.s., so the trivial solution of (2.1) is unstable.
We can prove the conclusion in the case \(x_{0}<0\) similarly. So we omit it here.
Finally Theorem 2 is proved. □
3.2 High dimensional situation I
Let \(\Gamma_{1}=\partial b(0)-\frac{1}{2}A^{2}\) in equation (2.13). Then we have the following results.
Theorem 3
If \(\partial b(0)\) and A are commutative, and the real parts of all the eigenvalues of \(\Gamma_{1}\) are negative, then the trivial solution of (2.1) is exponentially stable.
Proof
Finally we complete the proof of Theorem 3 from Theorem 1. □
Theorem 4
If \(\partial b(0)\) and A are commutative, and there is a positive one in the real part of all the eigenvalues of \(\Gamma_{1}\), then the trivial solution of (2.1) is unstable.
Proof
We omit the proof of Theorem 4 because we can obtain it in a completely parallel way to the proof of Theorem 2. □
When the real part of all the eigenvalues of \(\Gamma_{1}\) is 0, we cannot judge the stability of the trivial solution of (2.1).
Example 1
It is easy to see that if the multiplication of all the eigenvalues of A is 1, then the trivial solution of (2.1) is stable. If not, the trivial solution of (2.1) is unstable.
3.3 High dimensional situation II
Theorem 5
If \(\partial b(0)\), \(A_{1}, \ldots, A_{d}\) are commutative and the real parts of all the eigenvalues of \(\Gamma_{d}\) are negative, then the trivial solution of (2.1) is exponentially stable.
Theorem 6
If \(\partial b(0)\), \(A_{1}, \ldots, A_{d}\) are commutative and there is a positive one in the real parts of all the eigenvalues of \(\Gamma_{d}\), then the trivial solution of (2.1) is unstable.
Declarations
Acknowledgements
The author is thankful to the referees for their helpful suggestions and necessary corrections in the completion of this paper.
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Authors’ Affiliations
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