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Little seems to be known about evaluating the stochastic stability of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) via stochastic Lyapunov technique. The objective of this paper is to work with stochastic stability criterions for such systems. By defining a new derivative operator and constructing some suitable stochastic Lyapunov function, we establish some sufficient conditions for two types of stability, that is, stability in probability and moment exponential stability of a class of nonlinear SDEs driven by fBm. We will also give an example to illustrate our theory. Specifically, the obtained results open a possible way to stochastic stabilization and destabilization problem associated with nonlinear SDEs driven by fBm.

Fractional Brownian motion (fBm) is a family of Gaussian stochastic processes that appears naturally in the modeling of many situations. Kolmogorov [

According to the books [

Recently, some sufficient and necessary conditions for reducing the nonlinear stochastic systems driven by fBm to the linear ones were constructed in our previous work [

We organize this paper as follows. In Section

We consider the

It is convenient herein for us to give a few necessary notations. Let

Define a new derivative operator

Compared with the classical Itô (or Stratonovich) SDEs [

Since the kernel function

It turns out that there are at least three different types of stochastic stability: stability in probability, moment stability, and almost sure stability. We focus on the first two types in this paper since we already studied the third one in our previous paper [

We now give the definitions of stability in probability and the

The trivial solution of (

The trivial solution of (

The trivial solution of (

Assume that

We now extend the stochastic Lyapunov function techniques to the SDEs driven by fBm.

If there exists a positive-definite function

By the definition of a positive-definite function, we know that

Indeed, by the continuity of

Let

If there is a positive-definite, decrescent function

By the condition that

Let

If there is a positive-definite, decrescent, radially unbounded function

By the proof of Theorem

Next we focus on the

Assume that there exist a function

Fix any

By fractional Itô formula, we can derive that, for

Note that Hölder inequality implies

It should also be pointed out that when

It would be very hard to study the stochastic stability of some SDEs driven by fBm via Lyapunov function approach because of the nonlocal property of the kernel function

The greatest disadvantage of the stochastic Lyapunov technique is that no universal method has been given which enables you to find a Lyapunov function or determine that no such function exists.

Through the above discussion, we have established some stochastic-Lyapunov-function-based stability criterions for the SDEs (

Precisely, the Ornstein-Uhlenbeck process reads [

Equation (

We construct the time-dependent stochastic Lyanpunov function

First we check the positive-definite property of function (

Then we check the nonnegative property of the operator

Equation (

We construct the same stochastic Lyapunov function

Now we verify the decrescent property of (

Next we only need to check the nonnegative property of operator

Therefore, we prove this proposition according to Theorem

Equation (

We construct the same stochastic Lyapunov function

So we only need to check that function (

Therefore, we prove this proposition according to Theorem

Assume that

We first consider

On the other hand, by (

According to Theorem

When

Therefore, (

The nonlinear Ornstein-Uhlenbeck model driven by an fBm is given by

Equation (

We construct the time-dependent stochastic Lyapunov function

First we check the positive-definite property of function (

Then we check the nonnegative property of the operator

Equation (

We construct the same stochastic Lyapunov function

Now we verify the decrescent property of (

Next we only need to check the nonnegative property of operator

Therefore, we prove this proposition according to Theorem

Assume that

We first consider

On the other hand, by (

According to Theorem

When

Therefore, (

The aim of the time-dependent term in our constructed stochastic Lyapunov function is to eliminate the effect of nonlocal kernel function.

In this paper, we have established the stochastic Lyapunov techniques for SDEs driven by fBm. The obtained results are very effective to verify the two important types of stability, that is, stability in probability and moment exponential stability, for a given stochastic systems driven by fBm. Also, it opens a possible way to stochastic stabilization and destabilization problem associated with SDEs driven by fBm.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work was partly supported by the National Natural Science Foundation of China (nos. 11301090, 11271139, and 61104138), the Fundamental Research Funds for the Central Universities (no. 2014ZB0033), and Science and Technology Planning Project of Tianhe District, Guangzhou (no. 201301YG027).