Lyapunov Techniques for Stochastic Differential Equations Driven by Fractional Brownian Motion

and Applied Analysis 3 focus on the first two types in this paper since we already studied the third one in our previous paper [23]. We now give the definitions of stability in probability and the pth moment exponential stability, which are the same as those in [24]. Definition 3. The trivial solution of (2) is said to be stochastically stable or stable in probability if, for every pair of ε ∈ (0, 1) and r > 0, there exists a δ = δ(ε, r, t 0 ) > 0, such that P { 󵄨󵄨󵄨󵄨x (t; t0, x0) 󵄨󵄨󵄨󵄨 < r, ∀t ≥ t0} ≥ 1 − ε, (11) whenever |x 0 | < δ. Otherwise, it is said to be stochastically unstable. Definition 4. The trivial solution of (2) is said to be stochastically asymptotically stable if it is stochastically stable and, moreover, for every ε ∈ (0, 1), there exists a δ = δ(ε, t 0 ) > 0, such that P { lim t→∞ x (t; t 0 , x 0 ) = 0} ≥ 1 − ε, (12) whenever |x 0 | < δ. Definition 5. The trivial solution of (2) is said to be stochastically asymptotically stable in the large if it is stochastically stable and, moreover, for x 0 ∈ R P { lim t→∞ x (t; t 0 , x 0 ) = 0} = 1. (13) Definition 6. Assume that p > 0. The trivial solution of (2) is said to be pth moment exponentially stable if there is a pair of positive constants λ and C such that E [ 󵄨󵄨󵄨󵄨x (t; t0, x0) 󵄨󵄨󵄨󵄨 p ] ≤ C 󵄨󵄨󵄨󵄨x0 󵄨󵄨󵄨󵄨 p e −λ(t−t0), t ≥ t 0 , (14) for all x 0 ∈ R. 3. Main Results We now extend the stochastic Lyapunov function techniques to the SDEs driven by fBm. Theorem7. If there exists a positive-definite functionV(x, t) ∈ C(S h × [t 0 ,∞);R + ) such that L H V (x, t) ≤ 0 (15) for all (x, t) ∈ S h × [t 0 ,∞), then the trivial solution of (2) is stochastically stable. Proof. By the definition of a positive-definite function, we know that V(0, t) ≡ 0, and there is a function μ ∈ K, such that V (x, t) ≥ μ (|x|) , ∀ (x, t) ∈ Sh × [t0,∞) . (16) Let ε ∈ (0, 1) and r > 0be arbitrary.Without loss of generality, we assume that r < h. Indeed, by the continuity of V(x, t) and the fact that V(0, t 0 ) = 0, we can find a δ = δ(ε, r, t 0 ) > 0, such that


Introduction
Fractional Brownian motion (fBm) is a family of Gaussian stochastic processes that appears naturally in the modeling of many situations.Kolmogorov [1] was the first to consider this process and called it "Wiener Spirals." Later, Hurst [2,3] studied the long-term water flow characteristics of the Nile River and the parameter  then got the name "Hurst parameter." Mandelbrot and Van Ness [4] established a stochastic integral representation in terms of a standard Brownian motion.Since the introduction of the above mentioned pioneering work, fBm has played an increasingly important role in many fields of application such as hydrology, economics, and telecommunications (see [5] for a review).
According to the books [6,7], the standard fBm (  (),  ≥ 0) is defined as a self-similar centered Gaussian process with covariance function Cov (  () , where Hurst parameter 0 <  < 1.When  = 1/2, one recovers of course the usual Brownian motion, so this is a natural one-parameter family of generalizations of the "standard" Brownian motion.When  ̸ = 1/2, it was proved in [8] that fBm is not semimartingale.Therefore, the beautiful classical theory of stochastic analysis [9] is not applicable to stochastic differential equations (SDEs) driven by fBm with  ̸ = 1/2.It is a significant and challenging problem to extend the results in the classical stochastic analysis to these fBm ones.Over the last years some new techniques have been developed in order to define stochastic integrals with respect to fBm [10][11][12][13][14][15][16][17][18][19][20][21].For example, stochastic integral of deterministic functions with respect to fBm is called Wiener integral, which was defined for the first time in [10].The stochastic integral of Stratonovich type for fBm was defined in [11,12].However, the stochastic integral ∫  0 ()  (), introduced in [11,12], does not satisfy in general the following property:  ∫  0 ()  () = 0, which is important in the modeling problem by stochastic differential equations with fractional Gaussian noise as the driving random process.Motivated by this situation, Duncan et al. [13] defined a new stochastic integral of Itô type for fBm with Hurst parameter in the interval (1/2, 1).This stochastic integral is the limit of Riemann sums defined by means of the Wick products rather than ordinary products.In this paper, we adopt this stochastic integral definition.Then Elliott and Van der Hoek in [16] extended this fractional Itô calculus theory to all Hurst parameter  ∈ (0, 1) and applied it to develop option pricing in a fractional Black-Scholes market.This newly developed theory of stochastic integration with respect to fBm, based on white-noise theory and (Malliavin-type) differentiation, was introduced in [17].For other definitions of stochastic integrals for fBm and their relations, we refer to the book [6] for further details.
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 [22], which provide an effective approach to solve some linear and nonlinear fBm-driven stochastic systems.Indeed, necessary and sufficient conditions were established for stochastic stability of the Black-Scholes model driven by fBm by means of the Lyapunov exponents and the exact form of the solutions [23].Unfortunately, it is in general not possible to give explicit expressions for the solutions to SDEs and numerical solution is a cumbersome affair.It is therefore of great interest to study qualitative properties of SDEs driven by fBm without solving the equations.Therefore, the scope of this paper is to extend the stochastic Lyapunov function technique to SDEs driven by fBm without solving the considered equations.
We organize this paper as follows.In Section 2, we briefly introduce some necessary notations and stochastic stability concepts associated with SDEs driven by fBm.In Section 3, we state the main results on stability of SDEs driven by fBm via Lyapunov function technique.In Section 4, we apply the stability criterions to the Ornstein-Uhlenbeck process driven by fBm by constructing a time-dependent Lyapunov function.The conclusions are drawn in Section 5.

Stochastic Stability Concepts.
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 [23].
We now give the definitions of stability in probability and the th moment exponential stability, which are the same as those in [24].Definition 3. The trivial solution of ( 2) is said to be stochastically stable or stable in probability if, for every pair of  ∈ (0, 1) and  > 0, there exists a  = (, ,  0 ) > 0, such that whenever | 0 | < .Otherwise, it is said to be stochastically unstable.
Definition 5.The trivial solution of ( 2) is said to be stochastically asymptotically stable in the large if it is stochastically stable and, moreover, for Definition 6. Assume that  > 0. The trivial solution of ( 2) is said to be th moment exponentially stable if there is a pair of positive constants  and  such that for all  0 ∈ R  .

Main Results
We now extend the stochastic Lyapunov function techniques to the SDEs driven by fBm.
Proof.By the definition of a positive-definite function, we know that (0, ) ≡ 0, and there is a function  ∈ K, such that Let  ∈ (0, 1) and  > 0 be arbitrary.Without loss of generality, we assume that  < ℎ.
Indeed, by the continuity of (, ) and the fact that (0,  0 ) = 0, we can find a  = (, ,  0 ) > 0, such that It is not difficult to see that  < .Now fix the initial value  0 ∈ S  arbitrarily and write (;  0 ,  0 ) as () for simplicity.
Let  be the first exit time of () from S  ; that is, For  ≥  0 , it follows that Taking the expectation on both sides and utilizing the condition   (, ) ≤ 0, we have Note that if  ≤ .Hence, by ( 16), we further get Together with (17), (20), and ( 22), we have Letting  → ∞, we get That is, Thus the proof is established.
Proof.By the condition that   (, ) is negative definite, there exists a function  ∈ K, such that Then it follows from the definition of K-class functional that This, together with the positive-definite property of function (, ), satisfies the required conditions in Theorem 7. It means that the trivial solution is stochastically stable.So we only need to show that, for any  ∈ (0, 1), there is a  = (,  0 ) > 0 such that whenever | 0 | < .Note that the assumptions on function (, ) mean that (0, ) ≡ 0, and moreover, there are three functions  1 ,  2 ,  3 ∈ K such that for all (, ) ∈ S ℎ × [ 0 , ∞).
Let  ∈ (0, 1) be arbitrary, and by Theorem 7, there is a  = (,  0 ) > 0, such that For any  0 ∈ S  , write (;  0 ,  0 ) = () simply.Let 0 <  < | 0 | be arbitrary and choose 0 <  <  sufficiently small for Define the stopping times For any  ≥  0 , it follows from (29) that On the other hand, it yields It then follows from (33) and ( 34) that Letting  → ∞, we have On the other hand, it follows from (30) and (32) that Hence, which yields Choose  sufficiently large for Then Now, define two stopping times For any  ≥ , it follows Noting that on  ∈ {  ≥  ℎ ∧ }, we get Utilizing (30) and the fact that {  ≤ } ⊂ {  <  ℎ ∧ } we further have Together with (31), we deduce that Letting  → ∞, we have It follows from (41) that This implies that Since  is arbitrary, we must have as required.The proof is complete.
Proof.By the proof of Theorem 8, the trivial solution of equation is stochastically stable.So we only need to show that for all  0 ∈ R  ; fix any  0 and write (;  0 ,  0 ) = () again.
Let  ∈ (0, 1) be arbitrary; since (, ) is radially unbounded, we can find an ℎ > | 0 | sufficiently large for lim Define the stopping time For any  ≥  0 , it follows that But, it follows from (53) that It then follows from (55) that Let  → ∞; we have That is, From here, we can show in the same way as the proof of Theorem 8 that Since  is arbitrary, the required equation ( 52) must hold and thus the proof is complete.
Next we focus on the th moment exponential stability of (2) and always let  > 0. Now we establish a sufficient criterion for the th moment exponential stability by using a stochastic Lyapunov function.
Remark 11.Note that Hölder inequality implies for 0 <  < .Then th moment exponential stability implies th moment exponential stability.In particular, when  = 2, it is usually said to be exponentially stable in square-mean sense.
Remark 12.It should also be pointed out that when  = 1/2, the statements in these theorems reduce to Itô SDEs [24]; when (, ) = 0, these statements reduce to the corresponding deterministic ones.
Remark 13.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 (⋅, ⋅).
Remark 14.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.

Two Examples
Through the above discussion, we have established some stochastic-Lyapunov-function-based stability criterions for the SDEs (2) driven by fBm.Now we apply the obtained criterions to check the stochastic stability of Ornstein-Uhlenbeck process driven by fBm and nonlinear radial Ornstein-Uhlenbeck driven by fBm.To do this, the procedure is to construct some suitable stochastic Lyapunov function and verify the required properties of a new derivative operator   .