Exponential stability of SDEs driven by fBm with Markovian switching

In this paper, we focus on the exponential stability of stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter $ H\in(1/2, 1) $. Based on the generalized Ito formula and representation of the fBm, some sufficient conditions for exponential stability of a class of SDEs with additive fractional noise are given. Besides, we present a criterion on the exponential stability for the fractional Ornstein-Uhlenbeck process with Markov switching. A numerical example is provided to illustrate our results.


1.
Introduction. To characterize the continuous dynamical system changes with the discrete state, the following stochastic differential equations (SDEs) have been developed dX t = f (X t , r t )dt + σ(X t , r t )dB t , where {r t } t≥0 is a Markov chain taking values in S = {1, 2, ..., N } and {B t } t≥0 is a standard Brownian motion. The process {X t , r t } is called a switching diffusion or a diffusion with switching. In the past thirty years, stability of stochastic hybrid systems has been considered extensively. For example, Yuan and Mao [30] consider the moment exponential stability of stochastic hybrid delayed systems with Lévy noise in mean square. Mao [18] discusses the exponential stability of general nonlinear stochastic hybrid systems. Some sufficient conditions for asymptotic stability in distribution of SDEs with Markovian switching are given by Yuan and Mao [29]. Most recently, Tan [27] focuses on the exponential stability of fractional stochastic systems with distributed delay driven by fractional Brownian motion. There are lots of work having been dedicated to Markovian switching. See [24,3,7,17,32,23] and so forth. It is well known that if H > 1/2, {B H t } t≥0 exhibits long range dependence and self-similarity. Because of these properties, {B H t } t≥0 has been suggested as a useful tool in many fields, especially mathematical finance, network traffic analysis and pricing of weather derivatives. For example, fBm is used to model the dynamics of temperature in [6], and in [25], it is used to model the electricity prices in the liberated Nordic electricity market. However, some statisticians find that it is better to model the pricing with hybrid system(see, e.g., [12,28]). Hence, it is a natural question that under what conditions, SDEs driven by fBm with Markov switching have some exponential stability.
The main purpose of this paper is to consider the pth moment exponential stability of stochastic hybrid systems driven by fractional Brownian motion of the form: dX t = f (X t , t, r t )dt + σ(t, r t )dB H t , where {r t } t≥0 is a Markov chain taking values in S = {1, 2, ..., N }, {B H t } t≥0 is a standard fractional Brownian motion. Moreover f : R×R + ×S → R, σ : R + ×S → R.
This equation can be regarded as the result of the following N equations: switching from one to another according to the movement of {r t } t≥0 . Note that for each fixed i, σ(t, i) is nonrandom. From [10,8,21,14], we know that there exists a R−valued global solution satisfying Eq.(2), rather than Eq.(1), under suitable conditions, for each fixed i ∈ S. Therefore, our first goal is to obtain the existence and uniqueness of the solution for Eq.(1). Then we discuss the pth moment exponential stability for Eq.(1).
Throughout this paper, unless otherwise specified, the fBm We let (Ω, F, P, F t ) be the complete probability space, with the filtration described above. Also let C denote a general constant. Let C 2,1 (R×R + ×S; R) denote the family of all real value functions f (x, t, i) on R×R + ×S which are continuously twice differentiable in x and once differentiable in t. The Markov chain {r t } t≥0 is assumed to be independent of {B H t } t≥0 . Almost every sample path of the Markov chain {r t } t≥0 is assumed to be a right-continuous step function with a finite number of simple jumps in any finite subinterval of R + .
The rest of the paper is organized as follows. In Section 2, we shall briefly revisit some basic facts regarding Markovian switching, stochastic integration and the Itô formula with respect to fBm, and some preliminaries for our main results. In Section 3, we shall show the existence and uniqueness of the solution for Eq.(1) firstly. Then we will discuss the sufficient conditions to guarantee the pth moment exponential stability. Next, in Section 4, we shall use the theory of Poisson equation and Mmatrix to establish some criteria for the exponential stability. Then in Section 5, we will discuss the stability of switching fractional Ornstein-Uhlenbeck process. Finally, a numerical example will be given in Section 6.

Preliminaries.
2.1. Markov chain. Let {r t } t≥0 be a right-continuous Markov chain which takes values in a finite state space S = {1, 2, ..., N }. The generator Γ = (γ ij ) N ×N is given by Here γ ij is the transition rate from i to j if i = j while γ ii = − i =j γ ij .
Theorem 2.1. [2] Let P (t) = (P ij (t)) N ×N be the transition probability matrix and Γ = (γ ij ) N ×N be the generator of a finite Markov chain. Then Let ∆ ij be consecutive, left closed, right open intervals each having the length γ ij such that According to [11,26], a continuous-time Markov chain {r t } t≥0 with generator Γ = (γ ij ) N ×N can be represented as a stochastic integral with respect to a Poisson random measure. Then with initial condition r 0 = i 0 , where ν(dt × dy) is a Poisson random measure with intensity dt × m(dy). Here m(·) is the Lebesgue measure on R.

2.2.
Fractional Brownian motion and Wick product. Given a finite time interval [0, T ] with arbitrary fixed horizon T > 0, and let {B H t } t≥0 be a one-dimension standard fBm with Hurst parameter H ∈ (1/2, 1), i.e. a centered Gaussian process with covariance function: has the following Wiener integral representation: where {W t } t≥0 is a Wiener process and K H (t, s) is the kernel function defined by is the Beta function, and t > s. For more details about fBm, we refer the reader to [21,22,1].
Let I be the set of all finite multi-indices α = (α 1 , . . . , α n ) for some n ≥ 1 of non-negative integers. Denote |α| = α 1 + · · · + α n , and α! = α 1 ! · · · α n !. For n ≥ 0, define the Hermite polynomials by and Hermite functionsh Let S(R) denote the Schwartz space of rapidly decreasing infinitely differentiable real-valued functions, and denote the dual space of S(R) by S (R). Define the product of Hermite polynomials. Consider a square integrable random variable Thus, according to [21,13], every F (ω) admits a unique representation Definition 2.2. (Wick Product) For F (ω) = α∈I c α H α (ω) and G(ω) = β∈I d β H β (ω). Their Wick product is defined by and let if the limit exists in L p . Moreover if there exists a process (D φ s F s , s ≥ 0) such that and for each sequence of partitions π n , n ∈ N + such that π n → 0, as n → ∞. Moreover Next, we define a stochastic integral with respect to fBm considered in [5].
What's more, by Definition 3.4.1 in [5], the stochastic integral can be extended as follows H . Here (S) * H is the fractional Hida distribution space defined by Definition 3.1.11 in [5]. And in this extension, the integral on an interval [0, T ] can be defined by can find from textbooks (cf., Chapter 3 of Karatzas and Shreve [16]).
2.4. The Itô formula. At first, we shall review the results in [9] on the Itô formula for fBm. Then we will extend them to SDEs driven by fBm with Markovian switching.
Assume that there exists an γ > 1 − H and C > 0 such that where |u − v| ≤ δ for some δ > 0 and lim 0≤u,v≤t,|u−v|→0 Here D φ s x s is the Malliavin derivative defined in Definition 2.2.
In particular, for the process X φ is a deterministic function, then for each fixed i ∈ S, we have Substituting (5) into (4), we get In the sequel of our paper, unless otherwise specified, we let the coefficients of Eq.(1) satisfy the conditions in Theorem 2.5, for each fixed i ∈ S. Let V (X t , t, r t ) ∈ C 2,1 (R × R + × S; R + ). Then we will discuss an Itô's formula which reveals how V maps (X t , t, r t ) into a new process V (X t , t, r t ). Here {X t } t≥0 is a stochastic process with the stochastic differential (1).
3. Some properties of solutions of Eq.(1). In this section, we will consider the existence and uniqueness of the solution of Eq.(1). Besides, the pth moment exponential stability conditions will be presented.

Existence and uniqueness.
To ensure the existence and uniqueness of the solution, we shall impose the following basic assumptions.
(i) f is measurable, and there exists K > 0 such that (ii) There existsK > 0 such that  Proof. Recall that the Markov chain (r t ) t≥0 can be rewritten as where r τ k = i, and τ k+1 − τ k is exponentially distributed. The jump η k+1 = r τ k+1 − r τ k is independent of the past. According to [21,14], there exists a unique global solution to Eq.(2), for each i ∈ S,

Note the unique solution {X
(i) t } t≥0 is a stochastic process without Markov switching. We denote the unique solution by {X X0,i t } t≥0 . For each k ∈ N, t ∈ [τ k , τ k+1 ), we have r t = j ∈ S. Thus, we obtain a sequence of solutions {X Xτ k ,rτ k t } {t≥0, k∈N} . According to Lemma 3.1 of [31], we construct the solution to Eq.(1) as follows.
For t ∈ [0, τ 1 ), we define where x 0 is the initial value.

Using condition (i) one can show that
Then, according to the Gronwall inequality, (12) implies that Letting T → ∞, together with the continuity of sample path, one has P (X t =X t for all t ≥ 0) = 1.
The proof is complete.

Exponential stability.
In the sequel of this section, we will state one of the main criteria of this paper.
Theorem 3.2. Let Assumption 1-2 hold. If there exists a function V ∈ C 2,1 (R × R + ; R + ) and positive constants a 1 , a 2 , b and p ≥ 1, such that for all x ∈ R, t ≥ 0, i ∈ S. Then the solution of Eq.(1) is pth moment exponential stable, i.e.

Consequently, sup
Letting T → ∞ gives sup t≥0 E|X t | p ≤ e −ηt a 1 EV (X 0 , 0), and the required assertion follows. The proof is complete 4. Main results. In this section we shall use the theory of Poisson equation and M-matrix to establish some criteria for the exponential stability. These criteria can be more useful and can be verified much more easily than the general one in the previous section. For the convenience of the reader, we will introduce some useful notation and basic properties on M-matrix firstly. Let B be a vector or matrix. By B ≥ 0 we mean that all elements of B are nonnegative. By B 0, we mean that all elements of B are positive. By B > 0 we mean B ≥ 0 and at least one element of B is positive. Moreover, we also write It is easy to see that if A is a non-singular M-matrix then it has nonpositive off-diagonal and positive diagonal entries, that is a ii > 0, and a ij ≤ 0, i = j.
There are many conditions equivalent to the statement that A is a nonsingular M-matrix. Now we cite some of them for the use of this paper, and refer to [4] for more details.

3.
A is semipositive; that is, there exists x 0 in R N such that Ax 0.
4.Every real eigenvalue of A is positive.

5.
A is inverse-positive; that is A −1 exists and A −1 ≥ 0.
To discuss the stability, we impose the following assumption.
Here the constants β i could be positive or negative, which is different from Theorem 3.2. For the vector β = (β 1 , . . . , β N ) T , we use diag(β) = diag(β 1 , . . . , β N ) to denote the diagonal matrix generated by β as usual. Next, we will give a very simple criterion. Theorem 4.3. Assume that Assumptions 1-2 hold and there exists a function V ∈ C 2,1 (R × R + ; R + ) such that Assumption 3 holds and where (µ i ) i∈S is the invariant probability measure of {r t } t≥0 . Then the solution of Eq.(1) is pth moment exponential stable.
Note that (17) has the solution c = (c 1 , . . . , c N ) T . Then one has for i ∈ S, Then we get By [28], one has Making use of (19) and (20), we obtain that Substituting (18) into (21), we get where κ < 0. Making use of Theorem 3.2, we can show that the solution of Eq.(1) is pth moment exponential stable. The proof is complete.
In the sequel of this section, we shall use the theory of M-matrix to establish a criterion.
Proposition 1. Assume that Assumptions 1-2 hold and there exists a function V ∈ C 2,1 (R × R + ; R + ) such that Assumption 3 is satisfied and the matrix −(Γ + diag(β)) is a nonsingular M-matrix. Then the solution of Eq.(1) is pth moment exponential stable.
Proof. By (6), we have Note that min 1≤i≤N ( λi ξi ) > 0. Then making use of Theorem 3.2, we can show that the solution of Eq.(1) is pth moment exponential stable. The proof is complete.

5.
Fractional Ornstein-Uhlenbeck processes. In the previous sections, we provide some criteria for general SDEs driven by fractional Brownian motion with Markov switching. In this section, we will present a criterion for switching fractional Ornstein-Uhlenbeck process. Without loss generality, we consider the following Ornstein-Uhlenbeck process with Markov switching where {r t } t≥0 is a Markov chain taking values in S = {1, 2, ..., N }, α(i) and σ(i) are constants for each fixed i ∈ S. {B H t } t≥0 is a standard fractional Brownian motion, independent of {r t } t≥0 . In order to simplify the proof, we assume that x 0 = 0. We first provide a useful lemma. Then there exists constants C, c, α > 0 such that: for any initial condition r 0 and every t ≥ 0.
Proof. It is a consequence of Perron-Frobenius theorem and the study of eigenvalues. For further details, see also the proofs of [3] and Lemma 2.7 of [7]. Now we are able to give the desired criterion.
Proof. It is well known that for each fixed i ∈ S, Let Υ(t) = t 0 α(r s )ds, σ = max{|σ(i)|, i ∈ S}, and X t be the global solution of Eq.(22). By Theorem 3.1 and [15], one has We then see from (23) that Thus, by Hölder's inequality, we can derive that where m > 1, 1/m + 1/n = 1. According to Lemma 5.1, (24) implies where α > 0, C is a general positive constant. Making use of Theorem 1.1 of [20] and (25), in a similar way, we can derive that for any T > 0 and t ∈ [0, T ],  E|X t | p e (α−α)pt ≤ C.
Consequently, the required assertion follows. The proof is complete.
6. An example. In this section we shall give a numerical example to illustrate our results. Consider a one-dimension stochastic process with Markovian switching of the form: dX t = −α(r t )X t dt + σ(t, r t )dB H t , X 0 = 1, Note also that there exists ε sufficiently small and t 0 such that for t > t 0 , 2ε|X t | 2 > e −ηt .

LITAN YAN, WENYI PEI AND ZHENZHONG ZHANG
It then follows from Theorem 4.3 that Eq.(26) is second moment exponential stable. Fig.(1)-(2) show a single path of the solution and the corresponding solution's norm square, respectively.