Let { r t } t ≥ 0 be a right-continuous Markov chain taking values in a finite state space S = { 1,2 , … , N } . The generator Q = ( q i j ) N × N is given by P { r t + △ = j ∣ r t = i } = q i j △ + o ( △ ) , if i ≠ j , 1 + q i j △ + o ( △ ) , if i = j , where △ > 0.

Here q _ij is the transition rate from i to j if i ≠ j. According to [22, 23], a continuous-time Markov chain { r t } t ≥ 0 with generator Q = ( q i j ) N × N can be represented as a stochastic integral with respect to a Poisson random measure. Then we have d r t = ∫ R h ( r t − , y ) ν ( d t × d y ) , 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 .

Throughout this paper, unless otherwise specified, the Markov chain { r t } t ≥ 0 has the invariant probability measure μ = ( μ i ) i ∈ S and is assumed to be independent of B t H 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 time interval [0, T]. The generator Q = ( q i j ) N × N is assumed to be irreducible and conservative, i.e., q _i ≔ − q _ii = ∑ _i≠j q _ij < ∞. For more details about Markovian switching we further refer the reader to [24–26].

We recall some of the basic results of fBm briefly, which will be needed throughout this paper. For more details about fBm we refer the reader to [16, 17, 27, 28]. If H ∈ (0, 1/2) ∪ (1/2, 1), then the (standard) fractional Brownian motion with Hurst parameter H is a continuous centered Gaussian process B t H t ≥ 0 with E ( B t H ) = 0 and covariance function: R H ( s , t ) = E ( B s H B t H ) = 1 2 ( | s | 2 H + | t | 2 H − | s − t | 2 H ) , s , t ≥ 0 .

To simplify the representation, it is always assumed that B 0 H = 0 .

Besides, B t H t ≥ 0 has the following Wiener integral representation: B t H = ∫ 0 t K H ( t , s ) d W s , where { W t } t ≥ 0 is a Wiener process and K ^H (t, s) is the kernel function defined by K H ( t , s ) = c H s 1 2 − H ∫ 0 t ( u − s ) H − 3 2 u H − 1 2 d u , in which c H = H ( 2 H − 1 ) B ( 2 − 2 H , H − 1 2 ) 1 2 , where B(⋅, ⋅) is the Beta function, and s < t. In this paper, B t H t ≥ 0 generates a filtration { F t , t ≥ 0 } with F t = σ { B s H , t ≥ 0 } . Denote ( Ω , F , P , F t ) the complete probability space, with the filtration described above.

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 !.

Define the Hermite polynomials: h n ( x ) = ( − 1 ) n e x 2 d n d x n ( e − x 2 ) , n ≥ 0 , and Hermite functions: h ̃ n ( x ) = π − 1 4 ( n ! ) − 1 2 h n ( x ) e − x 2 4 , n ≥ 0 .

Let S ( R ) denote the Schwartz space of rapidly decreasing infinitely differentiable R -valued functions. Denote the dual space of S ( R ) by S ′ ( R ) . Define H α ( ω ) = ∏ i = 1 n h α i ( ⟨ h ̃ i ( x ) , ω ⟩ ) , the product of Hermite polynomials. Consider a square integrable random variable F = F ( ω ) ∈ L 2 ( S ′ ( R ) , F , P ) .

According to [17, 29], every F(ω) has a unique representation: F ( ω ) = ∑ α ∈ I c α H α ( ω ) , besides, ‖ F ‖ L 2 ( ω ) 2 = ∑ α ∈ I α ! c α 2 < ∞ .

Definition 2.1

(Wick Product) For F , G ∈ L 2 ( S ′ ( R ) , F , P ) , set 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 F ⋄ G ( ω ) = ∑ α , β ∈ I a α b β H α + β ( ω ) = ∑ γ ∈ I ∑ α + β = γ a α b β H γ ( ω ) .

Let L p ≔ L p ( Ω , F , P ) be the space of all random variables Ω → R , such that ‖ F ‖ p = E ( | F | p ) 1 / p < ∞ , and let L ϕ 2 ( R + ) = { f | f : R + → R , | f | ϕ 2 ≔ ∫ 0 ∞ ∫ 0 ∞ f ( s ) f ( t ) ϕ ( s , t ) d s d t < ∞ } , where ϕ(s, t) = H(2H − 1)|s − t|^2H−2.

Definition 2.2

The ϕ-derivative of F ∈ L ^p in the direction of Φ _g is defined by D Φ g F ( ω ) = lim δ → 0 1 δ F ω + δ ∫ 0 ⋅ ( Φ g ) ( u ) d u − F ( ω ) , if the limit exists in L ^p . Moreover if there exists a process ( D s ϕ F s , s ≥ 0 ) such that D Φ g F = ∫ 0 ∞ D s ϕ F s g s d s a . s . , for all g ∈ L ϕ 2 , then F is said to be ϕ-differentiable.

According to [16, 30], let A ( 0 , T ) be the family of stochastic process on [0, T] such that F ∈ A ( 0 , T ) if E | F | ϕ 2 < ∞ and F is ϕ-differentiable, the trace of ( D s ϕ F t , 0 ≤ s ≤ T , 0 ≤ t ≤ T ) exists and E ∫ 0 T ( D s ϕ F s ) 2 d s < ∞ , and for each sequence of partitions π n , n ∈ N such that |π _n | → 0, as n → ∞. Moreover ∑ i = 0 n − 1 E ∫ t i ( n ) t i + 1 ( n ) | D s ϕ F t i ( n ) π − D s ϕ F s | d s 2 → 0 , and E | F π − F | ϕ 2 → 0 , as n → ∞. Here π n : 0 = t 0 ( n ) < t 1 ( n ) < … < t n ( n ) = T , and | π n | = max i ∈ { 0,1 , … , n − 1 } { t i + 1 ( n ) − t i ( n ) } .

Now we define the B t H -integral considered in [16].

Definition 2.3

Let { F t } t ≥ 0 be a stochastic process such that F ∈ A ( 0 , T ) . Define ∫ 0 T F s d B s H by ∫ 0 T F s d B s H = lim | π | → 0 ∑ i = 0 n − 1 F t i π ⋄ ( B t i + 1 H − B t i H ) , where |π| =  max _{i∈{0,1,…,n−1}}{t _i+1 − t _i }.

Remark 2.1

: According to Theorem 3.6.1 in [16], if F s ∈ A ( 0 , T ) , then the stochastic integral satisfies E ∫ 0 T F s d B s H = 0 , and E ∫ 0 T F s d B s H 2 = E ∫ 0 T D s ϕ F s d s 2 + ∣ 1 [ 0 , T ] F ∣ ϕ 2 What’s more, according to Definition 3.4.1 in [16], the stochastic integral can be extended by ∫ R F t d B t H ≔ ∫ R F t ⋄ W H ( t ) d t , where F : R → ( S ) H * is a given function such that F _t ⋄ W ^H (t) is dṭ − integrable in ( S ) H * . Here ( S ) H * is the fractional Hida distribution space defined by Definition 3.1.11 in [16]. In particular, the integral on [0, T] can be defined by ∫ 0 T F t d B t H = ∫ R F t I [ 0 , T ] ( t ) d B t H .

To ensure the existence and uniqueness, we impose the following assumptions.

Assumption 3.1

Let f = f ( x , t , i ) : R × R + × S → R satisfy the hypothesises:

1) For each fixed i ∈ S , f(x, t, i) is measurable in all the arguments.

Assumption 3.2

Let σ = σ ( t , i ) : R + × S → R satisfy the hypothesises:

1) For each fixed i ∈ S , σ(t, i) is nonrandom;

Lemma 3.1

: Let Assumptions 3.1 , 3.2 hold. Then Eq. 1 has a unique solution.

Proof: The existence and uniqueness can be proved similar to that for Theorem 2.6 in [31], so we omit it here.

Next, we first review the results in [16, 30] on the Itô formula with respect to fBm. Then we extend it to SDEs driven by fBm with Markovian switching.

Lemma 3.2

[16] (The Itô Formula) Let (F _u , 0 ≤ u ≤ T) be a stochastic process in A ( 0 , T ) . Assume that there exists an α > 1 − H and C > 0 such that E | F u − F v | 2 ≤ C | u − v | 2 α , where |u − v| ≤ δ for some δ > 0 and lim 0 ≤ u , v ≤ t , | u − v | → 0 E | D u ϕ ( F u − F v ) | 2 = 0 .

Let sup_0≤s≤T |G _s | < ∞ and g ̃ = g ̃ ( x , t ) ∈ C 2,1 ( R × R + ; R ) with bounded derivatives. Moreover, for η t = ∫ 0 t F u d B u H , it is assumed that E ∫ 0 T | F s D s ϕ η s | d s < ∞ and ( ∂ g ̃ ∂ x ( s , η s ) F s , s ∈ [ 0 , T ] ) is in A ( 0 , T ) . Denote x t = x 0 + ∫ 0 t G u d u + ∫ 0 t F u d B u H , x 0 ∈ R for t ∈ [0, T]. Let ( ∂ g ̃ ∂ x ( x s , s ) F s , s ∈ [ 0 , T ] ) ∈ A ( 0 , T ) , E sup 0 ≤ s ≤ t | G s | < ∞ . Then for t ∈ [0, T], g ̃ ( x t , t ) = g ̃ ( x 0 , 0 ) + ∫ 0 t ∂ g ̃ ∂ s ( x s , s ) d s + ∫ 0 t ∂ g ̃ ∂ x ( x s , s ) G s d s + ∫ 0 t ∂ g ̃ ∂ x ( x s , s ) F s d B s H + ∫ 0 t ∂ 2 g ̃ ∂ x 2 ( x s , s ) F s D s ϕ x s d s .

Here D s ϕ x s is the Malliavin derivative defined in Definition 2.2 .

In particular, for the process X t ( i ) = X 0 ( i ) + ∫ 0 t f ( X s ( i ) , s , i ) d s + ∫ 0 t g ( X s ( i ) , s , i ) d B s H , with each fixed i ∈ S , we have that F ( X t ( i ) , t , i ) = F ( X 0 ( i ) , 0 , i ) + ∫ 0 t ∂ F ∂ s ( X s ( i ) , s , i ) d s + ∫ 0 t ∂ F ∂ x ( X s ( i ) , s , i ) f ( X s ( i ) , s , i ) d s + ∫ 0 t ∂ F ∂ x ( X s ( i ) , s , i ) g ( X s ( i ) , s , i ) d B s H + ∫ 0 t ∂ 2 F ∂ x 2 ( X s ( i ) , s , i ) g ( X s ( i ) , s , i ) D s ϕ X s ( i ) d s , (2)

Formally, d F ( X t ( i ) , t , i ) = F t ( X t ( i ) , t , i ) d t + F x x ( X t ( i ) , t , i ) g ( X t ( i ) , t , i ) D s ϕ X s ( i ) d t + F x ( X t ( i ) , t , i ) f ( X t ( i ) , t , i ) d t + F x ( X t ( i ) , t , i ) g ( X t ( i ) , t , i ) d B t H ,

Let L ( i ) F ( X t ( i ) , t , i ) = F t ( X t ( i ) , t , i ) + F x ( X t ( i ) , t , i ) f ( X t ( i ) , t , i ) + F x x ( X t ( i ) , t , i ) g ( X t ( i ) , t , i ) D s ϕ X s . (3)

Substituting Eq. 3 into Eq. 2, we get F ( X t ( i ) , t , i ) = F ( X 0 ( i ) , 0 , i ) + ∫ 0 t L ( i ) F ( X s ( i ) , s , i ) d s + ∫ 0 t F x ( X s ( i ) , s , i ) g ( X s ( i ) , s , i ) d B s H . (4)

In the sequel of this paper, unless otherwise specified, we let the coefficients of Eq. 1 satisfy the conditions in Lemma 3.2 , for each fixed i ∈ S . Set V ( X t , t , r t ) ∈ C 2,1 ( R × R + × S ; R + ) . Next we consider the Itô formula which reveals how V maps (X _t , t, r _t ) into a new process V(X _t , t, r _t ), where { X t } t ≥ 0 is a stochastic process with the stochastic differential Eq. 1.

Lemma 3.3

If V ( X t , t , r t ) ∈ C 2,1 ( R × R + × S ; R + ) , then for any 0 ≤ s < t, E V ( X t , t , r t ) = E V ( X s , s , r s ) + E ∫ s t A V ( X u , u , r u ) d u + E ∫ s t V x ( X u , u , r u ) g ( X u , u , r u ) d B u H (5) where A V is defined by A V ( x , t , i ) = L ( i ) V ( x , t , i ) + ∑ j = 1 N γ i j V ( x , t , j ) .

Proof: This result can be obtained similarly to that in [31] and we therefore omit it. For further details we also refer to [2, 23].

Theorem 4.1

Let { X t } t ≥ 0 be the solution of Eq. 7 with H ∈ (1/2, 1), h = 1/2 − H .

1) If ∑ i ∈ S μ i α ( i ) − ( 1 − p ) b ̲ 2 2 < 0 , then lim sup t → ∞ 1 t log ( E | X t | p ) < 0 .

Proof. According to [16], without too many calculations, we obtain that { X t } t ≥ 0 has the following form: X t = x 0 ⁡ exp ∫ 0 t σ ( r s ) d B s H + ∫ 0 t α ( r s ) d s − 1 2 ∫ R ( M s ( σ ( t , r s ) I [ 0 , t ] ( s ) ) ) 2 d s , (8) where M _s is the operator M acting on the variable s. Let x ₀ ≠ 0. It follows from Eq. 8 that E | X t | p = E | x 0 | exp ∫ 0 t σ ( t , r s ) d B s H + ∫ 0 t α ( r s ) d s − 1 2 ∫ R ( M s ( σ ( t , r s ) I [ 0 , t ] ( s ) ) ) 2 d s p (9)

We then see from Eq. 9 that E | X t | p = E exp p ∫ 0 t α ( r s ) d s − 1 − p 2 ∫ R ( M s ( σ ( t , r s ) I [ 0 , t ] ( s ) ) ) 2 d s ζ t , (10) where ζ t = | x 0 | p ⁡ exp ∫ 0 t p σ ( s , r s ) d B s H − p 2 2 ∫ R ( M s ( σ ( t , r s ) I [ 0 , t ] ( s ) ) ) 2 d s .

Noting that ζ _t is the solution to the equation d ζ t = p σ ( t , r t ) ζ t d B t H , with initial value ζ ₀ = |x ₀| ^p . Thus ζ t = | x 0 | p + ∫ 0 t p σ ( t , r s ) d B s H , which yields E ζ t = E | x 0 | p + ∫ 0 t p σ ( t , r s ) d B s H = | x 0 | p . (11)

Substituting Eq. 11 into Eq. 10 gives E | X t | p = E ⁡ exp p ∫ 0 t α ( r s ) d s − 1 − p 2 ∫ R ( M s ( σ ( r s ) I [ 0 , t ] ( s ) ) ) 2 d s | x 0 | p . (12)

Note that ∫ R ( M s ( b ̲ s h I [ 0 , t ] ( s ) ) ) 2 d s ≤ ∫ R ( M s ( σ ( t , r s ) I [ 0 , t ] ( s ) ) ) 2 d s ≤ ∫ R ( M s ( b ̄ s h I [ 0 , t ] ( s ) ) ) 2 d s .

Consequently, by Definition 4.1 and [16], one has b ̲ 2 t ≤ ∫ R ( M s ( σ ( r s ) I [ 0 , t ] ( s ) ) ) 2 d s ≤ b ̄ 2 t . (13)

Making use of Eqs 12, 13, we obtain that E ⁡ exp p ∫ 0 t α ( r s ) d s − 1 − p 2 b ̄ 2 t | x 0 | p ≤ E | X t | p ≤ E ⁡ exp p ∫ 0 t α ( r s ) d s − 1 − p 2 b ̲ 2 t | x 0 | p .

Therefore, by Lemma 4.1 and Eq. 12, the required assertions follow. The proof is complete.

Theorem 4.2

Let { X t } t ≥ 0 be the solution of Eq. 7 with H ∈ (0, 1/2), h = 1/2 − H .

1) If ∑ i ∈ S μ i α ( i ) < ( 1 − p ) b ̲ 2 2 , then lim sup t → ∞ 1 t log ( E | X t | p ) < 0 .

Proof: Similar to Theorem 4.1 , we write the solution as follows. E | X t | p = E ⁡ exp p ∫ 0 t α ( r s ) d s + p − 1 2 ∫ R ( M s ( σ ( r s ) I [ 0 , t ] ( s ) ) ) 2 d s | x 0 | p . (14)

Note that M _s is the operator M acting on the variable s, where M f ( x ) = C H ∫ R f ( x − t ) − f ( x ) | t | 3 / 2 − H d t .

According to [16], we also have that b ̲ 2 t ≤ ∫ R ( M s ( σ ( t , r s ) I [ 0 , t ] ( s ) ) ) 2 d s ≤ b ̄ 2 t . (15)

Consequently, by Lemma 4.1 , the result follows. The proof is complete.

Remark 4.1

In the above Theorems 4.1 , 4.2 , the parameter h is supposed to be H − 1/2. Noting that by Eqs 13, 15 and together with the Definition 4.1 , the stability of solution for Eq. 7 with h < 1/2 − H or h > 1/2 − H can be deduced respectively without too many difficulties.

Remark 4.2

Take H = 1/2. It’s easy to show that if ∑ i ∈ S μ i α ( i ) = α < ( 1 − p ) σ ̲ 2 2 , then lim sup t → ∞ 1 t log ( E | X t | p ) < 0 , and if ∑ i ∈ S μ i α ( i ) = α > ( 1 − p ) σ ̄ 2 2 , then lim t → ∞ E | X t | p = ∞ , which coincide with the results of SDEs driven by Brownian motion in [4, 32].

To proceed, we need to introduce the definition of almost sure stability and a useful lemma.

Definition 4.2

The equilibrium point x = 0 is said to be almost surely exponential stable if lim sup t → ∞ 1 t log | X t | < 0 a . s . for any x 0 ∈ R .

Lemma 4.2

(Law of the iterated logarithm) For a standard fBm B t H t ≥ 0 , we have that lim sup t → ∞ B t H t H log ⁡ log t = C H , (16) where C _H > 0 is a suitable constant.

Proof: By [33], we have lim sup t → 0 + B t H t H log ⁡ log t − 1 = c H , where c _H is a suitable constant. Then the thesis follows by the self-similarity of fBm and a change of variable t → 1/t.

For the sake of clarity, we firstly set h = 0. Namely, let us consider d X t = α ( r t ) X t d t + b ( r t ) X t d B t H , X 0 = x 0 . (17)

Noting that Eq. 17 is exactly the geometry fBm with Markovian Switching. We proceed to discuss the almost sure exponential stability about it.

Theorem 4.3

1) If 0 < H < 1/2, the equilibrium point x = 0 of the system Eq. 17 is almost surely exponential stable when ∑ i ∈ S μ i α ( i ) < 0 , but unstable when ∑ i ∈ S μ i α ( i ) > 0 ; 2) If H = 1/2, the equilibrium point x = 0 of the system Eq. 17 is almost surely exponential stable when ∑ i ∈ S μ i α ( i ) < 1 2 b ̲ 2 , but unstable when ∑ i ∈ S μ i α ( i ) > 1 2 b ̄ 2 ; 3) If 1/2 < H < 1, the equilibrium point x = 0 of the system Eq. 17 is almost surely exponential stable for all parameters α(i) and σ(i), i ∈ S .