Stability of Hybrid SDEs Driven by fBm

In this paper, the exponential stability of stochastic differential equations driven by multiplicative fractional Brownian motion (fBm) with Markovian switching is investigated. The quasi-linear cases with the Hurst parameter H ∈ (1/2, 1) and linear cases with H ∈ (0, 1/2) and H ∈ (1/2, 1) are all studied in this work. An example is presented as a demonstration.


INTRODUCTION
In the natural world, it is a common phenomena that many practical systems may face random abrupt changes in their structures and parameters, such as environmental variance, changing of subsystem interconnections and so on. To deal with these abrupt changes, Markovian switching systems, a particular class of hybrid systems, are investigated and widely used [1,2]. Especially in signal processing, financial engineering, queueing networks, wireless communications and so on (see, e.g. [1,3]).
In recent years, much attention has been paid to the stability of stochastic hybrid systems. For example, Mao [4] considers the exponential stability of general nonlinear stochastic hybrid systems. In [5], the criteria of moment exponential stability are obtained for stochastic hybrid delayed systems with Lévy noise in mean square. Zhou [6] investigates the pth moment exponential stability of the same systems. Some sufficient conditions for asymptotic stability in distribution of SDEs with Markovian switching are reported in [7]. See also [8,9] for more results about Markovian switching.
On the other hand, it is generally known that if H ∈ (0, 1/2) and H ∈ (1/2, 1), B H t t ≥ 0 has a long range dependence, which means if we put r(n) cov(B H 1 , (B H n+1 − B H n )), then ∞ n 1 r(n) ∞. Besides, the process B H t t ≥ 0 is also self-similar for any H ∈ (0, 1). Since the pioneering work of Hurst [10,11] and Mandelbrot [12], the fractional Brownian motion has been suggested as a useful tool in many fields such as mathematical finance [13,14] and weather derivatives [15]. Even though fractional Brownian motion is not a semimartingale, more and more financial models have been extended to fBm (see, e.g. [16,17]). Therefore, in this paper, the risk assets are described by hybrid stochastic systems driven by multiplicative fBm. Then it is a natural and interesting question that under what conditions, this stochastic systems have some exponential stability. For the sake of clarity, we only consider the one dimensional cases. For more details about fractional noise, we refer the reader to [18][19][20][21].
The main purpose of this paper is to discuss the exponential stability of a risky asset, with price dynamics: dX t f(X t , t, r t )dt + g(X t , t, r t )dB H t , where g(X t , t, r t ) σ(t, r t )X t , {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(x, t, r t ): R × R + × S → R and σ(t, r t ): R + × S → R.
In this paper, the initial value x 0 is assumed to be deterministic, otherwise more calculations about Wick product are required.
Equation 1 can be regarded as the result of the following N fractional stochastic differential equations: switching from one to another according to the movement of {r t } t ≥ 0 .
Throughout this paper, unless otherwise specified, we let C denote a general constant and p denote a non-negative constant. Let C 2,1 (R × R + × S; R) denote the family of all real value functions on R × R + × S which are continuously twice differentiable with respect to the first variables and once differentiable with respect to the second variables.
This paper is organized as follows. For the convenience of the reader, we briefly recall some of the basic results in Section 2. In Section 3, we investigate the solution and an extended Itô's Formula for the general hybrid fractional stochastic differential Equation 1. Section 3 is devoted to the linear cases. In this section the moment exponential stability and almost sure exponential stability are discussed respectively. In Section 4, some useful criteria for the exponential stability with respect to quasi-linear cases are presented. Finally, a numerical example and graphical illustration are presented in Section 6.

Markov Chain
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 ij ) N×N is given by 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 ij ) N×N can be represented as a stochastic integral with respect to a Poisson random measure. Then we have with initial condition r 0 i 0 , where ](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 H t t ≥ 0 . Almost every sample path of the Markov chain {r t } t ≥ 0 is assumed to be a rightcontinuous step function with a finite number of simple jumps in any finite time interval [0, T]. The generator Q (q ij ) N×N is assumed to be irreducible and conservative, i.e., q i d − q ii i≠j q ij < ∞. For more details about Markovian switching we further refer the reader to [24][25][26].

Fractional Brownian Motion and Wick Product
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 H t t ≥ 0 with E(B H t ) 0 and covariance function: To simplify the representation, it is always assumed that B H 0 0. Besides, B H t t ≥ 0 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 s < t. In this paper, B H t t ≥ 0 generates a filtration {F t , t ≥ 0} with F t σ{B H s , t ≥ 0}. Denote (Ω, F , P, F t ) the complete probability space, with the filtration described above.
Define the Hermite polynomials: h n (x) (−1) n e x 2 d n dx n (e −x 2 ), n≥ 0, and Hermite functions: 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 the product of Hermite polynomials. Consider a square integrable random variable According to [17,29], every F(ω) has a unique representation: Frontiers in Physics | www.frontiersin.org November 2021 | Volume 9 | Article 783434

Malliavin Derivative
Let L p dL p (Ω, F , P) be the space of all random variables Ω → R, such that and let for all g ∈ L 2 ϕ , then F is said to be ϕ-differentiable. According to [16,30], let A(0, T) be the family of stochastic process and for each sequence of partitions π n , n ∈ N such that |π n | → 0, as n → ∞. Moreover [16].
What's more, according to Definition 3.4.1 in [16], the stochastic integral can be extended by 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

HYBRID FRACTIONAL SYSTEMS
In this section, firstly, we consider the existence and uniqueness of solution for Eq. 1. Then, an extended Itô's Formula is presented.

Existence and Uniqueness
To ensure the existence and uniqueness, we impose the following assumptions. Proof: The existence and uniqueness can be proved similar to that for Theorem 2.6 in [31], so we omit it here.

The Itô Formula
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.
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 Formally, Substituting Eq. 3 into Eq. 2, we get 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, EV(X t , t, r t ) EV(X s , s, r s ) + E t s AV(X u , u, r u )du +E t s V x (X u , u, r u )g(X u , u, r u )dB H u where AV is defined by 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].

LINEAR HYBRID FRACTIONAL SYSTEMS
There are many models for financial markets with fBm (see, e.g. [16]). The simplest nontrivial type of market is the fBm version of the classical Black Scholes market, in which linear fractional SDEs is used. Thus, we would like to give some new criteria for switching linear fractional SDEs with H ∈ (0, 1 2 ) or H ∈ ( 1 2 , 1). At first, we present a definition and a useful lemma. (6) where Here Γ(·) denotes the classical Gamma function. According to [16], Eq. 6 can be restated as follows.

Mf(x) f(x).
For H ∈ (1/2, 1), we have Then there exists constants C, c > 0 such that: Proof: It is a consequence of Perron-Frobenius theorem and the study of eigenvalues. See Proposition 4.1 in [25], Proposition 4.2 in [25], and Lemma 2.7 in [26], for further details.
In Eq. 1, let us consider the case g(x, t, r t ) σ(t, r t )x t h b(r t )x, f(x, t, r t ) α(r t )x, where α(i) and b(i) are constants for each i ∈ S. This means that we are considering the following linear equation: Set b max{|b(i)|, i ∈ S} and b min{|b(i)|, i ∈ S}. x 0 is the deterministic initial value. For the sake of clarity, we firstly set h 1/2 − H.

pth Moment Exponential Stability
Proof. According to [16], without too many calculations, we obtain that {X t } t ≥ 0 has the following form: where M s is the operator M acting on the variable s. Let x 0 ≠ 0. It follows from Eq. 8 that We then see from Eq. 9 that where Noting that ζ t is the solution to the equation with initial value ζ 0 |x 0 | p . Thus Substituting Eq. 11 into Eq. 10 gives Note that Consequently, by Definition 4.1 and [16], one has Making use of Eqs 12, 13, we obtain that Therefore, by Lemma 4.1 and Eq. 12, the required assertions follow. The proof is complete. , then lim sup Proof: Similar to Theorem 4.1, we write the solution as follows.
Note that M s is the operator M acting on the variable s, where According to [16], we also have that Consequently, by Lemma 4.1, the result follows. The proof is complete.

Almost Sure Exponential Stability
To proceed, we need to introduce the definition of almost sure stability and a useful lemma. for any x 0 ∈ R.
where C H > 0 is a suitable constant. Proof: By [33], we have lim sup where c H is a suitable constant. Then the thesis follows by the selfsimilarity of fBm and a change of variable t → 1/t. For the sake of clarity, we firstly set h 0. Namely, let us consider 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) > Making use of Eq. 18, we get Especially, when H 1/2, we have that Therefore, the required results follows. The proof is complete.

QUASI-LINEAR HYBRID FRACTIONAL SYSTEMS
We now apply the extended Itô Formula in Section 3 to discuss the stability for quasi-linear fractional SDEs with Markovian switching.
Theorem 5.1. : Let Assumptions 3.1, 3.2 hold. If there exists a function V ∈ C 2,1 (R × R + × S; R + ) and positive constants a 1 , a 2 , b and p ≥ 1, such that for all X t ∈ R, t ≥ t 0 , i ∈ S. Then the solution of Eq. 1 is pth moment exponential stable. More precisely, lim sup Proof: According to Lemma 3.1, Eq. 1 has a unique solution.
Applying the conditions Eq. 19, 20, together with the generalized ItôEq. 5 and Remark 2.1, we obtain that for any t ∈ [0, T] Thus we obtain that Dividing both sides of Eq. 21 by a 1 e ηt , noting that λa 2 − b < 0, we get and the required assertion follows. The proof is complete.
In the sequel of this section, we give another useful criterion and prove it briefly.
Theorem 5.2. Assume that Eq. 1 has a unique solution and there exist a function V ∈ C 2,1 (R × R + × S; R + ) and positive constants b 1 , b 2 , p ≥ 1 and β i ∈ R such that for all x ∈ R, t ≥ t 0 , i ∈ S, Then Eq. 1 is pth moment exponential stable.
Then, for any t ∈ [0, T] we get According to [1,31], one has Making use of Eqs 25, 26, we obtain that Substituting Eq. 24 into Eq. 27, we get where κ < 0. Making use of Theorem 5.1, the desired criterion follows.
On the other hand, we can prove it in another way. Set η > 0 and λ ∈ (η, − κ). Define U(X t , t, i) e λt 1 − θc i U(X t , t, i). Thus we obtain that b 1 e ηt E|X t | p ≤ V(x 0 , 0, i 0 ) + E t 0 e λs b 2 (λ + κ)|X t | p ds, Dividing both sides of Eq. 28 by b 1 e ηt , noting that b 2 (λ + κ) < 0, we get Therefore, we obtain the required assertion lim sup The proof is complete.
Frontiers in Physics | www.frontiersin.org November 2021 | Volume 9 | Article 783434 In this section we give a numerical example to illustrate our results. Consider a risky asset, with the price dynamics: dX t f(X t , t, r r )dt + σ(t, r t )X t dB H t , X 0 1, on t ≥ 0. Here we take H 0.7 and f(x, t, i) −4x, σ(t, i) f(x, t, i) [2 − sin(x)]x, σ(t, i) e −t , if i 2.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Note that for all i ∈ S, dX t f(X t , t, i)dt + σ(t, i)X t dB H t satisfy the hypothesises (i)-(v). Then, by Lemma 3.1, it is easy to show that Eq. 29 has a unique solution {X t } t ≥ 0 as well. Set V(x, t, i) x 2 , for i 1, 2.
By Theorem 5.2, it's clear that the solution of Eq. 29 is second moment exponential stable. Figures 1, 2 show a single path of the solution and the solution's norm square, respectively.

DATA AVAILABILITY STATEMENT
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