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ORIGINAL RESEARCH article

Front. Phys., 02 November 2021
Sec. Interdisciplinary Physics
https://doi.org/10.3389/fphy.2021.783434

Stability of Hybrid SDEs Driven by fBm

www.frontiersin.orgWenyi Pei1,2,3* and www.frontiersin.orgZhenzhong Zhang4
  • 1School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, China
  • 2Collaborative Innovation Center of Statistical Data Engineering, Technology and Application, Zhejiang Gongshang University, Hangzhou, China
  • 3College of Information Science and Technology, Donghua University, Shanghai, China
  • 4Department of Statistics, Donghua University, Shanghai, China

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.

1 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), BtHt0 has a long range dependence, which means if we put

r(n)=cov(B1H,(Bn+1HBnH)),

then n=1r(n)=. Besides, the process BtHt0 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 [1821].

The main purpose of this paper is to discuss the exponential stability of a risky asset, with price dynamics:

dXt=f(Xt,t,rt)dt+g(Xt,t,rt)dBtH,X0=x0>0,(1)

where g(Xt, t, rt) = σ(t, rt)Xt, {rt}t0 is a Markov chain taking values in S={1,2,,N}, BtHt0 is a standard fractional Brownian motion. Moreover, f(x,t,rt):R×R+×SR and σ(t,rt):R+×SR. In this paper, the initial value x0 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:

dXt=f(Xt,t,i)dt+g(Xt,t,i)dBtH,1iN,X0=x0>0,

switching from one to another according to the movement of {rt}t0.

Throughout this paper, unless otherwise specified, we let C denote a general constant and p denote a non-negative constant. Let C2,1(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.

2 Preliminaries

2.1 Markov Chain

Let {rt}t0 be a right-continuous Markov chain taking values in a finite state space S={1,2,,N}. The generator Q=(qij)N×N is given by

P{rt+=jrt=i}=qij+o(),ifij,1+qij+o(),ifi=j,

where △ > 0.

Here qij is the transition rate from i to j if ij. According to [22, 23], a continuous-time Markov chain {rt}t0 with generator Q=(qij)N×N can be represented as a stochastic integral with respect to a Poisson random measure. Then we have

drt=Rh(rt,y)ν(dt×dy),

with initial condition r0 = i0, 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 {rt}t0 has the invariant probability measure μ=(μi)iS and is assumed to be independent of BtHt0. Almost every sample path of the Markov chain {rt}t0 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=(qij)N×N is assumed to be irreducible and conservative, i.e., qi≔ − qii = ij qij < . For more details about Markovian switching we further refer the reader to [2426].

2.2 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 BtHt0 with E(BtH)=0 and covariance function:

RH(s,t)=E(BsHBtH)=12(|s|2H+|t|2H|st|2H),s,t0.

To simplify the representation, it is always assumed that B0H=0.

Besides, BtHt0 has the following Wiener integral representation:

BtH=0tKH(t,s)dWs,

where {Wt}t0 is a Wiener process and KH(t, s) is the kernel function defined by

KH(t,s)=cHs12H0t(us)H32uH12du,

in which cH=H(2H1)B(22H,H12)12, where B(⋅, ⋅) is the Beta function, and s < t. In this paper, BtHt0 generates a filtration {Ft,t0} with Ft=σ{BsH,t0}. Denote (Ω,F,P,Ft) the complete probability space, with the filtration described above.

Let I be the set of all finite multi-indices α = (α1, …, αn) for some n ≥ 1 of non-negative integers. Denote |α| = α1 + ⋯ + αn, and α! = α1!⋯αn!.

Define the Hermite polynomials:

hn(x)=(1)nex2dndxn(ex2),n0,

and Hermite functions:

h̃n(x)=π14(n!)12hn(x)ex24,n0.

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

Hα(ω)=i=1nhαi(h̃i(x),ω),

the product of Hermite polynomials. Consider a square integrable random variable

F=F(ω)L2(S(R),F,P).

According to [17, 29], every F(ω) has a unique representation:

F(ω)=αIcαHα(ω),

besides,

FL2(ω)2=αIα!cα2<.

Definition 2.1. (Wick Product) For F,GL2(S(R),F,P), set F(ω)=αIcαHα(ω) and G(ω)=βIdβHβ(ω). Their Wick product is defined by

FG(ω)=α,βIaαbβHα+β(ω)=γIα+β=γaαbβHγ(ω).

2.3 Malliavin Derivative

Let LpLp(Ω,F,P) be the space of all random variables ΩR, such that

Fp=E(|F|p)1/p<,

and let

Lϕ2(R+)={f|f:R+R,|f|ϕ200f(s)f(t)ϕ(s,t)dsdt<},

where ϕ(s, t) = H(2H − 1)|st|2H−2.

Definition 2.2. The ϕ-derivative of FLp in the direction of Φg is defined by

DΦgF(ω)=limδ01δFω+δ0(Φg)(u)duF(ω),

if the limit exists in Lp. Moreover if there exists a process (DsϕFs,s0) such that

DΦgF=0DsϕFsgsdsa.s.,

for all gLϕ2, 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 FA(0,T) if E|F|ϕ2< and F is ϕ-differentiable, the trace of (DsϕFt,0sT,0tT) exists and E0T(DsϕFs)2ds<, and for each sequence of partitions πn,nN such that |πn| → 0, as n. Moreover

i=0n1Eti(n)ti+1(n)|DsϕFti(n)πDsϕFs|ds20,

and

E|FπF|ϕ20,

as n. Here πn:0=t0(n)<t1(n)<<tn(n)=T, and |πn|=maxi{0,1,,n1}{ti+1(n)ti(n)}.Now we define the BtH-integral considered in [16].

Definition 2.3. Let {Ft}t0 be a stochastic process such that FA(0,T). Define 0TFsdBsH by

0TFsdBsH=lim|π|0i=0n1Ftiπ(Bti+1HBtiH),

where |π| =  maxi∈{0,1,…,n−1}{ti+1ti}.

Remark 2.1.: According to Theorem 3.6.1 in [16], if FsA(0,T), then the stochastic integral satisfies E0TFsdBsH=0, and

E0TFsdBsH2=E0TDsϕFsds2+1[0,T]Fϕ2

What’s more, according to Definition 3.4.1 in [16], the stochastic integral can be extended by

RFtdBtHRFtWH(t)dt,

where F:R(S)H* is a given function such that FtWH(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

0TFtdBtH=RFtI[0,T](t)dBtH.

3 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.

3.1 Existence and Uniqueness

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

Assumption 3.1. Let f=f(x,t,i):R×R+×SR satisfy the hypothesises:

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

2) For each fixed iS, there exists a constant C > 0, such that |f(x,t,i)f(y,t,i)|C|xy|,x,yR,tR+.

3) For each fixed iS, there exists a constant C > 0, such that

|f(x,t,i)|C(1+|x|),(x,t)R×R+.

Assumption 3.2. Let σ=σ(t,i):R+×SR satisfy the hypothesises:

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

2) For each fixed iS, σ(t,i)L1H(R+).

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.

3.2 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.

Lemma 3.2. [16] (The Itô Formula) Let (Fu, 0 ≤ uT) be a stochastic process in A(0,T). Assume that there exists an α > 1 − H and C > 0 such that

E|FuFv|2C|uv|2α,

where |uv| ≤ δ for some δ > 0 and

lim0u,vt,|uv|0E|Duϕ(FuFv)|2=0.

Let sup0≤sT|Gs| < and g̃=g̃(x,t)C2,1(R×R+;R) with bounded derivatives. Moreover, for ηt=0tFudBuH, it is assumed that E0T|FsDsϕηs|ds< and (g̃x(s,ηs)Fs,s[0,T]) is in A(0,T). Denote xt=x0+0tGudu+0tFudBuH, x0R for t ∈ [0, T]. Let (g̃x(xs,s)Fs,s[0,T])A(0,T), Esup0st|Gs|<. Then for t ∈ [0, T],

g̃(xt,t)=g̃(x0,0)+0tg̃s(xs,s)ds+0tg̃x(xs,s)Gsds+0tg̃x(xs,s)FsdBsH+0t2g̃x2(xs,s)FsDsϕxsds.

Here Dsϕxs is the Malliavin derivative defined in Definition 2.2.In particular, for the process Xt(i)=X0(i)+0tf(Xs(i),s,i)ds+0tg(Xs(i),s,i)dBsH, with each fixed iS, we have that

F(Xt(i),t,i)=F(X0(i),0,i)+0tFs(Xs(i),s,i)ds+0tFx(Xs(i),s,i)f(Xs(i),s,i)ds+0tFx(Xs(i),s,i)g(Xs(i),s,i)dBsH+0t2Fx2(Xs(i),s,i)g(Xs(i),s,i)DsϕXs(i)ds,(2)

Formally,

dF(Xt(i),t,i)=Ft(Xt(i),t,i)dt+Fxx(Xt(i),t,i)g(Xt(i),t,i)DsϕXs(i)dt+Fx(Xt(i),t,i)f(Xt(i),t,i)dt+Fx(Xt(i),t,i)g(Xt(i),t,i)dBtH,

Let

L(i)F(Xt(i),t,i)=Ft(Xt(i),t,i)+Fx(Xt(i),t,i)f(Xt(i),t,i)+Fxx(Xt(i),t,i)g(Xt(i),t,i)DsϕXs.(3)

Substituting Eq. 3 into Eq. 2, we get

F(Xt(i),t,i)=F(X0(i),0,i)+0tL(i)F(Xs(i),s,i)ds+0tFx(Xs(i),s,i)g(Xs(i),s,i)dBsH.(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 iS. Set V(Xt,t,rt)C2,1(R×R+×S;R+). Next we consider the Itô formula which reveals how V maps (Xt, t, rt) into a new process V(Xt, t, rt), where {Xt}t0 is a stochastic process with the stochastic differential Eq. 1.

Lemma 3.3. If V(Xt,t,rt)C2,1(R×R+×S;R+), then for any 0 ≤ s < t,

EV(Xt,t,rt)=EV(Xs,s,rs)+EstAV(Xu,u,ru)du+EstVx(Xu,u,ru)g(Xu,u,ru)dBuH(5)

where AV is defined by

AV(x,t,i)=L(i)V(x,t,i)+j=1NγijV(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].

4 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(0,12) or H(12,1). At first, we present a definition and a useful lemma.

Definition 4.1. Let H ∈ (0, 1). The operator M is defined on functions fS(R) by

Mf(x)=ddxCH(H1/2)R(ts)|tx|H32f(t)dt(6)

where

CH=2ΓH12cosπ2H121Γ(2H+1)sin(πH)12.

Here Γ(⋅) denotes the classical Gamma function.According to [16], Eq. 6 can be restated as follows.For H ∈ (0, 1/2), we have

Mf(x)=CHRf(xt)f(x)|t|3/2Hdt.

For H = 1/2, we have

Mf(x)=f(x).

For H ∈ (1/2, 1), we have

Mf(x)=CHRf(t)|tx|3/2Hdt.

Lemma 4.1. Let {rt}t0 be a right-continuous Markov chain which takes values in a finite state space S={1,2,,N}. Assume that it is irreducible and positive recurrent with invarient measure μ. If α():SR is a function verifying

αiSμ(i)α(i)>0.

Then there exists constants C, c > 0 such that:

ceαtEe0tα(rs)dsCeαt,

for any initial condition r0 and every t ≥ 0.

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, rt) = σ(t, rt)x = thb(rt)x, f(x, t, rt) = α(rt)x, where α(i) and b(i) are constants for each iS. This means that we are considering the following linear equation:

dXt=α(rt)Xtdt+σ(t,rt)XtdBtH,X0=x0.(7)

Set b̄=max{|b(i)|,iS} and b̲=min{|b(i)|,iS}. x0 is the deterministic initial value. For the sake of clarity, we firstly set h = 1/2 − H.

4.1 pth Moment Exponential Stability

Theorem 4.1. Let {Xt}t0 be the solution of Eq. 7 with H ∈ (1/2, 1), h = 1/2 − H.

1) If iSμiα(i)(1p)b̲22<0, then lim supt1tlog(E|Xt|p)<0.

2) If iSμiα(i)(1p)b̄22>0, then limtE|Xt|p=.

Proof. According to [16], without too many calculations, we obtain that {Xt}t0 has the following form:

Xt=x0exp0tσ(rs)dBsH+0tα(rs)ds12R(Ms(σ(t,rs)I[0,t](s)))2ds,(8)

where Ms is the operator M acting on the variable s. Let x0 ≠ 0. It follows from Eq. 8 that

E|Xt|p=E|x0|exp0tσ(t,rs)dBsH+0tα(rs)ds12R(Ms(σ(t,rs)I[0,t](s)))2dsp(9)

We then see from Eq. 9 that

E|Xt|p=Eexpp0tα(rs)ds1p2R(Ms(σ(t,rs)I[0,t](s)))2dsζt,(10)

where

ζt=|x0|pexp0tpσ(s,rs)dBsHp22R(Ms(σ(t,rs)I[0,t](s)))2ds.

Noting that ζt is the solution to the equation

dζt=pσ(t,rt)ζtdBtH,

with initial value ζ0 = |x0|p. Thus

ζt=|x0|p+0tpσ(t,rs)dBsH,

which yields

Eζt=E|x0|p+0tpσ(t,rs)dBsH=|x0|p.(11)

Substituting Eq. 11 into Eq. 10 gives

E|Xt|p=Eexpp0tα(rs)ds1p2R(Ms(σ(rs)I[0,t](s)))2ds|x0|p.(12)

Note that

R(Ms(b̲shI[0,t](s)))2dsR(Ms(σ(t,rs)I[0,t](s)))2dsR(Ms(b̄shI[0,t](s)))2ds.

Consequently, by Definition 4.1 and [16], one has

b̲2tR(Ms(σ(rs)I[0,t](s)))2dsb̄2t.(13)

Making use of Eqs 12, 13, we obtain that

Eexpp0tα(rs)ds1p2b̄2t|x0|pE|Xt|pEexpp0tα(rs)ds1p2b̲2t|x0|p.

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

Theorem 4.2. Let {Xt}t0 be the solution of Eq. 7 with H ∈ (0, 1/2), h = 1/2 − H.

1) If iSμiα(i)<(1p)b̲22, then lim supt1tlog(E|Xt|p)<0.

2) If iSμiα(i)>(1p)b̄22, then limtE|Xt|p=.

Proof: Similar to Theorem 4.1, we write the solution as follows.

E|Xt|p=Eexpp0tα(rs)ds+p12R(Ms(σ(rs)I[0,t](s)))2ds|x0|p.(14)

Note that Ms is the operator M acting on the variable s, where

Mf(x)=CHRf(xt)f(x)|t|3/2Hdt.

According to [16], we also have that

b̲2tR(Ms(σ(t,rs)I[0,t](s)))2dsb̄2t.(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 iSμiα(i)=α<(1p)σ̲22, then lim supt1tlog(E|Xt|p)<0, and if iSμiα(i)=α>(1p)σ̄22, then limtE|Xt|p=, which coincide with the results of SDEs driven by Brownian motion in [4, 32].

4.2 Almost Sure Exponential Stability

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 supt1tlog|Xt|<0a.s.

for any x0R.

Lemma 4.2. (Law of the iterated logarithm) For a standard fBm BtHt0, we have that

lim suptBtHtHloglogt=CH,(16)

where CH > 0 is a suitable constant.

Proof: By [33], we have

lim supt0+BtHtHloglogt1=cH,

where cH 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

dXt=α(rt)Xtdt+b(rt)XtdBtH,X0=x0.(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 iSμiα(i)<0, but unstable when iSμiα(i)>0; 2) If H = 1/2, the equilibrium point x = 0 of the system Eq. 17 is almost surely exponential stable when iSμiα(i)<12b̲2, but unstable when iSμiα(i)>12b̄2; 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), iS.

Proof: Define

λ=lim supt1tlog|Xt|.

From Eqs 8, 16, we have

λ=lim supt1tlog|Xt|=lim supt1tlog|x0exp[0tσ(rs)dBsH+0tα(rs)ds12R(Ms(σ(rs)I[0,t](s)))2ds]|=limtiSμiα(i)12tR(Ms(σ(rs)I[0,t](s)))2ds.

By Definition 4.1 and [16], one has

b̲2t2HR(Ms(b(rs)I[0,t](s)))2dsb̄2t2H.(18)

Making use of Eq. 18, we get

λ=iSμiα(i),0<H<1/2;,1/2<H<1.

Especially, when H = 1/2, we have that

iSμiα(i)12b̄2λiSμiα(i)12b̲2.

Therefore, the required results follows. The proof is complete.

Remark 4.3. Making use of Eq. 18, one can discuss the almost sure exponential stability for Eq. 7 with h ≠ 0. The proofs are similar to Theorem 4.3 and are omitted.

5 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 VC2,1(R×R+×S;R+) and positive constants a1, a2, b and p ≥ 1, such that

a1|Xt|p|V(Xt,t,i)|a2|Xt|p,(19)
L(i)V(Xt,t,i)b|Xt|p,(20)

for all XtR, tt0, iS.Then the solution of Eq. 1 is pth moment exponential stable. More precisely,

lim supt1tlog(E|Xt|p)<0.

Proof: According to Lemma 3.1, Eq. 1 has a unique solution. Denote it {Xt}t0. Set

U(Xt,t,i)=eλtV(Xt,t,i),

where λ(η,ba2), η > 0. Making use of Definition 2.3 and Lemma 3.2, one has AU=eλt(λV+AV) and (Uxg,s[0,T])A(0,T).Applying the conditions Eq. 19, 20, together with the generalized ItôEq. 5 and Remark 2.1, we obtain that for any t ∈ [0, T]

a1eηtE|Xt|pEU(Xt,t,i)=EV(X0,0,r0)+E0tAUds+E0tUxgsH=EV(X0,0,r0)+E0tL(rs)Uds=EV(X0,0,r0)+E0teλs(λV+AV)dsEV(X0,0,r0)+E0teλs(λa2b)|Xt|pds.

Thus we obtain that

a1eηtE|Xt|pEV(X0,0,r0)+E0teλs(λa2b)|Xt|pds.(21)

Dividing both sides of Eq. 21 by a1eηt, noting that λa2b < 0, we get

E|Xt|peηta1EV(X0,0,r0)+eηta1E0teλs(λa2b)|Xt|pdseηta1EV(X0,0,r0).

Consequently,

supt[0,T]a1eηtE|Xt|pEV(X0,0,r0).

Letting T gives

supt0E|Xt|peηta1EV(X0,0,r0),

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 VC2,1(R×R+×S;R+) and positive constants b1, b2, p ≥ 1 and βiR such that for all xR, tt0, iS,

b1|x|p|V(x,t,i)|b2|x|p,
L(i)V(x,t,i)βiV(x,t,i),

and

iSμiβi<0

Then Eq. 1 is pth moment exponential stable.

Proof: Set β̄i=1θβi, where θ ∈ (0, 1). Let δ=iSμiβ̄i=μβ̄. Let 1 denote the vector which all elements are 1. Then,

μ(β̄+δ1)=μβ̄+δ=δ+δ=0.(22)

By [1], Eq. 22 implies the Poisson equation:

Qc=β̄+δ1.(23)

Note that Eq. 23 has the solution c=(c1,,cN)T. Hence,

δ=β̄ij=1Nqijcj,iS.(24)

For each iS, set U(x, t, i) = (1 − θci)V(x, t, i), where θ ∈ (0, 1) is already defined and sufficiently small satisfying 1 − θci > 0.Then, for any t ∈ [0, T] we get

AU(x,t,i)=(1θci)L(i)V(x,t,i)+ijqij(U(x,t,j)U(x,t,i))=(1θci)L(i)V(x,t,i)θV(x,t,i)ijqij(cjci)(1θci)θV(x,t,i)β̄iijqijcjci(1θci).(25)

According to [1, 31], one has

ijqijcjci(1θci)=ijqijcj+ijqijθcicjci1θci=j=1Nqijcj+ijqijci(cjci)1θciθ=j=1Nqijcj+o(θ).(26)

Making use of Eqs 25, 26, we obtain that

AU(x,t,i)(1θci)θV(x,t,i)β̄ij=1Nqijcj+o(θ).(27)

Substituting Eq. 24 into Eq. 27, we get

AU(x,t,i)(1θci)θV(x,t,i)[o(θ)δ]=κU(x,t,i),

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

Ū(Xt,t,i)=eλt1θciU(Xt,t,i).

Compute

b1eηtE|Xt|pEŪ(Xt,t,i)=EU(X0,0,i0)+E0tAŪds+E0tŪxgdBsH=EU(X0,0,i0)+E0teλs(λU+AU)dsEU(X0,0,i0)+E0teλs(λ+κ)Uds=EV(x0,0,i0)+E0teλs(λ+κ)VdsEV(x0,0,i0)+E0teλs(λ+κ)b2|Xt|pds.

Thus we obtain that

b1eηtE|Xt|pV(x0,0,i0)+E0teλsb2(λ+κ)|Xt|pds,(28)

Dividing both sides of Eq. 28 by b1eηt, noting that b2(λ + κ) < 0, we get

E|Xt|peηtb1EV(X0,0,r0)+eηtb1E0teλsb2(λ+κ)|Xt|pdseηtb1EV(X0,0,r0).

Therefore, we obtain the required assertion

lim supt1tlog(E|Xt|p)<0.

The proof is complete.

6 Example

In this section we give a numerical example to illustrate our results.

Example 1. Let {rt}t0 be a right-continuous Markov chain taking values in S={1,2} with invariant probability measure μ1=μ2=12.Consider a risky asset, with the price dynamics:

dXt=f(Xt,t,rr)dt+σ(t,rt)XtdBtH,X0=1,(29)

on t ≥ 0. Here we take H = 0.7 and

f(x,t,i)=4x,σ(t,i)=1t+1,if i=1,f(x,t,i)=[2sin(x)]x,σ(t,i)=et,if i=2.

Note that for all iS, dXt=f(Xt,t,i)dt+σ(t,i)XtdBtH satisfy the hypothesises (i)-(v). Then, by Lemma 3.1, it is easy to show that Eq. 29 has a unique solution {Xt}t0 as well. Set V(x, t, i) = x2, for i = 1, 2.Noting that for some t0 > 0 sufficiently large and all t > t0, we have

L(1)V(x,t,1)=Vx(x,t,1)f(Xt,t,1)+Vxx(x,t,1)1t+1xDsϕx8x2+21t+1xxHt2H1=8x2+o(1)x2β1x2,

and

L(2)V(x,t,2)=Vx(x,t,2)f(Xt,t,2)+Vxx(x,t,2)etxDsϕx=2x2[2sin(x)]+o(1)x26x2+o(1)x2β2x2.

Compute

iSμiβi=12(8+6)+o(1)<0.

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.

FIGURE 1
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FIGURE 1. A single path of solution.

FIGURE 2
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FIGURE 2. Norm square trajectory.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author Contributions

WP contributed to conception and design of the study. WP wrote the first draft of the manuscript. ZZ and WP wrote sections of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

Funding

The research of WP was supported by the Characteristic and Preponderant Discipline of Key Construction Universities in Zhejiang Province (Zhejiang Gongshang University-Statistics).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

The authors are grateful to thank the reviewers for careful reading of the paper and for helpful comments that led to improvement of the first version of this paper.

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Keywords: stochastic differential equation (SDEs), stability, fractional brownian motion, markovian switching, hybrid system

Citation: Pei W and Zhang Z (2021) Stability of Hybrid SDEs Driven by fBm. Front. Phys. 9:783434. doi: 10.3389/fphy.2021.783434

Received: 26 September 2021; Accepted: 13 October 2021;
Published: 02 November 2021.

Edited by:

Ming Li, Zhejiang University, China

Reviewed by:

Xichao Sun, Bengbu University, China
Yaozhong Hu, University of Alberta, Canada

Copyright © 2021 Pei and Zhang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Wenyi Pei, peiwenyi@163.com

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