Abstract
This work is mainly concerned with the exponential stability of time-changed stochastic functional differential equations with Markovian switching. By expanding the time-changed Itô formula and the Razumikhin theorem, we obtain the exponential stability results for the time-changed stochastic functional differential equations with Markovian switching. What’s more, we get many useful stability results by applying our new results to several important types of functional differential equations. Finally, an example is given to demonstrate the effectiveness of the main results.
1 Introduction
The research for stochastic differential equations (SDEs) is a mature field, which plays an important role in modeling dynamic system considering uncertainty noise in many applied areas such as economics and finance, physics, engineering and so on. Many qualitative properties of the solution of stochastic functional differential equations (SFDEs) have been received much attention. In particular, the stability or asymptotic stability of SFDEs has been studied widely by more and more researchers ([1, 2, 3, 4, 5]).
Recently, Chlebak et al.[6] discussed sub-diffusion process and its associated fractional Fokker-Planck-Kolmogorov equations. The fractional partial differential equations are well known to be connected with limit process arising from continuous-time random walks. The limit process is time-changed Lev́y process, which is the first hitting time process of a stable subordinator (see [7, 8, 9] for details). The existence and stability of SDE with respect to time-changed Brownian motion recently have received much attention([10, 11]). Wu [12, 13] established the time-changed Itô formula of time-changed SDE, and then obtained the stability results. Subsequently, Nane and Ni [14] established the Itô formula for time-changed Lévy noise, then discussed the stability of the solution.
However, to the best of our knowledge, there are no results for the time-changed stochastic functional differential equations with Markovian switching published till now. Motivated strongly by the above, in this paper, we will study the stability of time-changed SFDEs with Markovian switching. By applying the time-changed Itô formula and Lyapunov function, we present the Razumikhin-type theorem([15, 16]) of the time-changed SFDEs with Markovian switching. More precisely, we consider the following SFDEs with Markovian switching driven by time-changed Brownian motions:
on t ≥ 0 with {x(θ) : − τ ≤ θ ≤ 0} = ξ ∈
In the remaining parts of this paper, further needed concepts and related background will be presented in Section 2. In Section 3, the exponential stability results of the time-changed SFDEs with Markovian switching will be given. Many useful types of results of stochastic delay differential equations and stochastic differential equations are presented in Section 4 and Section 5 respectively. Finally, an example is given to show the availability of the main results.
2 Preliminary
Throughout this paper, let (Ω, 𝔉, {𝔉}t≥0, P) be a complete probability space with the filtration {𝔉}t≥0 which satisfies the usual condition(i.e. {𝔉}t≥0 is right continuous and 𝔉 contains all the P-null sets in 𝔉). Let {U(t), t ≥ 0} be a right continuous with left limit (RCLL) increasing Lévy process that is called subordinator starting from 0. For a subordinator U(t), in particular, is a β -stable subordinator if it is a strictly increasing process denoted by Uβ(t) and characterized by Laplace transform
For an adapted β -stable subordinator Uβ(t), define its generalized inverse as
which means the first hitting time process. And Et is continuous since Uβ(t) is strictly increasing.
Let Bt be a standard Brownian motion independent on Et, define the following filtration as
where σ1 ∨ σ2 denotes the σ -algebra generated by the union of σ -algebras σ1 and σ2. It concludes that the time-changed Brownian motion BEt is a square integrable martingale with respect to the filtration {𝔉Et}t≥0. And its quadratic variation satisfies < BEt, BEt > = Et.([17])
Let r(t), t ≥ 0 be a right continuous Markov chain on the probability space taking values in a finite state space S = {1, 2, …, N} with generator Γ = (γij)N×N by
where Δ > 0, γij is the transition rate from i to j if i ≠ j and γii =
For the future use, we formulate the following generalized time-changed Itô formula.
Lemma 2.1
(The generalized time-changed Itô formula) Suppose Uβ(t) is a β -stable subordinator and Et is its associated inverse stable subordinator. Let x(t) be a 𝔉Et adapted process defined in (1.1). If V : ℝn × ℝ+ × ℝ+ × S → ℝ is a C2,1,1 (ℝn × ℝ+ × ℝ+ × S; ℝ) function, let
and
then with probability one
where μ(ds, dl) = ν(ds, dl) − m(dl)ds is a martingale measure, ν(ds, dl) is a Poisson random measure with density dt × m(dl), in which m is the Lebesgue measure on ℝ.
Proof
Let y = [x, t1, t2]T = [x, t, Et]T, and G(y(t), r(t)) = V(xt, t, Et, r(t)). Based on the computation rules ([8]), we have
Applying the multi-dimensional Itô formula([18]) to G(y(t), r(t)) yields that
This completes the proof. □
Corollary 2.1
Suppose Uβ(t) is a β -stable subordinator and Et is its associated inverse. Let x(t) be an 𝔉Et adapted process defined in (1.1). If V:ℝn × ℝ+ × ℝ+ × S → ℝ is a C2,1,1(ℝn × ℝ+ × ℝ+ × S; ℝ) function, then for any stopping time 0 ≤ t1 ≤ t2 < ∞
where L1 and L2 are defined in the lemma above.
In this paper, the following hypothesis is imposed on the coefficients h, f and g.
Both h, f : ℝn × ℝ+ × ℝ+ × S → ℝn and g : ℝn × ℝ+ × ℝ+ × S → ℝn×m are Borel-measurable functions. They satisfy the Lipschitz condition. That is, there is L > 0 such that
for all t ≥ 0, i ∈ S and ϕ1, ϕ2 ∈ C([ − τ, 0];ℝn).
If x(t) is an RCLL and 𝔉Et-adapted process, then h(xt, t, Et, r(t)), f(xt, t, Et, r(t)), g(xt, t, Et, r(t)) ∈ 𝓛(𝔉Et), where 𝓛(𝔉Et) denotes the class of RCLL and 𝔉Et-adapted process.
3 Main results
In this section, we aim to establish the stability results of the system equation (1.1). Firstly, we have to guarantee the existence of the solution of the equation (1.1).
Lemma 3.1
Under the conditions of (H1) and (H2), for any initial data {x(θ) : − τ ≤ θ ≤ 0} = ξ ∈
Proof
Let T > 0 be arbitrary. It is known that ([18]) there is a sequence {τk}k≥0 of stopping times such that 0 < τ0 < τ1 < ⋯ < τk → ∞ and r(t) is constant on each interval [τk, τk+1), that is, for each k ≥ 0,
We first consider the equation on t ∈ [0, τ1 ∧ T], it becomes
with initial data x0 = ξ ∈
with initial data xτ1∧T given above. Again we know the equation has a unique continuous solution on [τ1 ∧ T − τ, τ2 ∧ T]. Repeating the progress, we can see the equation has a unique solution x(t) on [−τ, T]. Since T is arbitrary, the existence and uniqueness have been proved. □
Now, let us consider the exponential stability of equation (1.1). We fix the Markov chain r(t) and let the initial data ξ vary in
Theorem 3.1
Let (H1) and (H2) hold. Let λ1, λ2, p, c1, c2, α be all positive numbers and q > 1. Assume that there exists a function V(x, t, Et, i) ∈ C2,1,1(Rn × [−τ, ∞) × [0, ∞) × S; R+) such that
and for all t > 0,
provided ϕ = {ϕ(θ; − τ ≤ θ ≤ 0)} satisfying
for all −τ ≤ θ ≤ 0. Then for all ξ ∈
where γ = min{λ1, λ2, log(q)/τ}. In other words, the trivial solution of equation (1.1) is pth moment exponentially stable and the pth moment Lyapunov exponent is not greater than − γ.
Proof
For the initial data ξ ∈
Since r(t) is right continuous, the fact that both Et and x(t) is continuous and
To show this, we know that for each t ≥ 0, either U(t) > 𝓔[eγ(t+Et)V(x(t), t, Et, r(t))] or U(t) = 𝓔[eγ(t+Et)V(x(t), t, Et, r(t))].
If U(t) > 𝓔[eγ(t+Et)V(x(t), t, Et, r(t))], it follows from the right continuity of 𝓔[eγ(t+Et)V(x(t), t, Et, r(t))] that for each l > 0 sufficiently small
Noting that
if l + θ > 0, by (3.6), we have
Therefore, U(t + l) ≤ U(t). On the other hand, if l + θ ≤ 0, we set θ′ = l + θ, then
Therefore, for each t > 0, U(t + l) ≤ U(t) and D+ U(t) ≤ 0.
If U(t) = 𝓔[eγ(t+Et)V(x(t), t, Et, r(t))], by the definition of U(t), one obtains that for −τ ≤ θ ≤ 0,
it follows that
If 𝓔 [eγEtV(x(t), t, Et, r(t))] = 0, from (3.1) we can see that
which yields that x(t + θ) = 0, −τ ≤ θ ≤ 0. Since h(0, t, Et, i) = 0, f(0, t, Et, i) = 0 and g(0, t, Et, i) = 0 a.s. for all −τ ≤ θ ≤ 0, one obtains that x(t + l) = 0 a.s. for all l > 0, hence U(t + l) = 0 and D+ U(t) = 0.
On the other hand, if 𝓔 [eγEtV(x(t), t, Et, r(t))] > 0, one can see that
for all −τ ≤ θ ≤ 0 since eγτ < q. It follows from the condition (3.2) that
It means that
then
By the right continuity of the process involved one can see that for all l > 0 sufficiently small,
By the generalized time-changed Itô formula, we get that
Then U(t + l) ≤ U(t) for l > 0 sufficiently small.
Since
here we set θ′ = θ + l, if l + θ > 0, then 𝓔 [eγ(t+θ′+Et+θ′)V(x(t+θ′), t + θ′, Et+θ′, r(t + θ′))] ≤ U(t) from (3.7), otherwise, since U(t) = 𝓔[eγ(t+Et)V(x(t), t, Et, r(t))], then
so, by the definition of supremum, U(t + l) = U(t) for l > 0 sufficiently small and D+ U(t) = 0. Therefore, the inequality (3.5) has been proved. It follows that
this means
Since ε is arbitrary, the required inequality (3.4) must hold. The proof is completed. □
4 Stochastic delay differential equations with Markovian switching
In this section, as a special case of equation (1.1), we consider the time-changed stochastic delay differential equation with Marking switching as follows,
on t ≥ 0 with x0 = ξ ∈
and
We impose the following hypotheses:
Both H, F : ℝn × ℝ+ × ℝ+ × S → ℝn and G : ℝn × ℝ+ × ℝ+ × S → ℝn×m are Borel-measurable functions. They satisfy the Lipschitz condition. That is, there is L > 0 such that
for all t ≥ 0, i ∈ S and x, y, x, y ∈ ℝn.
If x(t) is an RCLL and 𝓕Et-adapted process, then H(x(t), x(t − δ(t)), t, Et, r(t)), F(x(t), x(t − δ(t)), t, Et, r(t)), G(x(t), x(t − δ(t)), t, Et, r(t)) ∈ 𝓛(𝔉Et), where 𝓛(𝔉Et) denotes the class of RCLL and 𝔉Et-adapted process.
If we define, for (ϕ, t, Ei, i) ∈ C([−τ, 0];ℝn) × ℝ+ × ℝ+ × S,
then the equation (4.1) becomes the equation (1.1) and (H3) (H4) imply (H1) (H2). So, by Lemma 3.1, the equation (4.1) has a unique global solution which is again denoted by x(t; ξ). Furthermore, assume that H(0, 0, t, Et, i) = 0, F(0, 0, t, Et, i) = 0, G(0, 0, t, Et, i) = 0.
If V ∈ C2,1,1(ℝn × [−τ, ∞) × [0, ∞) × S;ℝ+), define L1V and L2V from ℝn × ℝn × ℝ+ × ℝ+ × S to ℝ respectively by
Furthermore, we denote
Theorem 4.1
Let (H3) and (H4) hold. Let λ1, λ2, p, c1, c2, α be all positive numbers and q > 1. Assume that there exists a function V(x, t, Et, i) ∈ C2,1,1(Rn × [−τ, ∞) × [0, ∞) × S; R+) such that
and for all t > 0,
provided X, Y ∈
Then for all ξ ∈
where γ = min{λ1, λ2, log(q)/τ}. In other words, the trivial solution of equation (4.1) is pth moment exponentially stable and the pth moment Lyapunov exponent is not greater than −γ.
Proof
Let ϕ = {ϕ(θ) : − τ ≤ θ ≤ 0} ∈
Then, from (4.3) we have
which is (3.2). Hence the conditions in Theorem 3.1 are satisfied and the conclusions follow. Applying the Theorem 3.1, the proof is completed. □
Theorem 4.2
Let (H3) and (H4) hold. Let p, c1, c2, α be all positive numbers and λ1j > λ2j ≥ 0, j = 1, 2. Assume that there exists a function V(x, t, Et, i) ∈ C2,1,1(Rn × [−τ, ∞) × [0, ∞) × S; R+) such that
and for all t > 0,
Then the trivial solution of equation (4.1) is pth moment exponentially stable and the pth moment Lyapunov exponent is not greater than −γ, where γ = min{λ11 − qλ21, λ12 − qλ22, log(q)/τ} with q > 1.
Proof
For t ≥ 0, q < λ1j/λ2j, j = 1, 2 and X, Y ∈
we can arrive that
that is, (4.3) is satisfied with λj = λ1j − qλ2j, j = 1, 2. Then the conclusion follows form Theorem 4.1. □
5 Example
Let Et be generalized inverse of an β -stable subordinator Uβ(t). Let B(t) be a scalar Brownian motion and {r(t)} be a right-continuous Markov chain taking values in S = {1, 2} with generator Γ = {rij}2×2, here
Assume that B(t) and r(t) are independent. Then let us consider the following one-dimensional linear stochastic differential equation with Markovian switching
where
The equation (5.1) can be regarded as the result of
and
switching to each other according to the movement of the Markovian chain r(t).
We define the function V : ℝ × ℝ+ × ℝ+ × S → ℝ+ by
with ci = 1, c2 = c ∈ (0,
Using the following inequality
we can see that
Choose p = 2, 2 < c < 3, then
By the Theorem 4.2 we conclude that the trivial solution of the equation (5.1) is pth moment exponentially stable.
6 Conclusions
The stochastic differential equations(SDEs) driven by time-changed Brownian motions is a new research area for recent years. In this paper, we have studied the exponential stability of the time-changed SDEs with Markovian switching, by expanding the time-changed Itô formula and the time-changed Razumikhin theorem. Our result generalizes that of SDEs in the literature. Due to the more construction of SDEs with time-change than the usual SDEs, our result is not a trivial generalization.
Acknowledgement
The Project is supported by the National Natural Science Foundation of China (11861040, 11771198, 11661053).
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