Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching

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 ∈ CF0b ([−τ, 0];ℝⁿ), where h, f, g are appropriately specified later.

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 U_β(t) and characterized by Laplace transform

For an adapted β -stable subordinator U_β(t), define its generalized inverse as

which means the first hitting time process. And E_t is continuous since U_β(t) is strictly increasing.

Let B_t be a standard Brownian motion independent on E_t, 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 B_Et is a square integrable martingale with respect to the filtration {𝔉_Et}_t≥0. And its quadratic variation satisfies < B_Et, B_Et > = E_t.([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 = (γ_ij)_N×N by

where Δ > 0, γ_ij is the transition rate from i to j if i ≠ j and γ_ii = −∑i≠jγij . We assume that the Markov chain r(t) is independent on Brownian motion, it is well known that almost each sample path of r(t) is a right-continuous step function.

For the future use, we formulate the following generalized time-changed Itô formula.

In this paper, the following hypothesis is imposed on the coefficients h, f and g.

1. Both h, f : ℝⁿ × ℝ₊ × ℝ₊ × S → ℝⁿ and g : ℝⁿ × ℝ₊ × ℝ₊ × S → ℝ^n×m are Borel-measurable functions. They satisfy the Lipschitz condition. That is, there is L > 0 such that

   |h(ϕ1,t1,t2,i)−h(ϕ2,t1,t2,i)|∨|f(ϕ1,t1,t2,i)−f(ϕ2,t1,t2,i)|∨|g(ϕ1,t1,t2,i)−g(ϕ2,t1,t2,i)|≤L||ϕ1−ϕ2||

   for all t ≥ 0, i ∈ S and ϕ₁, ϕ₂ ∈ C([ − τ, 0];ℝⁿ).

2. If x(t) is an RCLL and 𝔉_Et-adapted process, then h(x_t, t, E_t, r(t)), f(x_t, t, E_t, r(t)), g(x_t, t, E_t, r(t)) ∈ 𝓛(𝔉_Et), where 𝓛(𝔉_Et) denotes the class of RCLL and 𝔉_Et-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).

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 CF0b ([−τ, 0];ℝⁿ). The solution of equation (1.1) is denoted as x(t; ξ) throughout this paper. Assume that h(0, t, E_t, i) = 0, f(0, t, E_t, i) = 0, g(0, t, E_t, i) = 0, so the equation (1.1) have a trivial solution x(t; 0) = 0. Next, we establish a new Razumikhin theorem on p-th moment exponential stability for the time-changed SFDEs with Markovian switching.

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 ∈ CF0b ([−τ, 0];ℝⁿ), where δ:ℝ₊ → [0, τ] is Borel measure while

and

We impose the following hypotheses:

1. Both H, F : ℝⁿ × ℝ₊ × ℝ₊ × S → ℝⁿ and G : ℝⁿ × ℝ₊ × ℝ₊ × S → ℝ^n×m are Borel-measurable functions. They satisfy the Lipschitz condition. That is, there is L > 0 such that

   |H(x,y,t1,t2,i)−H(x¯,y¯,t1,t2,i)|∨|F(x,y,t1,t2,i)−F(x¯,y¯,t1,t2,i)|∨|G(x,y,t1,t2,i)−G(x¯,y¯,t1,t2,i)|≤L(|x−x¯|+|y−y¯|)

   for all t ≥ 0, i ∈ S and x, y, x, y ∈ ℝⁿ.

2. If x(t) is an RCLL and 𝓕_Et-adapted process, then H(x(t), x(t − δ(t)), t, E_t, r(t)), F(x(t), x(t − δ(t)), t, E_t, r(t)), G(x(t), x(t − δ(t)), t, E_t, r(t)) ∈ 𝓛(𝔉_Et), where 𝓛(𝔉_Et) denotes the class of RCLL and 𝔉_Et-adapted process.

If we define, for (ϕ, t, E_i, 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, E_t, i) = 0, F(0, 0, t, E_t, i) = 0, G(0, 0, t, E_t, i) = 0.

If V ∈ C^2,1,1(ℝⁿ × [−τ, ∞) × [0, ∞) × S;ℝ⁺), define L₁V and L₂V from ℝⁿ × ℝⁿ × ℝ⁺ × ℝ⁺ × S to ℝ respectively by

Furthermore, we denote LFtp (Ω, ℝⁿ) as the family of all 𝔉_t-measurable ℝⁿ-valued random variables X such that E|X|^p < ∞. Meanwhile, we set

5 Example

Let E_t be generalized inverse of an β -stable subordinator U_β(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 = {r_ij}_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