Stability of neutral pantograph stochastic differential equations with generalized decay rate

In this paper, we investigate the stability of highly nonlinear hybrid neutral pantograph stochastic differential equations (NPSDEs) with general decay rate. By applying the method of the Lyapunov function, the pth moment and almost sure stability with general decay rate of solution for NPSDEs are derived. Finally, an example is presented to show the effectiveness of the proposed methods.


Introduction
Hybrid stochastic differential delay equations (SDDEs) have been widely used to model many practical systems in science and industry, where the systems may experience abrupt changes in their structure and parameters. One of the important issues in the study of hybrid SDDEs is the analysis of stability and boundedness (see, e.g. Hu et al., 2021;Li et al., 2018;Mao, 1999Mao, , 2002Mao, , 2007Wu et al., 2018). Most of the papers in this area only considered stability problems where the underlying systems were either linear or nonlinear with the linear growth condition (see e.g. Mao, 1991;Mao & Yuan, 2006;Qi et al., 2021). However, many hybrid SDDEs do not satisfy this linear growth condition (namely they are highly nonlinear). Recently, some significant results have been done for highly nonlinear stochastic delay systems. For instance, Fei et al. (2017) explored delay dependent stability of hybrid SDDEs under the polynomial growth condition, while Fei et al. (2018) gave the structured robust stability criteria for highly nonlinear hybrid SDDEs.
A pantograph differential equation was used to investigate how an electric current is collected by the pantograph of an electric locomotive. Pantograph stochastic differential equations (PSDEs) as a class of special SDDE which have unbounded delay have been frequently applied in many practical areas, including electrodynamics and the collection of current by the pantograph of an electric locomotive. In recent years, the stability of PSDEs without linear growth condition has also attracted the interest of many researchers. For example, Hu et al. (2013) obtained polynomial stability of a class nonlinear hybrid CONTACT Yingjuan Yang yangyingjuan@ahpu.edu.cn PSDE. Shen, Mei, et al. (2019) explored the structured robust stability of highly nonlinear hybrid PSDEs. You et al. (2015) investigated the exponential stability of solutions for highly nonlinear hybrid PSDEs which experience abrupt changes in their parameters. On the other hand, many practical stochastic systems may not only depend on present and past states but also involve derivatives with delays. The neutral stochastic differential equations (NSDEs) have been developed to cope with such a situation, a great deal of results about the stability for NSDEs have been reported in the literature (see e.g. Shen, Fei, et al., 2019;Shen et al., 2018;Song & Shen, 2013;Wu et al., 2010;Zhang et al., 2019). However, to the best of our knowledge, there are only few results about the stability for NPSDEs. For example,  investigated the exponential stability of NPSDEs by both Lyapunov functional and M-matrix method. Liu and Deng (2018) discussed pth moment exponential stability of highly nonlinear NPSDEs driven by Lévy noise. In fact, the speed of the solution decays to zero is different. However, these stability concepts can be generalized to the stability with general decay rate (see, e.g. Li & Deng, 2017;Song & Shen, 2013;Wu et al., 2010). Consider a hybrid pantograph stochastic differential equation where B(t) is a scalar Brownian motion, r(t) is a rightcontinuous Markov. Mao et al. (2019) have studied almost sure stability of NPSDEs (1) with Markov switching. However, the results in Mao et al. (2019) may have some limitations in application. We now state a theorem of Mao et al. (2019) to show the limitations.
However, Theorem 3.4 in Mao et al. (2019) requires not only the function V(x, i, t) has the same degree for each i ∈ S but also that the diffusion operator LV(x, y, i, t) be bounded by a polynomial with the same degree for every i ∈ S (the notation will be explained in Section 2). These restrictive requirements make Theorem 3.4 to be applied to hybrid NPSDEs which have different nonlinear structures in different modes. To explain more clearly, let us, for example, consider a scalar hybrid NPSDE If we define the Lyapunov function as where θ 1 , θ 2 are positive numbers. It is easy to show No matter what parameters θ 1 and θ 2 we choose, we cannot get the form as condition (2) has. This problem prevents Theorem 1.1 from being used. Due to this NPSDE has different nonlinear structures in different modes, it is natural to use different types of Lyapunov functions in different modes. For example, let us define the Lyapunov function as In this paper, we will establish the stability results under these weaker conditions.
The key contributions of our paper are highlighted below: (1) This paper investigates stability for highly nonlinear hybrid NPSDEs with general decay rate. A significant amount of new mathematics has been developed to deal with the difficulties due to the neutral term.
(2) In the paper of Mao et al. (2019), there are some restrictive conditions which may exclude many nonlinear NPSDEs. In this paper, we will loosen this restrictive condition to cover a much wider class NPS-DEs.
(3) The stability criteria can be applied to a class of hybrid NPSDEs which have the same structures in different modes and a class of hybrid NPSDEs which have different structures in different modes.
To develop our theory, we will introduce some necessary notations in Section 2. Our main results on stability with general decay rate will be discussed in Section 3. We will present an example in Section 4 to illustrate our theory. We will finally conclude our paper in Section 5.

Notation
Let ( , F, {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is increasing and right continuous while F 0 contains all P-null sets). Let B(t) = (B 1 (t), . . . , B m (t)) T be an m-dimensional Brownian motion defined on the probability space. Let r(t) be a continuous Markov chain on the probability space taking values in a finite state space S = {1, 2, . . . , N} with generator = (γ ij ) N×N given by where > 0 and γ ij ≥ 0 is the transition rate from i to j if i = j while γ ii = − j =i γ ij . We assume that the Markov chain r(·) is independent of the Brownian motion B(·). We also denote by |x| the Euclidean norm for x ∈ R n . If A is a vector or matrix, its transpose is denoted by on t ≥ 0 with initial date x(0) = x 0 , r(0) = r 0 , where the coefficients f : R n × R n × S × R + → R n and g : R n × R n × S × R + → R n×m are Borel measurable functions. For the purpose of stability with general decay rate, we need the following definition of ψ-type function.
It is obvious that the functions ψ(t) = e t and ψ(t) = 1 + t are all ψ-type functions since they satisfy the three conditions of Definition 2.1.
As a standing hypothesis, we assume the coefficients are locally Lipschitz continuous.

Stability of PNSDEs with general decay rate
With the notations and assumptions introduced in above, the following theorem shows the existence and uniqueness of the global solution. The theorem forms the foundation of this paper.
Theorem 3.1: Let Assumptions 2.2 and 2.3 hold. Assume there are the functions V 1 , V 2 ∈ C(R n × R + ; R + ) and V ∈ C 2,1 (R n × S × R + ; R + ), as well as nonnegative constants K, K j , p j (1 ≤ j ≤ L) for some integer L such that

then for any given initial date, there is a unique global solution x(t) to the NPSDE (4).
Proof: Fix the initial data x 0 ∈ R n and r 0 ∈ S arbitrarily. Define where throughout this paper we set inf ∅ = ∞ (as usual ∅ denotes the empty set). It is easy to see that σ k is increasing as k → ∞ and lim k→∞ σ k = ∞ a.s. By the generalized Itô formula (see, e.g. Mao & Yuan, 2006), we obtain that
Before giving our results, we state the following lemma which plays a crucial role in this paper.

Lemma 3.2:
Define the quasi polynomial function G(x) = a m |x| m + · · · + a 2 |x| 2 , x ∈ R n , where |x| is the Euclidean norm of x, a m > 0, a i ≥ 0, i = 1, . . . , m − 1 and m ≥ 2 are positive integers. Let Assumption 2.3 hold, we have the following conclusion: Proof: For p ≥ 2, we apply the inequality which, together with (6) and 0 < λ < 1, shows that Hence, the proof is complete.
The key technique used in this paper is the method of the Lyapunov function. To study the stability of the NPSDE (4), we need to impose the following assumption which gives the our Lyapunov function.
Theorem 3.4: Let Assumptions 2.2, 2.3 and 3.3 hold, then for any given initial date, the solution of the NPSDE (4) has the properties that lim sup Proof: The condition (9) is stronger than Assumption 5, thus, there is a unique global solution for Equation (4). By the generalized Itô formula (see, e.g. Mao & Yuan, 2006), we obtain that
By Lemma 3.2, we have where C θ depends on the highest power of U 2 . Thus, we can obtain Now we choose ε sufficiently small such that α 3 ≥ α 4 + εrC θ + λεr, then by (8), we can deduce that which implies assertion (11).
Then, for any T > 0, we have This together with (12), we can get where C 1 is a positive constant. Letting T → ∞, we have that is By the generalized Itô formula, we have where M 1 (t) is a local martingale with the initial value M 1 (0) = 0. Applying the same argument on deriving (13), where M 2 (t) is a local martingale. Applying the nonnegative semi-martingale convergence theorem, we get Therefore, there is a finite positive random variable ξ such that which completes the proof.

An example
In this section we will discuss an example to illustrate our theory. Although our example is scalar hybrid NPSDEs, it will illustrate our theory fully.
The corresponding initial values are given by x(0) = x 0 ∈ R, r(0) = i 0 ∈ S, moreover, θ, D, f and g are defined by (3). By Theorem 3.4 and Corollary 3.5, we can conclude that the solution of the NPSDE (14)    This shows that solution of Equation (14) is almost surely exponentially stable. Figure 1 illustrates the sample path of the NPSDE (14) while Figure 2 illustrates the moment characteristic of t 0 x 4 (s) ds.

Conclusion
In this paper we have established the general decay rate stability criteria for hybrid NPSDEs by removing the linear growth condition. Our results not only can be applied to a class of hybrid NPSDEs which have the same structures in different modes but also can be applied to a class of hybrid NPSDEs which have different structures in different modes.

Disclosure statement
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.