Partial asymptotic stability of neutral pantograph stochastic differential equations with Markovian switching

In this paper, we investigate the partial asymptotic stability (PAS) of neutral pantograph stochastic differential equations with Markovian switching (NPSDEwMSs). The main tools used to show the results are the Lyapunov method and the stochastic calculus techniques. We discuss a numerical example to illustrate our main results.

we prove the PAS of solutions of NPSDEwMSs. In this sense, our results extend the analysis in [5] and [12] providing the neutral term and the delay in the case of the PSDE with Markovian switching.
Let us outline the framework of this paper. After preliminaries and notations (see Sect. 1), in Sect. 2, we recall some important notions and definitions. In Sect. 3, we establish the PAS for a class of NPSDEwMSs. Finally, in Sect. 4, we present a numerical example to show the applicability of our results.

Preliminaries and notations
for > 0. Here γ jk ≥ 0 is the transition rate from j to k if j = k, whereas We suppose that r and W are independent. Consider the following NPSDEwMS: where G(s, z(qs), m(s)) = (G 1 (s, z(qs), m(s)), G 2 (s, z(qs), m(s))) T ∈ R n . We assume that Let z = (z 1 , z 2 ) T ∈ R n be the solution of equation (2.1), where z 1 ∈ R k and z 2 ∈ R p , and k + p = n. We will impose the following assumptions on f , g, and G: Set G(s, 0, j) = 0 and κ = max j∈S κ j . Let C 1,2 ([qs 0 , +∞) × R n ×S; R + ) be the set of all nonnegative functions V (s, z, j) on [qs 0 , +∞) × R n ×S that are once continuously differentiable with respect to s and twice continuously differentiable with respect to z.

Main results
We discuss the PS in probability and PAS of equation (2.1).

Definition 3.1
(i) The solution z(s) = (z 1 (s), z 2 (s)) of equation (2.1) is called PS in probability with respect to z 1 if for all η > 0 and λ ∈ (0, 1), there exists δ 0 = δ 0 (λ, η, s 0 ) > 0 such that (ii) The solution z(s) = (z 1 (s), z 2 (s)) of equation (2.1) is called PAS in probability with respect to z 1 if it is stable in probability with respect to z 1 and for all Let K be the set of all continuous nondecreasing functions μ :

Then the solution of equation (2.1) is PS in probability with respect to z 1 .
Proof By Assumptions (A 1 )-(A 3 ) system (2.1) has a unique global solution z(s) for s ≥ s 0 (see [17]).
Let λ ∈ (0, 1) and η > 0 be arbitrary. We will assume that η < H. By the continuity of V (s, z, j) and the fact V (s 0 , 0, m(s 0 )) = 0 we can find ρ = ρ(λ, η, s 0 ) > 0 such that We can see that ρ < η. Fix an arbitrary initial condition ζ ∈ L p F s 0 ([qs 0 , s 0 ]; R n ) such that ζ < ρ. Let ϑ be the stopping time given by By the Itô formula, for every s ≥ s 0 , we have Using (ii) and equation (3.1), we obtain that Notice that if ϑ ≤ s, then Then by (i) we have and the proof is completed.

Theorem 3.2 Suppose that assumptions
Proof We will proceed as in the proof of Theorem 3.1 in [23] with necessary changes.
By Theorem 3.1 it is easy to prove that equation (2.1) is stable in probability with respect to z 1 .
Consequently, the solution of system (2.1) is asymptotically stable in probability with respect to z 1 .

Asymptotic instability of NPSDEwMS
We will state a theorem about the asymptotic instability with respect to all variables of NPSDEwMS.
Proof The proof is similar to that of Theorem 4.3 in [6].

Example and numerical solution
We now give a numerical example to illustrate the application of our results.
Let W (s) be a three-dimensional Brownian motion. Let m(s) be a right-continuous Markov chain taking values inS = {1, 2, 3} with = (γ jk ) 1≤j,k≤3 given by Moreover, we assume that W (s) and m(s) are independent. Consider the following NPS-DEwMS:  The simulation results clearly show that the trajectories of the corresponding stochastic system converge asymptotically to the equilibrium state for any given initial values, thus verifying the effectiveness of theoretical results.