Stochastic Functional Differential Equation under Regime Switching

We discuss stochastic functional differential equation under regime switching dx t f xt, r t , t dt q r t x t dW1 t σ r t |x t |βx t dW2 t . We obtain unique global solution of this system without the linear growth condition; furthermore, we prove its asymptotic ultimate boundedness. Using the ergodic property of the Markov chain, we give the sufficient condition of almost surely exponentially stable of this system.


Introduction
Recently, many papers devoted their attention to the hybrid system, they concerned that how to change if the system undergoes the environmental noise and the regime switching.For the detailed understanding of this subject, 1 is good reference.
In this paper we will consider the following stochastic functional equation: dx t f x t , r t , t dt q r t x t dW 1 t σ r t |x t | β x t dW 2 t .

1.1
The switching between these N regimes is governed by a Markovian chain r t on the state space S {1, 2, . . ., N}. x t ∈ C −τ, 0 ; R n is defined by x t θ x t θ ; θ ∈ −τ, 0 .C −τ, 0 ; R n denote the family of continuous functions from −τ, 0 to R n , which is a Banach space with the norm φ sup −τ≤θ≤0 |φ θ |. f : C −τ, 0 ; R n × S × R → R n satisfies local Lipschitz condition as follows.
Throughout this paper, unless otherwise specified, we let Ω, 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 right continuous and F 0 contains all P-null sets .Let W i t i 1, 2 , t ≥ 0, be the standard Brownian motion defined on this probability space.We also denote by R n {x ∈ R n : x i > 0 for all 1 ≤ i ≤ n}.Let r t be a right-continuous Markov chain on the probability space taking values in a finite state space S {1, 2, . . ., N} with the generator Γ γ uv N×N given by where δ > 0.Here γ uv is the transition rate from u to v and γ uv ≥ 0 if u / v while We assume that the Markov chain r • is independent on the Brownian motion W i • , i 1, 2; furthermore, W 1 and W 2 are independent.
In addition, throughout this paper, let C 2,1 R n × −τ, ∞ × S; R denote the family of all positive real-valued functions V x, t, k on R n × −τ, ∞ × S which are continuously twice differentiable in x and once in t.If for the following equation dx t f x t , r t , t dt g x, r t , t dB t , 1.5 where

1.7
Here we should emphasize that 1, Page 305 the operator V thought as a single notation rather than acting on V is defined on

Global Solution
Firstly, in this paper, we are concerned about that the existence of global solution of stochastic functional differential equation 1.1 .
In order to have a global solution for any given initial data for a stochastic functional equation, it is usually required to satisfy the local Lipschitz condition and the linear growth condition 1, 2 .In addition, as a generation of linear condition, it is also mentioned in 3, 4 with one-sided linear growth condition.The authors improve the results using polynomial growth condition in 5, 6 .After that, these conditions were mentioned under regime systems 7-9 .
Replacing the linear growth condition or the one-sided linear growth condition, we impose the so-called polynomial growth condition on the function f for 1.1 .

Assumption B.
For each i ∈ S, there exist nonnegative constants α, κ i , κ i , γ, K and probability measures μ, on −τ, 0 such that

2.2
Clearly then τ e ∞ a.s.for all t ≥ 0. That is, to complete the proof, also equivalent to prove that, for any t > 0, P τ m ≤ t → 0 as m → ∞.If this conclusion is false, there is a pair of constants T > 0 and ε ∈ 0, 1 such that So there exists an integer m 1 ≥ m 0 such that To prove the conclusion that we desired, for any p ∈ 0, 1/2 , define a C 2 -function: where {c k , 1 ≤ k ≤ N} is positive constant sequence.Applying the generalized It ô formula, where V is computed as γ kl V ϕ 0 , l .

2.7
Let q max k,l∈S {c l /c k }.For any k, l ∈ S, we get According to Assumption B, the first term in 2.7

2.10
By the Young inequality and noting that p ∈ 0, 1/2 , it is obvious that where the first inequality we have used the elementary inequality: for any a, b ≥ 0 and r ∈ 0, 1 , a b r ≤ a r b r .Therefore we have

2.15
Noting that p ∈ 0, 1/2 and 2β > α ≥ 0, by the boundedness property of polynomial functions, there exists a positive constant M k such that Taking expectation from two sides of 2.6 leads to and from 2.12 and 2.15 , we have 2.17 where we denote č max k∈S,1≤i≤n c i k .
By the Fubini theorem and a substitution technique, we may compute that

2.19
Therefore we rewrite 2.17 into

2.20
where K T is bounded and K T is independent of m.
By the definition of τ m , x τ m m or 1/m, so let Ω m {τ m ≤ T } for m ≥ m 1 and by 2.6 , noting that for every ω ∈ Ω m , there is some m such that x m τ m , ω equals either m or 1/m hence

2.21
Letting m → ∞ implies that lim sup So we must obtain τ ∞ ∞ a.s., as required.The proof is complete.

Asymptotic Boundedness
Theorem 2.1 shows that the solution of SDE 1.1 exists globally and will not explode under some reasonable conditions.In the study of stochastic system, stochastically ultimate boundedness is more important topic comparing with nonexplosion of the solution, which means that the solution of this system will survive under finite boundedness in the future.
Here we examine the 2pth moment boundedness.Here V ϕ, k is defined as before 2.7 .Now we consider the function

3.3
Similar to the proof of Theorem 2.1, then we know 3.3 is upper bounded; there exists constant

3.4
We have the following calculus transformation:

3.6
We therefore have from 3.4

3.10
This means that the solution is bounded in the 2pth moment; the stochastically ultimate boundedness will follow directly.

Stabilization of Noise
From Sections 2 and 3, we know that under the condition σ i / 0 and 2β > α, the Brownian noise σ i |x t | β x t dW 1 t can suppress the potential explosion of the solution and guarantee this global solution to be bounded in the sense of the 2pth moment.Clearly, the boundedness results are also dependent only on the choice of β under the condition σ i / 0 and independent of q i .This implies that the noise W 1 t plays no role to guarantee existence and boundedness of the global solution to 1.1 .This section is devoted to consider the effect of noise q i x t dW 1 t , we will show that the system 1.1 is exponential stability if for some sufficiently large q i .For the purpose of stability study, we impose the following the general polynomial growth condition: Assumption C. For each i ∈ S, there exist nonnegative constants α, κ i , κ i , γ, and K and probability measures μ, on −τ, 0 such that Proof.For any initial data ξ ∈ C −τ, 0 ; R n satisfying x 0 / 0, for sufficiently large positive number i 0 , such that |x 0 | > 1/i 0 .For each integer i ≥ i 0 , define the stopping time Clearly, ρ i is increasing as i → ∞ and ρ i → ρ ∞ a.s.If we can show that ρ ∞ ∞ a.s., the desired result P x t / 0 1 on t ≥ 0 follows.This is equivalent to proving that, for any t > 0, P ρ i ≤ t → 0 as i → ∞.
To prove this statement, define a C 2 -function Applying the It ô formula and taking the expectation yield where V 1 is defined as for any ϕ ∈ C −τ, 0 ; R n .By Assumption C and the Young inequality, the first term of 4.8 will be written as

4.11
Noting 2.9 and 2β > α, σ k / 0, using the boundedness property of polynomial functions, there exists a constant M k such that Furthermore, we may estimate that

4.12
It therefore follows that

4.13
We know that as required.The proof is completed.
This lemma shows that almost all the sample path of any solution of 1.1 starting from a non-zero state will never reach the origin.Because of this nice property, the Lyapunov functions we can choose need not be imposed globally but only in a deleted neighborhood of the origin.
Especially, the hybrid system always switch from any regime to another regime, so it is reasonable to assume that the Markov chain r t is irreducible.It means to the condition that irreducible Markov chain has a unique stationary probability distribution π π 1 , π 2 , . . ., π N ∈ R 1×N which can be determined by solving the following linear equation πΓ 0 subject to N k 1 π k 1 and π k > 0 for any k ∈ S, where Γ is generator Γ γ uv N×N .
Theorem 4.2.Suppose the Markov chain r t is irreducible, under Assumption A and C, if for δ ∈ 0, 1 , k ∈ S, σ k / 0 and 2β > α, the solution x t of SDE 1.1 with any initial data ξ ∈ C −τ, 0 ; R n satisfying x 0 / 0 has the property where

4.17
In particular, the nonlinear hybrid system 1.1 is almost surely exponentially stable if

4.22
This, together with Assumption C, denote q : max k∈S {q k } and κ : max k∈S {κ k }; noting the definition of 4.17 , we therefore have

4.24
Applying the strong law of large number Noting that n i 1 π i φ i − q 2 i /2 has the form However, as the result of Markovian switching, the overall behavior, that is SDE 1.1 will be almost surely exponentially stable as long as We can see the impact of the Markov chain r t .The distribution π π 1 , π 2 , . . ., π n of r t plays a very important role, which, combined with n i 1 π i φ i − 1/2 q 2 i < 0, determine that system 1.1 is almost surely exponentially stable.If r t spends enough time in the "good" states the state where φ i − q 2 i /2 < 0 for some i , even if there exist some "bad" states the states where φ i − q 2 i /2 > 0 for some i , the system 1.1 will still be almost surely exponentially stable.
|x| α 2p s θ dμ θ ds − t 0 e ε s τ |x| α 2p s ds ≤ e ετ 0 Under the conditions of Theorem 2.1, for any p ∈ 0, 1/2 , there exists a constant K p independent on the initial data such that the global solution x t of SDE 1.1 has the property that Lemma 3.1.Proof.First, Theorem 2.1 indicates that the solution x t of 1.1 almost surely remain in R n for all t ≥ −τ with probability 1. Applying the It ô formula to e εt V x, k and taking expectation yields EV x, k e −εt V ξ 0 , r 0 e −εt E t 0 e εs V x s , k εV x s , k ds.3.2 Definition 3.2.The solutions x t of SDE 1.1 are called stochastically ultimately bounded, if for any ∈ 0, 1 , there is a positive constant χ χ , such that the solution of SDE 1.1 with any positive initial value has the property that Consider another stochastic differential equation with Markovian switching, where r t is a Markov chain taking values in S {1, 2, 3}.Here subsystem of 1.1 is writtern as three different equations: 2 t θ dμ θ dt 2x t dW 1 t x 3 t dW 2 t , Let the generator of the Markov chain r t be By solving the linear equation πΓ 0 subject to N k 1 π k 1 and π k > 0 for any k ∈ S, we obtain the unique stationary distribution x x 2 t θ dμ θ dt 4x t dW 1 t 2x 3 t dW 2 t , Therefore, by Theorems 4.2, System 1.1 is almost surely exponentially stable in Case 1.