Abstract. This paper studies both nonautonomous stochastic differential equations and stochastic differential delay equations with Markovian switching. A new result on almost sure stability of stochastic differential equations is given. Moreover, we provide new conditions for tightness and almost sure stability of stochastic differential equations
SOME RESULTS ON ALMOST SURE STABILITY OF NON-AUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS WITH MARKOVIAN SWITCHING NGUYEN THANH DIEU Abstract. This paper studies both non-autonomous stochastic differential equations and stochastic differential delay equations with Markovian switching. A new result on almost sure stability of stochastic differential equations is given. Moreover, we provide new conditions for tightness and almost sure stability of stochastic differential equations. 1. SDDEs with Markovian switching Due to increasing demands from real systems and phenomena in which both continuous dynamics and discrete events are involved, hybrid models have been increasingly considered for decades. If random factors in terms of white noise and Markov chains are taken into account, stochastic differential equations with regime-switching will be uses for modeling. These equations have numerous applications in many branches of science and industry such as manufacturing systems, financial engineering, genetic technologies, ecology, see [2, 4, 7, 14, 15, 21, 22, 24, 25, 31, 32] among others. If we suppose that the discrete component is a finite Markov process that does not depend on the state of the continuous components, we have a stochastic differential equation (SDEs) with Markovian switching. This kind of differential equation have received a lot of attention (see e.g. [1, 5, 6, 10, 13, 18, 19, 26, 27]). However, it is pointed out that many systems arising from science and technology do not depend only the present but also on the past. This fact results in needs of thorough research on stochastic differential delay equations (SDDEs) with Markovian switching. For the two past decades, SDDEs with Markovian switching have been study very frequently in literature focusing mainly on types of stability and boundedness (see e.g. [3, 9, 8, 11, 13, 20, 23, 28]). Continuing these studies, this paper consider both non-autonomous SDEs and SDDEs with Markovian switching. 2. Almost sure stability of SDEs with Markovian switching In this section, we deal with sufficient condition for almost sure stability of SDEs with Markovian switching as follows. dX(t) = f X(t), r(t), t dt + g X(t), r(t), t dB(t) (2.1) on a probability space (Ω, F, (Ft )t≥0 , P) satisfying the usual conditions, where f : Rn × S × R+ → Rn ; g : Rn × S × R+ → Rn × Rm ; B(t) = 1991 Mathematics Subject Classification. 34K50, 34K20, 65C30, 60J10. Key words and phrases. Stochastic differential delay equations; Stability in distribution; Itˆ o’s formula; Markov switching. 1 2 NGUYEN THANH DIEU (B1 (t), . . . , Bm (t))T is an m-dimensional Brownian motion, r(t) is a Markov chain taking values in a finite state space S = {1, 2, . . . , N } with generator Γ = (γij )N ×N with γij > 0 if i = j and r(·) is independent of B(·). Moreover, B(t) and r(t) are Ft -adapted. Let C 2,1 (Rn × S × R+ ; R+ ) denote the family of nonnegative functions V on Rn × S × R+ which are twice continuously differentiable in x and once continuously differentiable in t. For V ∈ C 2,1 (Rn × S × R+ ; R+ ), we define n LV (x, i, t) =Vt (x, i, t) + γij V (x, j, t) + Vx (x, i, t)f (x, i, t) j=1 1 + trace[g T (x, i, t)Vxx (x, i, t)g(x, i, t)], 2 where Vt (x, i, t) = ∂V (x, i, t) ∂V (x, i, t) ,..., ∂x1 ∂xn 2 ∂ V (x, i, t) Vxx (x, i, t) = . ∂xk ∂xj n×n ∂V (x, i, t) , Vx (x, i, t) = ∂t , (2.2) Denote by X x0 ,i (t) the solution to Equation (2.1) with initial data X(0) = x0 and r(0) = i. For any two stopping times 0 ≤ τ1 ≤ τ2 < ∞, it follows from the generalized Itˆ o formula that EV (X x0 ,i (τ2 ), r(τ2 ), τ2 ) = EV (X x0 ,i (τ1 ), r(τ1 ), τ1 ) τ2 +E LV (X x0 ,i (s), r(s), s)ds τ1 provided that the integrations involved exist and are finite. In [12], the authors provided a criterion for stochastically asymptotically stable in the large of Equation (2.1) which is cited as the following theorem. Theorem 2.1. [12, Theorem 5.37, pp.205] Assume that there are functions V ∈ C 2,1 (Rn × R+ ×; R+ ), µ1 , µ2 ∈ K∞ and µ3 ∈ K such that µ1 (|x|) ≤ V (x, t, i) and LV (x, t, i) ≤ −µ3 (|x|) (2.3) for all (x, i, t) ∈ Rn × R+ × S. Then the trivial of equation (2.1) is stochastically asymptotically stable in the large. Although this theorem can be applied to many stochastic differential equations with Markovian switching as demonstrated in [12], the condition (2.3) seem to be restrictive, in which LV (x, t, i) is required to be uniformly upper bounded by a function of |x|, may not be satisfied for many equations. Motivated by this comment, the main goal of this section is to weaken the aforesaid hypotheses. We always impose the following assumption Assumption 2.1. Suppose that Equation (2.1) has a unique global solution for any initial value (x0 , i, t0 ) ∈ Rn × S × R+ ; f (0, i, t) = g(0, i, t) = 0 ∀i ∈ S, t ∈ R+ . Furthermore, for each integer k ≥ 1 there is a positive constant hk such that |f (x, i, t) − f (y, i, t)| + |g(x, i, t) − g(y, i, t)| ≤ hk (|x − y|) (2.4) ALMOST SURE STABILITY 3 for all |x| ∨ |y| ≤ k, i ∈ S, t ∈ R+ . Assumption 2.2. There are two functions V ∈ C 2,1 (Rn × S × R+ ; R+ ) and w ∈ C(Rn ; R+ ) such that w vanish at only 0 and that for any (x, i, t) ∈ Rn × S × R+ and LV (x, i, t) ≤ γ(t) − α(t)w(x) (2.5) where α(.), γ(.) are non-negative, continuous, bounded function satisfying t+T α(s)ds > 0 αT = inf t∈R+ for some T > 0 and t ∞ γ(t)dt < ∞. 0 Lemma 2.2. Under assumption 2.1, for any σ > 0, ε > 0, T > 0, there exists a δ > 0 satisfying for any t > 0, inf t≤s≤t+T P{|X x0 ,i (s)| ≥ δ} > 1 − ε provided that X x0 ,i (t) ≥ σ. Proof. Since f (0, i, t) = g(0, i, t) = 0, we have from (2.4) that |f (x, i, t)| + |g(x, i, t)| ≤ h1 |x| ∀ |x| ≤ 1. Note that, we always can construct a function ϕ(r) ∈ C 2 ((0, ∞); R+ ) satisfying ϕ(r) = 0 if r > 1 and ϕ(r) = 1r if 0 < r ≤ 1. Lϕ(x) = − |x|−3 xT f (x, i, t) 1 − |x|−3 xT f (x, i, t)|g(x, i, t)|2 + 3|x|−5 |xT g(x, i, t)|2 + 2 ≤|x|−2 ||f (x, i, t)| + |x−3 ||g(x, i, t)|2 ≤ (h1 + h21 )ϕ(x). It is easy to see that for |x| ≥ 1, we can find a constant H > 0 (we choose H > (h1 +h21 )) such that Lϕ(x) ≤ Hϕ(x). So, for all x = 0, Lϕ(x) ≤ Hϕ(x). for each k ∈ N, define the stopping time ςkt = inf{s ≥ t : |ϕ(X x0 ,i (s))| > k}. Applying Itˆ o’s formula for e−H(s−t) ϕ(X x0 ,i (s)) we have t Ee−H((t+T )∧ςk −t) ϕ(X x0 ,i ((t + T ) ∧ ςkt )) = Eϕ(X x0 ,i (t+T )∧ςkt (t)) + E e−H(s−t) − Hϕ(X x0 ,i (s)) + Lϕ(X x0 ,i (s)) ds t 1 . σ It can be implied from this inequality that lim ςkt > T almost surely. More≤ Eϕ(X x0 ,i (t)) ≤ k→∞ Eϕ(X x0 ,i (t over, letting k → ∞ we have + T )) ≤ this lemma follows directly from this estimate. eHT . The conclusion of σ Theorem 2.3. Let Assumptions 2.1 and 2.2 be satisfied. Then, for any initial value (x0 , i), we have P{ lim X x0 ,i (t) = 0} = 1. t→∞ 4 NGUYEN THANH DIEU Proof. It suffices to show that for any σ > 0, P{lim sup |X x0 ,i (t)| ≤ σ} = 1. t→∞ Let > 0 and set x0 ,i Aσ, (t)| ≤ }, bσ, = inf{w(x) : σ ≤ |x| ≤ }. t = {σ ≤ |X For each n ∈ N, define the stopping time Tn = inf{s > 0 : |X x0 ,i (s)| > n} ∧ t. In view of Itˆ o formula, Tn EV (X x0 ,i (Tn ), r(Tn ), Tn ) ≤ Tn γ(s)ds+EV (x0 , i, 0)−E 0 0 Tn Tn γ(s)ds + V (x0 , i, 0) − E ≤ α(s)w(X x0 ,i (s))ds 1Aσ, α(s)w(X x0 ,i (s))ds. (2.6) s 0 0 Consequently, t α(s)P(Aσ, s )ds ≤ 0 ≤ lim Tk 1 k→∞ bσ, E 0 t 1 bσ, E 0 1Aσ, α(s)w(X x0 ,i (s))ds s t 1Aσ, α(s)w(X x0 ,i (s))ds ≤ s γ(s)ds + V (x0 , i, 0). (2.7) 0 Letting t → ∞ we have ∞ α(s)P(Aσ, s )ds < ∞. (2.8) 0 Suppose that P(Aσ, s ) does not converge to 0, then, there is a sequence tn ↑ ∞ such that P{Aσ, tn } > , ∀n ∈ N. We can suppose without loss of generality that tn+1 > tn + T , where T is the constant satisfying αT = t+T inf t α(s)ds > 0. Using Lemma 2.2 and the Markovian property of t∈R+ X x0 ,i (t), we can find δ > 0 such that P(Aδ, s ) > , ∀ tn ≤ s ≤ tn + T . Hence 2 tn +T α(s)P(Aδ, s )ds ≥ tn Consequently tn +T 2 tn α(s) ≥ αT ∀n ∈ N. 2 ∞ α(s)P(Aδ, s )ds = ∞ 0 which is a contradiction since the inequality (2.8) holds for any σ, that is, it must holds for the pair (δ, ). We therefore conclude that > 0, lim P{Aσ, t } = 0. t→∞ That means for any σ > 0, P{lim sup |X x0 ,i (t)| ≤ σ} = 1. The proof is t→∞ complete. Example 2.1. Let us consider Equation (2.1) with S = {1, 2}, r(t) with the generator −1 1 Γ= . 2 −2 ALMOST SURE STABILITY 5 Let f (x, 1, t) = −(0, 5 + sin+ t)x, g(x, 1, t) = x, and 5x x , g(x, 2, t) = √ cos t 8 2 + where sin t = 0 ∨ sin t. It is easy to see that f and g satisfy the Assumption 2.1. Define V (x, 1, t) = x2 , V (x, 2, t) = 4x2 . By computation, f (x, 2, t) = − LV (x, 1, t) = −2x2 (0, 5 + sin+ t) + x2 = −2x2 sin+ t and LV (x, 2, t) = −5x2 + 2x2 cos2 t = −3x2 − 2x2 sin2 t. It is easy to see that, there is no nonnegative function w(x) vanishing only at 0 such that LV (x, i, t) ≤ −w(x). That mean the condition of Theorem 2.1 is not satisfied. However, we have LV (x, i, t) ≤ −2x2 sin+ t. This means that Assumption 2.2 holds. In view of Theorem 2.3, Equation (2.1) is almost surely asymptotic stability. Approximating the stochastic differential equations with Markovian switching by Euler-Maruyama method as in [29], we can simulate a sample path of solution to (2.1) with respect to this example. It is illustrated by Figure 1 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Figure 1. Trajectories of X(t) in Example 2.1. 3. Tightness and almost sure stability of SDDEs with Markovian switching In this section we consider the sufficient condition for the almost sure stability of SDDEs with Markovian switching as follows dX(t) = f X(t), X(t − τ ), r(t), t dt + g X(t), X(t − τ ), r(t), t dB(t) (3.1) where f : Rn × Rn × S × R+ → Rn ; g : Rn × Rn × S × R+ → Rn × Rm . In order to prevent possible confusion, the notations in this paper are the same as in [30] which will be reintroduced below for convenience. Denote by C([−τ ; 0]; Rn ) the family of continuous functions ϕ(·) from [−τ ; 0] to Rn with norm ϕ = sup−τ ≤θ≤0 |ϕ(θ)|. For a continuous Rn -valued stochastic process X(t) on t ∈ [−τ ; ∞), denote Xt = {X(t + θ) : −τ ≤ θ ≤ 0} for t ≥ 0 so Xt is a stochastic process with state space C([−τ ; 0]; Rn ). Moreover, for 6 NGUYEN THANH DIEU a subset A ⊂ Ω, we denote by 1A the indicator function of A, i.e. 1A = 1 if ω ∈ A and 1A = 0 otherwise. For V ∈ C 2,1 (Rn × S × R+ ; R+ ), we define n LV (x, y, i, t) =Vt (x, i, t) + γij V (x, j, t) + Vx (x, i, t)f (x, y, i, t) j=1 1 + trace[g T (x, y, i, t)Vxx (x, i, t)g(x, y, i, t)], 2 where Vt (x, i, t), Vx (x, i, t), Vxx (x, i, t) are defined by (2.2). Denote by X ξ,i (t) the solution to Equation (3.1) with initial data X0 = ξ ∈ C([−τ, 0]; Rn ) and r(0) = i. We also denote by ri (t) the Markov chain starting in i. For any two stopping times 0 ≤ τ1 ≤ τ2 < ∞, it follows from the generalized Itˆo formula that EV (X ξ,i (τ2 ), ri (τ2 )) = EV (X ξ,i (τ1 ), ri (τ1 )) τ2 +E LV (X ξ,i (s), X ξ,i (s − τ ), ri (s), s)ds τ1 provided that the integrations involved exist and are finite. In [28], the authors provided a criterion for stochastically asymptotically stable in the large of Equation (3.1) which is cited as the following theorem. Theorem 3.1. [28, Theorem 2.1, pp.344] Let Assumption 3.1 hold. Assume that there are functions V ∈ C 2,1 (Rn ×R+ ×; R+ ), γ ∈ L1 (R+ ; R+ ), w1 , w2 ∈ C(Rn ; R+ ) such that LV (x, y, i, t) ≤ γ(t) − w1 (x) + w2 (y) (3.2) for any (x, y, i, t) ∈ Rn × Rn × S × R+ , w1 (0) − w2 (0) = 0, w1 (x) > w2 (x) ∀ x = 0 (3.3) and lim y→∞ inf |x|≥y,i∈S,t∈R+ V (x, i, t) = ∞. (3.4) Then the trivial of equation (3.1) is stochastically asymptotically stable in the large. Although this theorem can be applied to many stochastic differential delay equations with Markovian switching as demonstrated in [28], the condition (3.2) is restrictive because the last two terms w1 (x) and w2 (y) are required to be independent of variable t. It is in fact not easy to construct a right Lyapunov function to satisfy this condition when we consider non autonomous SDDEs. Motivated by this comment, the main goal of this section is to weaken the hypotheses in Theorem 3.1. Assumption 3.1 (The local Lipschitz condition). For each integer k ≥ 1 there is a positive constant hk such that |f (x, y, i, t) − f (x, y, i)|2 + |g(x, y, i, t) − g(x, y, i)|2 ≤ hk (|x − x|2 + |y − y|2 ) (3.5) for all |x| ∨ |y| ∨ |x| ∨ |y| ≤ k, i ∈ S. ALMOST SURE STABILITY 7 Assumption 3.2. There is a positive numbers c1 and there are functions V ∈ C 2,1 (Rn × S × R+ ; R+ ), w1 ∈ C(Rn × [0; +∞); R+ ), w2 ∈ C(Rn × [−τ ; +∞); R+ ) and w ∈ C(Rn ; R+ ) such that lim y→∞ inf |x|≥y,i∈S,t∈R+ V (x, i, t) = ∞, w1 (x, t) − w2 (x, t) ≥ α(t)w(x); w2 (x, t) ≤ c1 V (x, i, t) + α(t)w(x) (3.6) (3.7) and that LV (x, y, i, t) ≤ γ(t) − w1 (x, t) + w2 (y, t − τ ) (3.8) for any (x, y, i, t) ∈ Rn × Rn × S × R+ where w(x) = 0 ∀x = 0 and α(.), γ(.) are non-negative, continuous function satisfying t+δ α(s)ds > 0, ∀δ > 0 α(δ) = inf t∈R+ and ∞ 0 γ(t)dt t < ∞. To establish new sufficient conditions for almost sure stability of Equation (3.1), we will give following lemma. Lemma 3.2. Let Assumptions 3.1 and 3.2 hold. For any ξ ∈ C([−τ, 0]; Rn ) and i ∈ S, there exists a unique global solution X ξ,i (t) to the equation (3.1) on [0, ∞). Moreover, a) there is M > 0 such that EV (X ξ,i (t), ri (t), t) ≤ M ∀ t ≥ 0; b) for any T > 0, ε > 0, there exists a positive integer H = H(T, ε) such that P Xsξ,i ≤ H ∀ s ∈ [t; t + T ] ≥ 1 − ε, ∀ t ≥ 0. Proof. Under Assumptions 3.1 and 3.2, the existence of a unique global solution follows from Theorem [13, 7.13, pp. 280]. For each k ∈ N, define the stopping time σk = inf{t ≥ 0 : |X ξ,i (t)| > k}. 8 NGUYEN THANH DIEU Applying the generalized Itˆ o formula to V (X ξ,i (t), ri (t), t) and then using Assumption 3.2, yields EV (X ξ,i (t ∧ σk ), ri (t ∧ σk ), t ∧ σk ) t∧σk = EV (X ξ,i (0), ri (0), 0) + E LV (X ξ,i (s), X ξ,i (s − τ ), ri (s), s)ds 0 t∧σk t γ(s)ds + EV (X ξ,i (0), ri (0), 0) − E ≤ α(s)w(X ξ,i (s))ds 0 0 t∧σk +E − w2 (X ξ,i (s), s) + w2 (X ξ,i (s − τ ), s − τ ) ds 0 t∧σk t γ(s)ds + V (ξ(0), i, 0) − E ≤ α(s)w(X ξ,i (s))ds 0 0 τ w2 (X ξ,i (s − τ ), s − τ ) ds +E 0 ∞ ≤ 0 w2 (ξ(s), s)ds := M < ∞. γ(s)ds + V (ξ(0), i, 0) + −τ 0 (3.9) Letting k → ∞ we obtain the item a). Now, we move on to the item b). Note that (3.9) implies that t α(s)w(X ξ,i (s))ds ≤ M ∀ t ≥ 0. E (3.10) 0 Since X ξ,i (s) = ξ(s) if s ≤ 0, it follows from (3.10) and (3.7) t w2 (X ξ,i (s), s)ds ≤ c1 (1 + τ )M. E (3.11) t−τ Let X ξ,i (s) be the solution with initial value (ξ, i). Define σkt = inf{s ≥ t : Xs > k}. Employing the generalized Itˆo formula and Assumption 3.2 again we have EV (X ξ,i ((T + t) ∧ σkt ), ri ((T + t) ∧ σkt ), (T + t) ∧ σkt ) (T +t)∧σkt t+T γ(s)ds + EV (X ξ,i (t), ri (t), t) − E ≤ α(s)w(X ξ,i (s))ds t t (T +t)∧σk +E − w2 (X ξ,i (s), s) + w2 (X ξ,i (s), s) ds t ∞ γ(s)ds + EV (X ξ,i (t), ri (t), t) + E ≤ 0 t∧σk w2 (X ξ,i (s), s)ds. t−τ (3.12) Applying item a) and (3.11) to (3.12) EV (X ξ,i ((T + t) ∧ σkt ), ri ((T + t) ∧ σkt ), (T + t) ∧ σkt ) ≤ (1 + c1 (1 + τ ))M. (3.13) Let H = H(K, T, ε) ∈ N satisfy inf |y|≥H,j∈S,t∈R+ V (y, j, t) ≥ 1 (1 + c1 + c1 τ )M . ε (3.14) ALMOST SURE STABILITY 9 Employing (3.14) and (3.13) yields 1 t (1 + c1 + c1 τ )M P{σH < t + T} ε ≤ inf |y|≥H,j∈S,t∈R+ t V (y, j, t) · P{σH < t + T} t t t ≤ EV X ξ,i ((t + T ) ∧ σH ), ri ((t + T ) ∧ σH ), (t + T ) ∧ σH (3.15) ≤ (1 + c1 + c1 τ )M. (3.16) t < t + T ) ≤ ε. The proof is complete. This implies that P(σH Using this lemma, we are able to show that ε1 , ε2 > 0, there exists a δ0 = δ0 (ε1 , ε2 , K) > 0 such that |X ξ,i (s2 ) − X ξ,i (s1 )| ≥ ε1 sup P ≤ ε2 , ∀ (ξ, i) ∈ K × S, t ≥ 0. t≤s1 ≤s2 ≤t+τ s2 −s1 0, lim P Aσ, t t→∞ → 0, where Aσ, = ω : Xtξ,i ≤ , |X ξ,i (t)| ≥ σ . For each n ∈ N, define the t stopping time Tn = inf{s > 0 : Xsξ,i > n} ∧ t. To simplify the notation, denote cσ, 2 = min{w(x) : σ ≤ |x| ≤ } > 0. We implies from (3.10) that ∞ α(s)P{Aσ, s }ds ≤ 0 ∞ 1 cσ, 2 α(s) 1{Aσ, } w(X ξ,i (s)) ds < ∞. E s 0 (3.18) Suppose that lim sup P{Aσ, t } > 0. t→∞ Thus, there exists a constant tn+1 − τ and that P{Aσ, tn } = P > 0 and a sequence {tn }∞ n=1 such that tn < Xtξ,i ≤ , |X ξ,i (tn )| ≥ σ > , ∀ n ∈ N. n (3.19) Due to the tightness, there is 0 < δ0 < τ such that P sup tn ≤s 0, ∀ n ∈ N. 4 }ds = ∞ 0 which is a contradiction since the inequality (3.18) holds for any σ, > 0, σ that is, it must holds for the pair ( , H1 ). We therefore conclude that 2 lim P{Aσ, t } = 0. t→∞ (3.24) ε be such large that P{ Xt ≤ } ≥ 1 − ∀t. In view of 2 ε (3.24), we can find T = T (σ, ε) such that P{Aσ, } ≤ for all t > T , which t 2 ξ,i implies that P{|X (t)| > σ} ≤ , ∀t > T . Since we can choose σ arbitrarily, we conclude that P{ lim X ξ,i (t) = 0} = 1. For any ε, σ > 0, let t→∞ Example 3.1. Let us consider Equation (3.1) with S = {1, 2}, r(t) having the generator −1 1 Γ= . 4 −4 while f (x, y, 1, t) = −(0, 5x + x cos2 t − 0, 5y), g(x, y, 1, t) = xy cos t, and 1 1 f (x, y, 2, t) = [y cos(t − τ ) − 2x], g(x, y, 2, t) = √ x sin t. 6 3 It is easy to see that f and g satisfy Assumption 3.1. Define V (x, 1, t) = x2 , V (x, 2, t) = 3x2 . By computation, LV (x, y, 1, t) = −x2 − 2x2 cos2 t + xy + x2 cos2 t = −x2 − x2 cos2 t + xy, ALMOST SURE STABILITY 11 and LV (x, y, 2, t) = x(y cos(t − τ ) − 2x) + x2 sin2 t = −x2 − x2 cos2 t + xy cos(t − τ ). Unfortunately, we can not apply Theorem 3.1. Let x2 x2 + x2 cos2 t, w2 (x, t) = . 2 2 w1 (x, t) = Then w1 (x, t) − w2 (x, t) = x2 cos2 t. Since xy ≤ x2 + y 2 x2 + y 2 ; xy cos(t − τ ) ≤ , 2 2 we have, for i = 1, 2 LV (x, y, 1, t) ≤ −x2 y2 − x2 cos2 t + = −w1 (x, t) + w2 (y, t − τ ). 2 2 So that Assumption 3.2 holds. In view of Theorem 3.3, Equation (3.1) is almost surely asymptotic stability. By using semi-implicit Euler method for stochastic differential delay equations with Markovian switching (see: [16, 17]), we can plot a sample path of solution to (3.1) in this case with time delay τ = 0.3. It is illustrated by Figure 2 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 Figure 2. Trajectories of X(t) in Example 3.1. Acknowledgements. Author would like to thank anonymous reviewers for their valuable comments which helped to improve the manuscript. This research was supported in part by the Foundation for Science and Technology Development of Vietnam’s Ministry of Education and Training. 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Appl. 357(1), 154-170 (2009) E-mail address: dieunguyen2008@gmail.com Department of Mathematics,Vinh University, 182 Le Dan, Vinh, Nghe An, Vietnam [...]... On stability in distribution of stochastic differential delay equations with Markovian switching Systems Control Lett 65,4349 (2014) [4] Ji, Y., Chizeck, H.J.: Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control IEEE Trans Automat Control, 35, 777-788 (1990) [5] Kac, I Ya.: Method of Lyapunov functions in problems of stability and stabilization of systems of. .. criteria for stochastic differential delay equations with Markovian switching Sci China Ser A 46, no 1, 129-138 (2003) [9] Mao, X., Shaikhet, L.: Delay-dependent stability criteria for stochastic differential delay equations with Markovian switching Stability and Control: Theory and Applications, 3(2), 88-102 (2000) [10] Mao, X.: Stability of stochastic differential equations with Markovian switching Stochastic. .. support and hospitality of VIASM 12 NGUYEN THANH DIEU References [1] Dang, N H.: A note on sufficient conditions for asymptotic stability in distribution of stochastic differential equations with Markovian switching Nonlinear Analysis 95 625-631 (2014) [2] Du, N H., Dang, N H.: Dynamics of Kolmogorov systems of competitive type under the telegraph noise J Differential Equations, 250(1) 386-409 (2011)... Balachandran, K.: Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations Nonlinear Anal Hybrid Syst 2(4), 1256-1263 (2008) [18] Shaikhet, L.: Stability of stochastic hereditary systems with Markov switching Theory of Stochastic Processes, 2(18)(3-4):180-185 (1996) [19] Shaikhet, L.: Numerical simulation and stability of stochastic systems with Markovian... Control Optim 41, 18201842 (2003) [25] Yin, G., Liu, R.H., Zhang, Q.: Recursive algorithms for stock liquidation: A stochastic optimization approach SIAM J Optim 13, 240-263 (2002) [26] Yuan, C.: Stability in terms of two measures for stochastic differential equations with Markovian switching Stoch Anal Appl 23(6), 1259-1276 (2005) [27] Yuan, C., Lygeros, J.: Stabilization of a class of stochastic differential... Computations Atlanta, Dynamic Publishers, 10(2), 199-208 (2002) [20] Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations Springer, Dordrecht, Heidelb erg, New York, London, 342p (2013) [21] Sethi S.P., Zhang, Q.: Hierarchical Decision Making in Stochastic Manufacturing Systems, Birkhauser, Boston, (1994) [22] Sethi, S P., Zhang, H., Zhang, Q.: Average-cost Control... equations with Markovian switching Systems Control Lett 54(9), 819-833 (2005) [28] Yuan, C., Mao, X.: Robust stability and controllability of stochastic differential delay equations with Markovian switching Automatica J IFAC 40(3), 343-354 (2004) [29] Yuan, C., Mao, X.: Convergence of the Euler-Maruyama method for stochastic differential equations with Markovian switching Math Comput Simulation, 64(2),... Asymptotic stability for stochastic differential delay equations with Markovian switching Funct Differ Equ 9 no 1-2, 201-220 (2002) [12] Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching Imperial College Press (2006) [13] Mao, X., Yuan, C.: Stochastic differential equations with Markovian switching Imperial College Press, London, xviii+409 pp (2006) [14] Mariton, M.: Jump... Zhang, H., Zhang, Q.: Average-cost Control of Stochastic Manufacturing Systems Springer, New York, NY (2005) ALMOST SURE STABILITY 13 [23] Wang, L., Wu, F.: Existence, uniqueness and asymptotic properties of a class of nonlinear stochastic differential delay equations with Markovian switching Stoch Dyn 9(2), 253-275 (2009) [24] Yin, G., Dey, S.: Weak convergence of hybrid filtering problems involving nearly... that Assumption 3.2 holds In view of Theorem 3.3, Equation (3.1) is almost surely asymptotic stability By using semi-implicit Euler method for stochastic differential delay equations with Markovian switching (see: [16, 17]), we can plot a sample path of solution to (3.1) in this case with time delay τ = 0.3 It is illustrated by Figure 2 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 Figure 2 Trajectories of X(t) in ... Example 2.1 Tightness and almost sure stability of SDDEs with Markovian switching In this section we consider the sufficient condition for the almost sure stability of SDDEs with Markovian switching... condition when we consider non autonomous SDDEs Motivated by this comment, the main goal of this section is to weaken the hypotheses in Theorem 3.1 Assumption 3.1 (The local Lipschitz condition)... γ(.) are non-negative, continuous function satisfying t+δ α(s)ds > 0, ∀δ > α(δ) = inf t∈R+ and ∞ γ(t)dt t < ∞ To establish new sufficient conditions for almost sure stability of Equation (3.1),