Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 01, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.01 STABILITY AND RATE OF DECAY FOR SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS WITH MARKOV SWITCHING SHUAISHUAI LU, XUE YANG Abstract. In this article, we present the almost sure asymptotic stability and a general rate of decay for solutions to stochastic differential equations (SDEs) with Markov switching. By establishing a suitable Lyapunov function and using an exponential Martingale inequality and the Borel-Cantelli theorem, we give sufficient conditions for the asymptotic stability. Also, we obtain sufficient conditions for the construction of two kinds of Lyapunov functions. Finally give two examples to illustrate the validity of our results. 1. Introduction As an important property of stochastic differential equations (SDEs), the sta- bility research has been a popular direction. Many results have been achieved on various stability problems of SDEs, see [2, 15, 16, 24, 26]. For instance, Mao in- troduced the concept of polynomial decay rate stability into stochastic differential systems. In the subsequent research, the concept of stability with general decay rate was extended in [4, 5, 6]. Recently, the concept of the partial practical stability of SDEs with general decay rate was introduced by [2]. Stochastic differential equation with Markov switching is an important type of stochastic equations [7, 22]. It plays an important role in prediction model, physics, ecological engineering, financial stock market, network control, etc., and can be used to explain the physical process of sudden change of environment or transformation models under different conditions [19, 21, 25, 28, 34, 38]. Therefore, it is very worth studying the stability of this kind of equation. The p-moment and exponential stability of SDEs with Markov switching has been studied in [17, 30, 31, 37]. In these references, Lyapunov methods are used to study stability. It is interesting to note that in [17], the theory of M -matrices is used to establish some sufficient criteria for the exponential stability and these criteria are much easier to determine than the results obtained using the Lyapunov methods. In the stability theory of SDEs with Markov switching, the almost sure stability is also very significant, see [18, 35, 36, 27]. A sufficient condition that the equation is almost sure stability is given in [35]. The important thing is that the sufficient condition given in [35] is 2020 Mathematics Subject Classification. 60H10, 34D99, 34F05. Key words and phrases. Almost sure asymptotic stability; Markov switching; exponential Martingale inequality; Lyapunov function; generalized Itô formula. ©2024. This work is licensed under a CC BY 4.0 license. Submitted July 19, 2023. Published January 3, 2024. 1 2 S. LU, X. YANG EJDE-2024/01 independent of the moment stability of the system. In addition, in [11, 14, 32, 33], some problems of discrete Markov switching systems are studied, and in [9, 10] the stability is studied for nonlinear stochastic delay systems with asynchronous Markov switching. Lyapunov functions are often used to prove the stability of differential systems. A function is called a Lyapunov candidate function if it has the possibility to prove the stability of the differential system at an equilibrium. With the development of Lyapunov’s first and second methods, more and more work is based on Lyapunov methods to study the stability of differential systems, see [1, 8, 12, 23]. In the analysis of asymptotic stability of stochastic differential systems with Markov switching, many article study the exponential asymptotic stability of so- lutions, i.e., if the deterministic or stochastic system is not stable, our goal is to add a noise term to make the solution path of the stochastic system exponentially stable. However, this type of results fails to be applied, for instance, when the de- terministic model is non autonomous. In this case, it may occur that the stability cannot always be exponential, or even sub exponential or super exponential. This fact has inspired this article. The main purpose of this article is to extend expo- nential stability to general decay stability, such as polynomial decay, logarithmic decay, sub-exponential decay, super exponential decay and so on. Using Lyapunov method, we establish sufficient conditions for the almost sure asymptotic stability of equations with the general decay rate. At the same time, this paper discusses two kinds of Lyapunov functions to prove the rationality of our theorems. Therefore, this paper generalizes the results in [17]. The structure of this article is as follows: In Section 2 we give the basic concepts of SDEs with Markov switching and related definitions. In Section 3, the sufficient conditions ensuring the almost sure asymptotic stability on a general decay rate of SDEs with Markov switching are given, and the relevant proofs are given by using Markov inequality, Borel-Cantelli theorem and exponential Martingale inequality. Then, in Section 4 we present two examples to illustrate the theoretical findings. 2. SDEs with Markov switching We assume that (Ω,F , {Ft}t≥0,P) is a complete probability space, {Ft}t≥0 is a filtration in the probability space. Then F0 is right continuous and contains all the P-null test sets. We’re going to use ‖ · ‖ for the Euclidean norm in Rn. If A is a matrix or a vector, AT is its transpose. If A is a matrix, the norm is expressed as ‖A‖ = √ trace(AAT ). If A is a symmetric matrix, its maximum and minimum eigenvalues are denoted by λmax(A) and λmin(A) respectively. We use m ∨ n to represent max{m,n} and m ∧ n to represent min{m,n}. Now consider the stochastic differential equation with Markov switching dx(t) = f(x(t), t, r(t))dt+ g(x(t), t, r(t))dW (t), (2.1) where f : Rn × R+ × S → Rn, g : Rn × R+ × S → Rn×w, EJDE-2024/01 STABILITY AND RATE OF DECAY FOR SOLUTIONS 3 and {W (t)}t≥0 is the w-dimensional Brownian motion and r(t)(t ≥ 0) is the right continuous Markov chain in finite state space S = {1, 2, 3, . . . , N}, whose composi- tion Γ = (γij)N×N is generated as follows: P{r(t+ ∆) = j|r(t) = i} = { γij∆ + o(∆), i 6= j, 1 + γij∆ + o(∆), i = j, where ∆ > 0. Here γij ≥ 0 is transition rate from i to state j if i 6= j while γii = − N∑ j 6=i (γij). Therefore, (2.1) can be rewritten as the result of the following N equations: dx(t) = f(x(t), t, i)dt+ g(x(t), t, i)dW (t), t ≥ 0, 1 ≤ i ≤ N. (2.2) In this article we assume that Markov chains r(t) and Brownian motion W (t) are independent each other. To ensure the existence and uniqueness solutions of (2.1), the following assumptions are made: (H1) f and g satisfy the following linear growth conditions and local Lipschitz conditions: (1) There is h > 0 such that for all (x, t, i) ∈ Rn × R+ × S, ‖f(x, t, i)‖ ∨ ‖g(x, t, i)‖ ≤ h(1 + ‖x‖); (2) For each k = 1, 2, 3, . . . , there is an hk > 0 such that ‖f(x, t, i)− f(y, t, i)‖ ∨ ‖g(x, t, i)− g(y, t, i)‖ ≤ hk‖x− y‖, for all t ≥ 0, i ∈ S and x, y ∈ Rn with ‖x‖ ∨ ‖y‖ ≤ k. It is known [20, Theorem 3.16 and Lemma 4.1] and [17, (H)]) that if system (2.1) satisfies (H1), then for any initial value x0 ∈ Rn, there exists a unique continuous solution x(t, t0, x0), denoted by x(t), and for any p > 0, E[sup{‖x(s)‖p : t0 ≤ s ≤ t}] <∞, t ≥ t0. Definition 2.1. Let C2,1(Rn×R+×S;R+) represent the family of all non-negative functions on Rn × R+ × S that are twice differentiable in x and continuously dif- ferentiable in t. Suppose V (x, t, i) ∈ C2,1(Rn × R+ × S;R+) has the following: Vt = ∂V (x, t, i) ∂t , Vx = ∂V (x, t, i) ∂x , Vxx = ( ∂2V (x, t, i) ∂xi∂xj )n×n. Then we define an operator L acting on V (x, t, i) and LV : Rn × R+ × S → R, where LV (x, t, i) = Vt(x, t, i) + Vx(x, t, i)f(x, t, i) + 1 2 trace[gT (x, t, i)Vxx(x, t, i)g(x, t, i)] + N∑ j=1 γijV (x, t, j). Definition 2.2. Let α(t) > 0 be such that α(t) → ∞ as t → ∞. A non-trivial solution x(t) of system (2.1) is almost sure asymptotic stable with decay function α(t) and order at least γ > 0, if its generalized Lyapunov exponent is less than or equal to −γ with probability one, i.e., lim sup t→ +∞ ln(‖x(t)‖) lnα(t) ≤ −γ, a.s. (2.3) 4 S. LU, X. YANG EJDE-2024/01 The following lemmas will play important roles in our derivation. Lemma 2.3. Assume (H1) and that f(0, t, i) = 0 and g(0, t, i) = 0. Then for all x0 ∈ Rn, such that x0 6= 0, we have P(x(t, t0, x0) 6= 0,∀t ≥ t0) = 1. The proof of the above lemma can be found in [17, Lemma 2.1]. We require that the assumptions of Lemma 2.3 hold for the rest of this article, i.e., assume that for all t ∈ R+ and i ∈ S, f(0, t, i) ≡ 0, g(0, t, i) ≡ 0. Lemma 2.4 (Exponential Martingale inequality). Let F (t) ∈ L2(R+,Rn) and T , ε, η be any positive numbers. Then P[sup{Y (t) : 0 ≤ t ≤ T} > η] ≤ e−εη, where Y (t) = ∫ t 0 F (s)dW (s)− ε 2 ∫ t 0 ‖F (s)‖2ds. The proof of the above lemma can be found in [18, Theorem 1.7.4]. With the above preparations, the main result of this paper is to seek sufficient conditions about the almost sure asymptotic stability on a general decay rate of SDEs with Markov switching. 3. Almost sure asymptotic stability of SDEs with Markov switching In this section, based on the research in [17] on SDEs with Markov switching, we discuss the almost sure asymptotic stability on a general decay rate of stochastic differential systems with Markov switching. We are in a position to state the first result. Theorem 3.1. Assume (H1) and that there exist a function V ∈ C2,1(Rn ×R+ × S;R+) a continuous function G(t) > 0, constants p ∈ N+, β > 0, σ > 0, m ≥ 0, M ≥ 0, such that for all t ≥ 0, x ∈ Rn and i ∈ S, the following conditions hold: (1) LV (x, t, i) ≤ −βV (x, t, i); (2) α(t)m‖x‖p ≤ G(t)V (x, t, i) and limt→+∞ lnG(t) lnα(t) = a, a ∈ R; (3) lim supt→+∞ t lnα(t) = M ; (4) ‖Vx(x, t, i)g(x, t, i)‖2 ≤ σ‖V (x, t, i)‖2. Let x0 ∈ Rn(x0 6= 0). Then lim sup t→+∞ ln(‖x(t)‖) lnα(t) < −γ∗, a.s., where γ∗ = m+βM−a p . Furthermore, if γ∗ > 0, the solution x(t) of (2.1) is deemed to converge to zero with decay function α(t) and order at least γ∗ with probability one. Proof. Since system (2.1) satisfies condition (H1), without loss of generality, we assume that the initial moment is t0 = 0, so for any initial value x0 ∈ Rn(x0 6= 0), there exists a unique continuous solution x(t) of system (2.1). And Lemma 2.3 states that almost all sample paths of x(t) will never arrive at the origin, i.e., x(t) 6= 0 almost surely for any t ≥ 0. EJDE-2024/01 STABILITY AND RATE OF DECAY FOR SOLUTIONS 5 For each n ≥ 1, we define a stopping time τn = inf{t ≥ 0 : ‖x(t)‖ > n}. The condition (H1) is satisfied, which means that the solution process of system (2.1) is non-explosive, and we can get τn → ∞ as n → ∞. Next, we use the generalized Itô formula to study eβtV (x(t), t, r(t)), then we obtain E[eβ(t∧τn)V (x(t ∧ τn), t ∧ τn, r(t ∧ τn))] = E[V (x(0), 0, r(0))] + E ∫ t∧τn 0 eβs[βV (x(s), s, r(s)) + LV (x(s), s, r(s))]ds ≤ D, where D = E[V (x(0), 0, r(0))]. Letting n→∞, we have E[V (x(t), t, r(t))] ≤ De−βt. (3.1) For all continuous functions V (x, t, i) ∈ C2,1(Rn×R+×S;R+), by the generalized Itô formula we have V (x(t), t, r(t)) = V (x(0), 0, r(0)) + ∫ t 0 LV (x(s), s, r(s))ds + ∫ t 0 Vx(x(s), s, r(s))g(x(s), s, r(s))dW (s) ≤ V (x(0), 0, r(0)) + ∫ t 0 Vx(x(s), s, r(s))g(x(s), s, r(s))dW (s). Hence we obtain that for any k ≥ 1, δ ∈ (0, 1 32σ ), E ( sup δ(k−1)≤t≤δk V (x(t), t, r(t)) ) ≤ E(V (x(δ(k − 1)), δ(k − 1), r(δ(k − 1)))) + E( sup δ(k−1)≤t≤δk ∫ t δ(k−1) Vx(x(s), s, r(s))g(x(s), s, r(s))dW (s)). An application of the Burkholder-Davis-Gundy inequality leads to E ( sup δ(k−1)≤t≤δk ∫ t δ(k−1) Vx(x(s), s, r(s))g(x(s), s, r(s))dW (s) ) ≤ √ 32E (∫ δk δ(k−1) ‖Vx(x(s), s, r(s))g(x(s), s, r(s))‖2ds )1/2 ≤ √ 32σE (∫ δk δ(k−1) ‖V (x(s), s, r(s))‖2ds )1/2 ≤ √ 32σδE ( sup δ(k−1)≤t≤δk V (x(t), t, r(t)) ) , (3.2) which yields E ( sup δ(k−1)≤t≤δk V (x(t), t, r(t)) ) ≤ 1 1− √ 32σδ E(V (x(δ(k−1)), δ(k−1), r(δ(k−1)))). 6 S. LU, X. YANG EJDE-2024/01 Then by (3.1) we have E [ sup δ(k−1)≤t≤δk V (x(t), t, r(t)) ] ≤ D 1− √ 32σδ e−βδ(k−1). By Markov’s inequality, for any given time t ≥ 0 and any % ∈ (0, βδ), we obtain P { ω : sup δ(k−1)≤t≤δk {V (x(t), t, r(t))} > e(%−βδ)k } ≤ De−βδ(k−1) (1− √ 32σδ)e(%−βδ)k = Deβδ (1− √ 32σδ) e−%k. (3.3) Since ∑∞ k=1 e −%k <∞, we have ∞∑ k=1 P{ω : sup δ(k−1)≤t≤δk {V (x(t), t, r(t))} > e(%−βδ)k} < +∞. Then we apply the Borel-Cantelli lemma to obtain that, for almost all ω ∈ Ω, ∃k̄ > 0, where k̄ only related to ω ∈ Ω, with P { ω : sup δ(k−1)≤t≤δk {V (x(t), t, r(t))} > e(%−βδ)k } = 0, ∀δ(k − 1) ≤ t ≤ δk, k ≥ k̄, i.e. V (x(t), t, r(t)) ≤ e(%−βδ)k, a.s. Further V (x(t), t, r(t)) ≤ e(%−βδ)k ≤ e (%−βδ) δ t, ∀δ(k − 1) ≤ t ≤ δk, k ≥ k̄. (3.4) By (3.4), when ∀δ(k − 1) ≤ t ≤ δk, k ≥ k̄, we obtain α(t)m(‖x(t)‖p) ≤ G(t)V (x(t), t, r(t)) ≤ G(t)e (%−βδ) δ t, which implies m lnα(t) + ln(‖x(t)‖p) ≤ lnG(t) + (%− βδ) δ t. Further ln(‖x(t)‖p) lnα(t) ≤ −m+ (%− βδ) δ t lnα(t) + lnG(t) lnα(t) . Now letting k → +∞ yields lim sup t→+∞ ln(‖x(t)‖p) lnα(t) ≤ lim t→+∞ (−m+ (%− βδ) δ t lnα(t) + lnG(t) lnα(t) ). By conditions (2), (3) and the property of α(t), we have lim sup t→+∞ ln(‖x(t)‖) lnα(t) ≤ − m+ (β − % δ )M − a p , a.s. Let %→ 0 and γ∗ := m+βM−a p , we have lim sup t→+∞ ln(‖x(t)‖) lnα(t) ≤ −γ∗, a.s. � EJDE-2024/01 STABILITY AND RATE OF DECAY FOR SOLUTIONS 7 Remark 3.2. (1) Assuming that all the conditions of [17, Theorem 3.2] hold, with m = 0, G(t) = 1 c1 , a = 0, M = 1, and β = λ c2 , we have that Theorem 3.1 holds and the Lyapunov exponent is less than or equal to γ∗ = m+βM−a p = λ pc2 , which is the same as the conclusion of [17]. In other words, we can obtain the same result and decay rate. Hence Theorem 3.1 is a generalization of the results in [17]. (2) The two important questions of the above Theorem are: one is the selection of the decay function, and the other is whether the Lyapunov function used in the theorem exists. The first question is generally easy to solve. For example, we can choose α(t) = O(et) or α(t) = O(ln(t + 1)), so we can get the almost sure exponential stability or the almost sure logarithmic stability. The key question is the selection of Lyapunov functions. In the following, we will construct suitable Lyapunov functions. It is rare to find the Lyapunov functions satisfying Theorem 3.1 when judging the stability of the system. However, in this article it is not complex to determine the stability of (2.1) with V (x, t, i) = (xTQix) p 2 (1 ≤ i ≤ N) and p > 0. For instance, let p = 1, x ∈ Rn and Qi are n-dimensional symmetric positive-definite matrices. From definition 2.1, we have Vt(x, t, i) = 0, Vx(x, t, i) = (xTQix)(−1/2)xTQi, Vxx(x, t, i) = (xTQix)(−1/2)Qi − 1 2 (xTQix)(−3/2)Qixx TQi, 1 2 trace[gT (x, t, i)Vxx(x, t, i)g(x, t, i)] = 1 2 (xTQix)(−3/2) trace[gT (x, t, i)Qig(x, t, i)] − 1 4 (xTQix)(−3/2)‖xTQig(x, t, i)‖2; therefore, LV (x, t, i) = Vt(x, t, i) + Vx(x, t, i)f(x, t, i) + 1 2 trace[gT (x, t, i)Vxx(x, t, i)g(x, t, i)] + N∑ j=1 γijV (x, t, j) = (xTQix)(−1/2)xTQif(x, t, i) + 1 2 (xTQix)(−1/2) trace[gT (x, t, i)Qig(x, t, i)] − 1 2 (xTQix)(−3/2)‖xTQig(x, t, i)‖2 + N∑ j=1 γij(x TQjx)1/2. (3.5) Corollary 3.3. Consider (2.1). Assume (H1) and let p = 1 in Theorem 3.1. In addition, for all t ≥ 0, x ∈ Rn and i ∈ S, assume the following conditions hold: (1) Z(x, t, i) ≤ −λ‖x‖2, where Z(x, t, i) = xTQif(x, t, i) + 1 2 trace[gT (x, t, i)Qig(x, t, i)] − 1 2 (xTQix)(−1)‖xTQig(x, t, i)‖2 + (xTQix)( 1 2 ) N∑ j=1 γij(x TQjx)1/2; 8 S. LU, X. YANG EJDE-2024/01 (2) lim supt→+∞ t lnα(t) = M ; (3) ‖xTQig(x, t, i)‖2 ≤ σ‖xTQix‖2. Then the conclusion in Theorem 3.1 holds. Proof. Let V (x, t, i) = (xTQix)1/2. We know that the specified Lyapunov functions V are not differentiable when the spatial variable (denoted x) is zero. However, by Lemma 2.3, we just need the Lyapunov function to be differentiable in Rn\{0} with respect to the variable x. Let Rn0 := Rn \ {0}, so it is obvious that (xTQix)1/2 ∈ C2,1(Rn0 × R+ × S;R+). Because matrix Qi is symmetric positive-definite matrix, we obtain λmax(Qi) ≥ λmin(Qi) > 0. Therefore, for all (x, t, i) ∈ Rn × R+ × S, [min{λmin(Qi) : 1 ≤ i ≤ N}]1/2‖x‖ ≤ (xTQix)1/2 ≤ [max{λmax(Qi) : 1 ≤ i ≤ N}]1/2‖x‖, and we have the following conclusions: (1) V (x, t, i) ≤ [max{λmax(Qi) : 1 ≤ i ≤ N}]1/2‖x‖. (2) (xTQix)1/2 ≥ [min{λmin(Qi) : 1 ≤ i ≤ N}]1/2‖x‖. For Theorem 3.1, we can make G(t) = [min{λmin(Qi) : 1 ≤ i ≤ N}](− 1 2 )α(t)m, so G(t)(xTQix)1/2 ≥ α(t)m‖x‖. Further lim t→+∞ lnG(t) lnα(t) = m. Corresponding to Theorem 3.1, we obtain a = m. (3) From condition (1) in this theorem, we obtain LV (x, t, i) = (xTQix)(−1/2)xTQif(x, t, i) + 1 2 (xTQix)(−1/2) trace[gT (x, t, i)Qig(x, t, i)] − 1 2 (xTQix)(−3/2)‖xTQig(x, t, i)‖2 + N∑ j=1 γij(x TQjx)1/2 ≤ − λ max{λmax(Qi) : 1 ≤ i ≤ N} (xTQix)1/2, (3.6) which implies β = λ max{λmax(Qi):1≤i≤N} . Summing up, from the analysis of (1), (2) and (3) we see that V (x(t), t, i) = (xT (t)Qix(t))1/2 satisfies all the conditions of Theorem 3.1, so lim sup t→+∞ ln(‖x(t)‖) lnα(t) ≤ −γ∗, a.s. By Theorem 3.1, we can take δ ∈ (0, 1 32σ ), thus γ∗ = βM . If M > 0, the solution x(t) of (2.1) converges to zero with decay function α(t) and order at least βM with probability one. � EJDE-2024/01 STABILITY AND RATE OF DECAY FOR SOLUTIONS 9 Remark 3.4. (1) If (H1) and Lemma 2.3 hold, we know that almost all sample paths of any solution of system (2.1) beginning from a nonzero state will never arrive at the origin, so we only need Lyapunov functions V ∈ C2,1(Rn0 × R+ × S;R+) in Theorem 3.1 and Corollary 3.3. The following Theorem 3.5 and Corollary 3.7 are equally applicable. (2) The above proof process shows that when analyzing the almost sure as- ymptotic stability of a stochastic differential system with Markov switching, the qualified matrices Qi(1 ≤ i ≤ N) can be selected to construct the Lyapunov func- tion according to Theorem 3.1. If it is a one-dimensional system, we need to choose Qi as N positive numbers, and the proof process is the same as Corollary 3.3. In the following section, we will further extend the application of Theorem 3.1. In Theorem 3.1, the condition (1) we want to ensure is LV (x, t, i) ≤ −βV (x, t, i), and then our main task is to generalize this condition to LV (x, t, i) ≤ h1(t)V (x, t, i) where h1(t) ∈ R for any t ∈ R+. To ensure the well-posedness of the above system, the following assumptions are made: (H2) f and g satisfy (1) There is a nonnegative function φ1(t) such that for all (x, t, i) ∈ Rn × R+ × S, ‖f(x, t, i)‖2 ∨ ‖g(x, t, i)‖2 ≤ φ1(t)(1 + ‖x‖2); (2) There is a nonnegative function φ2(t) such that for all t ≥ 0, i ∈ S and x, y ∈ Rn, ‖f(x, t, i)− f(y, t, i)‖ ∨ ‖g(x, t, i)− g(y, t, i)‖ ≤ φ2(t)‖x− y‖. Then, according to theabove conditions, there exists a unique global solution with initial values x0 ∈ Rn defined in an interval [t0, T ). Since we study the asymptotic behavior of solutions, we assume T = +∞. Theorem 3.5. Assume (H2), there exist continuous functions V ∈ C2,1(Rn0 × R+ × S;R+), G(t) ≥ 0, h1(t) ∈ R and h2(t) ≥ 0 for all t ∈ R+, constants p ∈ N+, m ≥ 0,M ≥ 0, ϑ1 ∈ R, ϑ2 ≥ 0, such that for all t ≥ t0, x ∈ Rn0 and i ∈ S, the following conditions hold: (1) α(t)m‖x‖p ≤ G(t)V (x, t, i) and limt→+∞ lnG(t) lnα(t) = a, a ∈ R; (2) LV (x, t, i) ≤ h1(t)V (x, t, i) and lim supt→+∞ ∫ t 0 h1(s)ds lnα(t) ≤ ϑ1; (3) ‖Vx(x, t, i)g(x, t, i)‖2 ≥ h2(t)V 2(x, t, i) and lim inft→+∞ ∫ t 0 h2(s)ds lnα(t) ≥ ϑ2; (4) lim supt→+∞ t lnα(t) = M . Let x0 ∈ Rn(x0 6= 0). Then lim sup t→+∞ ln(‖x(t)‖) lnα(t) ≤ −γ∗, a.s., where γ∗ = { 1 p (m− ϑ1 −M − a), M > 1 2ϑ2, 1 p (m− ϑ1 − a+ 1 2ϑ2 − 3 2 √ Mϑ2), M ≤ 1 2ϑ2. . Further, if γ∗ > 0, the solution x(t) of equation (2.1) is almost sure asymptotic stable with decay function α(t) and order at least γ∗ with probability one. 10 S. LU, X. YANG EJDE-2024/01 Proof. Since system (2.1) satisfies condition (H2), for any initial value x(t0) = x0 ∈ Rn(x0 6= 0) there exists a unique continuous solution x(t, t0, x0). Applying the Itô formula to ln V (·) along the trajectory x(·) for (2.1), for any t ≥ t0, ln (V (x(t), t, r(t))) = ln (V (x(t0), t0, r(t0))) +Mt + ∫ t t0 LV (x(s), s, r(s)) V (x(s), s, r(s)) ds − 1 2 ∫ t t0 ‖Vx(x(s), s, r(s))g(x(s), s, r(s))‖2 V 2(x(s), s, r(s)) ds, (3.7) where Mt = ∫ t t0 Vx(x(s), s, r(s))g(x(s), s, r(s)) V (x(s), s, r(s)) dW (s) is a continuous martingale. The next main task is to study Mt. We choose the standard initial value x(t0) = x0 and guarantee that E |x0|2 <∞. According to exponential Martingale inequality, letting T = k, ε = ε, η = k−1 ε , and ε ∈ (0, 1), k ∈ N+(k > 1). Then by Lemma 2.4 we have P [ sup 0≤t≤k {Mt − Y (t, ε)} > k − 1 ε ] ≤ e−(k−1), where Y (t, ε) = ε 2 ∫ t t0 ‖Vx(x(s), s, r(s))g(x(s), s, r(s))‖2 V 2(x(s), s, r(s)) ds. Since ∑∞ k=2 e −(k−1) < ∞, by Borel-Cantelli Lemma, we obtain that for almost all ω ∈ Ω, P [ lim inf t→+∞ ( sup 0≤t≤k {Mt − Y (t, ε)}) ≤ k − 1 ε ] = 1. In other words, there exists k̃ > 0 where k̃ only related to ω ∈ Ω such that for all k − 1 ≤ t ≤ k(k ≥ k̃), we have Mt ≤ ε 2 ∫ t t0 ‖Vx(x(s), s, r(s))g(x(s), s, r(s))‖2 V 2(x(s), s, r(s)) ds+ k − 1 ε , a.s. Using (2)–(4), it follows that by (3.7), ln (V (x(t), t, r(t))) ≤ ln (V (x(t0), t0, r(t0))) + t ε + ∫ t t0 h1(s)ds− (1− ε) 2 ∫ t t0 h2(s)ds. Further, for all k − 1 ≤ t ≤ k(k ≥ k̃), when k → +∞ we obtain lim sup t→+∞ ln V (x(t), t, r(t)) ln α(t) ≤ ϑ1 − (1− ε) 2 ϑ2 + 1 ε M. (3.8) By condition (1), we have m lnα(t) + p ln(‖x(t)‖) ≤ lnG(t) + ln (V (x(t), t, r(t))); further ln ‖x(t)‖ lnα(t) ≤ 1 p [ −m+ ln (V (x(t), t)) lnα(t) + lnG(t) lnα(t) ] . By (3.8) and the property of α(t), letting k →∞ yields lim sup t→+∞ ln(‖x(t)‖) lnα(t) ≤ −1 p (m− ϑ1 + (1− ε) 2 ϑ2 − 1 ε M − a). (3.9) EJDE-2024/01 STABILITY AND RATE OF DECAY FOR SOLUTIONS 11 Let γ(ε) = 1 p (m− ϑ1 + (1−ε) 2 ϑ2 − 1 εM − a), which shows that the order of decay depends on the parameter ε. The next main task is to find the optimal valued γ∗ = sup ε∈(0,1) γ(ε). Obviously we can obtain that dγ(ε) dε = 1 p ( 1 ε2 M − 1 2 ϑ2), which implies that γ∗ = 1 p (m− ϑ1 −M − a), M > 1 2 ϑ2, γ∗ = 1 p (m− ϑ1 − a+ 1 2 ϑ2 − 3 2 √ Mϑ2), M ≤ 1 2 ϑ2. Thus if γ∗ > 0, we obtain lim sup t→+∞ ln(‖x(t)‖) lnα(t) ≤ −γ∗, a.s., where γ∗ = { 1 p (m− ϑ1 −M − a), M > 1 2ϑ2, 1 p (m− ϑ1 − a+ 1 2ϑ2 − 3 2 √ Mϑ2), M ≤ 1 2ϑ2. � Remark 3.6. It should be noted that the Lyapunov function constructed accord- ing to Theorem 3.1 is independent of time t. Let Q : R+ → Rn×n be a C1 positive-definite function with Q(t) = Q(t)T [29]. We take the Lyapunov function as V (x, t, i) = xTQi(t)x, where x ∈ Rn. Corollary 3.7. Consider (2.1) and let f be C1(Rn0 × R+ × S;R+) in x. There exist continuous functions V ∈ C2,1(Rn0 ×R+ × S;R+), G(t) ≥ 0, ϕ1(t), ϕ2(t) ∈ R and h2(t) ≥ 0, constants p = 2, m ≥ 0, ã ∈ R, M ≥ 0, ϑ1 ∈ R, ϑ2 ≥ 0. For all t ≥ t0, x ∈ Rn0 and i ∈ S, the following conditions hold: (1) Q̇i(t) + ∂fT (x,t,i) ∂x Qi(t) +Qi(t) ∂f(x,t,i) ∂x ≤ ϕ1(t)Qi(t); (2) trace[gT (x, t, i)Qi(t)g(x, t, i)] + ∑N j=1 γijV (x, t, j) ≤ ϕ2(t)xTQi(t)x, h1(t) := ϕ1(t) + ϕ2(t) and lim supt→+∞ ∫ t 0 h1(s)ds lnα(t) ≤ ϑ1; (3) ‖xTQi(t)g(x, t, i)‖2 ≥ h2(t)‖xTQi(t)x‖2 and lim inft→+∞ ∫ t 0 h2(s)ds lnα(t) ≥ ϑ2; (4) lim supt→+∞ t lnα(t) = M and lim inft→+∞ lnQi(t) lnα(t) ≥ ã. where ‖Qi(t)‖ is the determinant of the matrix Qi(t) at time t. Then the conclusion in Theorem 3.5 holds. Proof. Let V (x, t, i) = xTQi(t)x. It is obvious that V (x, t, i) ∈ C2,1(Rn0 × R+ × S;R+). And by Definition 2.1, we obtain Vt(x, t, i) = xT Q̇i(t)x, Vx(x, t, i)f(x, t, i) = xTQi(t)f(x, t, i) + fT (x, t, i)Qi(t)x, Vxx(x, t, i) = Qi(t), 12 S. LU, X. YANG EJDE-2024/01 and LV (x, t, i) = Vt(x, t, i) + Vx(x, t, i)f(x, t, i) + 1 2 trace[gT (x, t, i)Vxx(x, t, i)g(x, t, i)] + N∑ j=1 γijV (x, t, j) = xT Q̇i(t)x+ xTQi(t)f(x, t, i) + fT (x, t, i)Qi(t)x + trace[gT (x, t, i)Qi(t)g(x, t, i)] + N∑ j=1 γijx TQj(t)x. (3.10) Analyzing the above equality, we have LV (x(t), t, i) = xT (t)Q̇i(t)x(t) + xT (t)Qi(t)[f(x(t), t, i)− f(0, t, i)] + [f(x(t), t, i)− f(0, t, i)]TQi(t)x(t) + trace[gT (x(t), t, i)Qi(t)g(x(t), t, i)] + N∑ j=1 γijx T (t)Qj(t)x(t). Since f be C1(Rn0 × R+ × S;R+) in x, by Lagrange Mean Value Theorem, there exists εt ∈ (0, x(t)) such that f(x(t), t, i)− f(0, t, i) = fx(εtx(t), t, i)x(t), which implies that LV (x(t), t, i) = xT (t)Q̇i(t)x(t) + xT (t)Qi(t)fx(εtx(t), t, i)x(t) + xT (t)Qi(t)f T x (εtx(t), t, i)x(t) + trace[gT (x(t), t, i)Qi(t)g(x(t), t, i)] + N∑ j=1 γijx T (t)Qj(t)x(t). By conditions (1) and (2), we have LV (x(t), t, i) ≤ (ϕ1(t) + ϕ2(t))x(t)TQi(t)x(t) = h1(t)V (x(t), t, i), ∀1 ≤ i ≤ N. We can make G(t) = α(t)m Qi(t) . To sum up, Corollary 3.7 satisfies all conditions of Theorem 3.5, which allow us to conclude that lim sup t→+∞ ln(‖x(t)‖) lnα(t) ≤ −γ∗, where γ∗ = { 1 2 (ã− ϑ1 −M), M > 1 2ϑ2, 1 2 (ã− ϑ1 + 1 2ϑ2 − 3 2 √ Mϑ2), M ≤ 1 2ϑ2. � EJDE-2024/01 STABILITY AND RATE OF DECAY FOR SOLUTIONS 13 4. Examples To illustrate the validity of our main results, we provide two examples. Example 4.1. Let W (t) be a standard Brownian motion on (Ω,F , {Ft}t≥0,P). We consider the following one-dimensional stochastic differential equation with Markovian switching, dx(t) = f(x(t), t, r(t))dt+ g(x(t), t, r(t))dW (t), (4.1) with initial value x0 ∈ Rn0 and r(t) be a right-continuous Markov chain takingvalues in S = {1, 2} with its generator Γ = ( −2 2 3 −3 ) . Let f(x, t, i) = { −3x, i = 1, 1 2x, i = 2, g(x, t, i) = { 3x, i = 1, 5x, i = 2, V (x, t, i) = { x2, i = 1, 4x2, i = 2. By Corollary 3.3, we can define a Lyapunov function (V (x, t, i))1/2, which implies Q1 = 1 nd Q2 = 4. Let α(t) = 2tet, m = 0 and p = 1, which shows that M = 1, G(t) = 1 and a = 0. By computing the Itô operator, we have Z(x, t, 1) = xf(x, t, 1) + 1 2 ‖g(x, t, 1)‖2 − 1 2 ‖x‖(−2)‖xg(x, t, 1)‖2 + ‖x‖(γ11 + 2γ11)‖x‖ = −‖x‖2, Z(x, t, 2) = 4xf(x, t, 2) + 2‖x‖(γ21 + 2γ21)‖x‖ ≤ −2‖x‖2; then, for each i ∈ S, we have Z(x, t, i) ≤ −‖x‖2, which implies λ = 1. In addition, we have ‖xTQ1g(x, t, 1)‖2 = 9x4, ‖xTQ2g(x, t, 2)‖2 = 400x4. Let σ = 100, so we obtain ‖xTQig(x, t, i)‖2 ≤ 100‖xTQix‖2. Then it is not hard to show that β = λ max{λmax(Qi):1≤i≤N} = 1 4 by Corollary 3.3, i.e., all conditions of Corollary 3.3 and (H1) are satisfied. Therefore, the solution of system (4.1) converges to zero with decay function 2tet and order at least 1/4 with probability one. Example 4.2. To show the validity of Theorem 3.5, let us consider the equation dx(t) = f(x(t), t, r(t))dt+ g(x(t), t, r(t))dW (t), (4.2) where r(t) be a right-continuous Markov chain taking values in S = {1, 2} with its generator Γ = ( −1 1 2 −2 ) . Let f(x, t, i) = { k1tx, i = 1, 1 2e −tx, i = 2, g(x, t, i) = { k2t 1/2x, i = 1, k3t 1/2x, i = 2, 14 S. LU, X. YANG EJDE-2024/01 with initial value x0 ∈ Rn0 and define a Lyapunov function V (x, t, i) = { tx2, i = 1, x2, i = 2. From the above definition, we can obtain that system (4.2) satisfies the condition (H2) and the assumptions of Lemma 2.3. We will be interested in analyzing the asymptotic behavior of solutions. Assuming initial time t0 ≥ 2. It is clear that Q1(t) = t when i = 1, hence we have Q̇1(t) + ∂fT (x, y, 1) ∂x Q1(t) +Q1(t) ∂f(x, y, 1) ∂x = ( 1 t + 2k1t)Q1, trace[gT (x, t, 1)Q1(t)g(x, t, 1)] + N∑ j=1 γ1jV (x, t, j) = ( 1 t + k22t− 1)tx2. By Corollary 3.7, for t sufficiently large, we obtain LV (x, t, 1) ≤ ( 2 t − 1 + 2k1t+ k22t)tx 2 ≤ (2k1t+ k22t)V (x, t, 1). When i = 2, we know Q2 = 1 and this shows that LV (x, t, 2) = 2xf(x, t, 2) + g2(x, t, 2) + 2tx2 − 2x2 = (e−t − 2 + 2t+ k23t)x 2 ≤ (2t+ k23t)V (x, t, 2). Therefore, h1(t) = (2k1t+ k22t) ∨ (2t+ k23t). We obtain LV (x, t, i) ≤ h1(t)V (x, t, i) for each i ∈ {1, 2}. On the other hand, we have ‖Vx(x, t, 1)g(x, t, 1)‖2 = 4k22t 3x4, ‖Vx(x, t, 2)g(x, t, 2)‖2 = 4k23tx 4. Letting h2(t) = 4k22t ∧ 4k23t, we obtain ‖Vx(x, t, i)g(x, t, i)‖2 ≥ h2(t)V 2(x, t, i) for all i ∈ {1, 2}. Taking α(t) = et 2 , m = 1, p = 2, k1 = 1 4 , k2 = 2, k3 = 2 and G(t) = { 1 t e t2 , i = 1, et 2 , i = 2, and M = 0, h1(t) = 6t, and h2(t) = 16t. We can check that assumptions in Theorem 3.5 hold with ã = 0, ϑ1 = 3, ϑ2 = 8, and M ≤ 1 2ϑ2, which implies γ∗ = 1 2 (ã−ϑ1 + 1 2ϑ2− 3 2 √ Mϑ2) = 1 2 . 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Zhang; Recursive algorithms for stock liquidation: a stochastic opti- mization approach, SIAM J. Optim., 13 (2002) 240-263. [35] C. Yuan, J. Lygeros; On the exponential stability of switching diffusion processes, IEEE Trans. Automat. Control, 50 (2005), 1422-1426. [36] C. Yuan, X. Mao; Robust stability and controllability of stochastic differential delay equations with Markovian switching, Automatica J. IFAC, 40 (2004), 343-354. [37] Q. Zhu; Razumikhin-type theorem for stochastic functional differential equations with LšŠvy noise and Markov switching, Internat. J. Control, 90 (2017), 1703-1712. [38] C. Zhu, G. Yin; On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl, 357 (2009) 154-170. Shuaishuai Lu College of Mathematics, Jilin University, Changchun 130012, China Email address: stluss@outlook.com Xue Yang College of Mathematics, Jilin University, Changchun, 130012, China Email address: xueyang@jlu.edu.cn 1. Introduction 2. SDEs with Markov switching 3. Almost sure asymptotic stability of SDEs with Markov switching 4. Examples Acknowledgments References