Asymptotic boundedness and stability of 1 solutions to hybrid stochastic differential 2 equations with jumps and the 3 Euler-Maruyama approximation

6 In this paper, we are concerned with the asymptotic properties and numerical anal7 ysis of the solution to hybrid stochastic differential equations with jumps. Applying the 8 theory of M-matrices, we will study the pth moment asymptotic boundedness and stabil9 ity of the solution. Under the non-linear growth condition, we also show the convergence 10 in probability of the Euler-Maruyama approximate solution to the true solution. Finally, 11 some examples are provided to illustrate our new results. 12

Moreover, by Doob-Meyer's decomposition theorem, there exists a unique {F t }-adapted mar-tingaleÑ (t, Z) and a unique {F t }-adapted natural increasing processN (t, Z) such that HereÑ (t, Z) is called the compensated Poisson random measure andN (t, Z) = π(Z)t is called 2 the compensator. For more details on the Poisson point process and Lévy jumps, see [1,30].  where ∆ > 0. Here γ ij ≥ 0 is the transition rate from i to j, i = j, While γ ii = − j =i γ ij . 6 As a standing hypothesis, we assume that the Markov chain r(t) is irreducible. Under this 7 condition, r(t) has a unique stationary distributionπ = (π 1 ,π 2 , · · · ,π N ) ∈ R 1×N satisfying 8 the following linear equationπΓ = 0 subject to N i=1π i = 1 andπ i > 0, ∀ i ∈ S. We assume 9 that the Markov chain r(.) is independent of the Brownian motion w(.) and Poisson random 10 measures N (., Z).
In this paper, the following hypotheses are imposed on the coefficients f, g, and h. 13 Assumption 2.1 Let p ≥ 2. For each integer d > 0, there exists a positive constant k d such where 2i for all i ∈ S, where C p i = Z (h i (v)) p/2 π(dv) < ∞. 13 Then for any given initial data x 0 and r 0 , there exists a unique global solution x(t) to equation 14 (2.1) such that x(t) ∈ L p for all t ≥ 0.
15 Corollary 2.4 Let Assumptions 2.1 and 2.2 hold. Assume also that one of the following 1 conditions holds: 2 (a) γ 1 > γ 2 ; 3 (b) γ 1 = γ 2 and 2α 2i > C i β 2i for all i ∈ S, where C i = Zh i (v)π(dv) < ∞. 4 Then for any given initial data x 0 and r 0 , there exists a unique global solution x(t) to equation 5 (2.1) such that in x(t) ∈ L 2 for all t ≥ 0.  To emphasize the main purpose of this paper, we shall leave the proof of the existence In this section, we shall use the theory of M-matrices to discuss the asymptotic behavior of 14 the solution, i.e., the asymptotic boundedness and stability in pth moment of the solution to 15 equation (2.1). 16 For the convenience of the reader, let us cite some useful results on M-matrices. For more detailed information, please see e.g. [24]. We will need a few more notations. If B is a vector or matrix, by B 0 we mean all elements of B are positive. If B 1 and B 2 are vectors or matrices with same dimensions we write B 1 B 2 if and only if B 1 − B 2 0. Moreover, we also adopt here the traditional notation by letting  Before we state our main results, we need the following useful lemma (see, e.g., [24]).

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Lemma 3.2 If A ∈ Z N ×N , then the following statements are equivalent: 1 (1) A is a nonsingular M-matrix. 2 (2) A is semi-positive; that is, there exists x 0 in R n such that Ax 0.

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(3) A −1 exists and its elements are all nonnegative.
Theorem 3.3 Let Assumptions 2.1 and 2.2 hold. Assume that is a nonsingular M-matrix and one of the following conditions holds: Then there is a positive constant C (independent of the initial data) such that for any initial 9 data x 0 and r 0 , the solution of equation (2.1) has the property that In other words, the hybrid SDEs with jumps (2.1) is asymptotically bounded in pth moment.

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Proof. As A p is a nonsingular M-matrix, by lemma 3.2, we see that θ = (θ 1 , · · · , θ N ) : By the generalized Itô formula, we have By the Young inequality Using the basic inequality |a + b + c| where (3.4) has been used.

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In either case (a) or (b), it is easy to see that there is a positive constant C 1 such that Taking the expectations on both sides of (3.5), we get where θ m = min i∈S θ i and θ M = max i∈S θ i . Dividing both sides by e εt and then letting t → ∞, 6 we obtain that 7 lim sup t→∞ E|x(t)| p ≤ C := C 1 /ε as required. The proof is therefore complete. initial data x 0 and r 0 such that Theorem 3.5 shows that equation (2.1) will be ultimately bounded with large probability, 9 while the following theorem estimates the limit of the average in the time of the pth moment.
for any initial data x 0 and r 0 , where Proof. This theorem can be proved in the same way as Theorems 2.3 and 3.3 were proved so 14 we omit its proof.

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Let us now proceed to discuss the asymptotic stability in the qth moment of the solution 16 to equation (2.1).

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Assumption 3.7 There exist positive constants k l , q l , l = 1, 2, 3 such that for all and Before we state the asymptotic stability result, we need one useful lemma. Let Assumption 3.7 hold. Assume moreover that for some q ≥ 2 such that p ≥ (q + q 1 ) ∨ is a nonsingular M-matrix and Then for any initial data x 0 and r 0 , the solution x(t) of equation (2.1) has the property that In other words, the hybrid SDEs with jumps (2.1) is asymptotically stable in the qth moment. Proof. Fix any initial data x 0 and r 0 . By Theorem 3.3, the solution is already asymp-10 totically bounded in L p . That is, there is a sufficiently large T 0 such that AsĀ q is a nonsingular M-matrix, by Lemma 3.2, we see thatθ = (θ 1 , · · · ,θ N ) := (3.14) Define the function V (x, i) =θ i |x| q . Applying the generalized Itô formula, we have Using the basic inequality |a + b| This, together with (3.14), implies But, by condition (3.11), Taking the expectations on both sides of (3.15), we get 16) whereθ m = min i∈Sθi andθ M = max i∈Sθi . Letting t → ∞ and then using the Fubini theorem, we obtain This of course implies that ∞ T 0 E|x(t)| q dt < ∞. 5 We now claim that E|x(t)| q is uniformly continuous on t ∈ [T 0 , ∞). By the generalized 6 Itô formula, we have that for any t > s > T 0 , Then, by Assumption 3.7, we have By the Young inequality (3.8), we show that By the Hölder inequality, there exists an δ > 0 such that Then, Assumption 3.7 implies that Recalling that p ≥ (q + q 1 ) ∨ (q + q 2 ) ∨ [0.5q(q 3 + 2)] and using (3.13), we get This implies that E|x(t)| q is uniformly continuous on [T 0 , ∞). Finally, by Lemma 3.8, we can obtain that lim t→∞ E|x(t)| q = 0 as required. The proof is therefore complete.
Remark 3.10 In this work, we consider the asymptotic boundedness and stability of the so- In this section, we will study the convergence of the EM approximate solutions for hybrid 14 SDEs with jumps (2.1) under the local Lipschitz condition and nonlinear growth condition.

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Before we define the EM approximate solution for equation (2.1), we need the property 16 of the embedded discrete Markov chain. The following lemma describes this property.

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For a given constant stepsize h > 0, we define the EM method for equation (2.1) as follows with initial value y 0 = x 0 and y n denotes the numerical approximation of x(t) with t n = nh.  To define the continuous-time approximate solution, let us introduce two step processes for t ∈ [t n , t n+1 ). Hence we define the continuous version y(t) as follows h(z(s),r(s), v)N (ds, dv).

(4.2)
It is not hard to verify that y(t n ) = y n , that is, y(t) coincides with the discrete solutions at 2 the grid-points.

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Let us define three stopping times Lemma 4.2 [13] Let φ : R + × Z → R n and assume that Then, there exists D p > 0 such that where a = f, g and M d is a constant independent of h.
whereC d is a constant independent of h.

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Proof. For simplicity, denote e(t) = x(t) − y(t). For any t 1 ∈ [0, T ], by the basic inequality Using the Hölder inequality, we obtain By Assumption 2.1, Lemma 4.3 and the basic inequality where C(d) is a constant independent of h, and in the computation below C(d) varies line-6 by-line. In the same way as Mao did in [24], we can show using lemma 4.2 that (4.5) Hence, Using the Burkholder-Davis-Gundy inequality and the Hölder inequality, we can derive that |g(x(t − ), r(t)) − g(z(t),r(t))| p dt.
In the same way as H 1 was estimated, we can then show To estimate H 3 , we first apply the basic inequality |a + b| p ≤ 2 p−1 (|a| p + |b| p ) to get By Lemma 4.2 and the Hölder inequality, we obtain In the same way as H 1 was estimated, we can then obtain Substituting (4.6), (4.7) and (4.8) into (4.4), we obtain that The Gronwall inequality implies that Therefore the proof is complete. Proof. We divide the whole proof into three steps.
The proof is therefore complete. Let N (dt, dv) be a Poisson random measures and σ-finite measure π(dv) is given by π(dv) = Of course, w(t), N (dt, dv) and r(t) are assumed to be indepen-7 dent.

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Consider the following scalar hybrid SDEs with jumps for x ∈ R. Obviously, the coefficient g satisfies the global Lipschitz condition and the linear 11 growth condition, while f, h satisfy the local Lipschitz condition but they do not satisfy the 12 linear growth condition. In fact, the coefficients f and g also satisfy the weak linear growth 13 conditions. Through a straight computation, we can have 14 x f (x, 1) 2) So the inequalities (5.2)-(5.5) show that Assumption 2.2 holds. Moreover, by the property of 1 normal distribute, we can obtain that π(Z) = 1 2 , and On the one hand, the matrix A 2 defined by (3.1) is It is easy to compute γ 2j θ j = 3.0643.
By Theorem 3.3, we can conclude that equation (5.1) is asymptotically bounded in mean square. 7 That is, On the other hand, similar to (4.2), we can obtain the EM approximate solution y(t) of , 0 ≤ v < ∞ is the density 4 function of a lognormal random variable. Of course N (dt, dv) and r(t) are assumed to be 5 independent.
The above conditions (5.13)-(5.16) imply that Hence, by Theorem 3.6, we can conclude that for any initial value x 0 , the solution x(t) of 4 equation (5.9) has the property that That is to say, the limit of the average in the time of the 2th moment is not greater than 6 0.3696.

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Consider the following scalar hybrid SDEs with pure jumps 12 with initial data x(0) = x 0 and r(0) = 1. Here for any x ∈ R. We note that equation (5.17) can be regarded as the result of the two equations shall see that due to the Markovian switching, the overall system (5.17) will be asymptotically 6 stable in 4th moment for certain γ. In fact, the coefficients f, g satisfy the local Lipschitz 7 condition but they do not satisfy the linear growth condition. Through a straight computation, the property of normal distribute, we can obtain that In this appendix, we shall prove Theorem 2.3. Proof of Theorem 2.3. Since the coefficients of equation (2.1) are locally Lipschitz continuous, for any given initial data x 0 and r 0 , there is a maximal local solution x(t) in L p on t ∈ [0, σ ∞ ), where σ ∞ is the explosion time (see, e.g., [40]). Fix any initial data x 0 and r 0 and find a sufficiently large k 0 for |x 0 | < k 0 . For each integer k ≥ k 0 , define the stopping time where, throughout this paper, we set inf φ = ∞ (as usual φ denote the empty set). Clearly, 5 τ k is increasing as k → ∞. Set τ ∞ = lim k→∞ τ k , whence τ ∞ ≤ σ ∞ a.s. Note if we can show 6 that τ ∞ = ∞ a.s., then σ ∞ = ∞ a.s. So we just need to show that τ ∞ = ∞ a.s. Define