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Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons

Focusing on hybrid diffusion dynamics involving continuous dynamics as well as discrete events, this article investigates the explicit approximations for nonlinear switching diffusion systems modulated by a Markov chain. Different kinds of easily implementable explicit schemes have been proposed to approximate the dynamical behaviors of switching diffusion systems with local Lipschitz continuous drift and diffusion coefficients in both finite and infinite intervals. Without additional restriction conditions except those which guarantee the exact solutions posses their dynamical properties, the numerical solutions converge strongly to the exact solutions in finite horizon, moreover, realize the approximation of long-time dynamical properties including the moment boundedness, stability and ergodicity. Some simulations and examples are provided to support the theoretical results and demonstrate the validity of the approach.


Introduction
The switching diffusion systems (SDSs) modulated by Markov chains involving continuous dynamics and discrete events, have drawn more and more attention to many researches. Much of the study originated from applications arising from biological systems, financial engineering, manufacturing systems, wireless communications (see, e.g., [1,2,3,4,5] and the references therein). Compared with those of the subsystems the dynamics of SDSs are seemingly much different. For instance, considering a predator-prey ecosystem switching between two environments randomly, Takeuchi et al. in [4] revealed that both subsystems develop periodically but switching between them makes them neither permanent nor dissipative. Pinsky and his coauthors in [6,7] provided several interesting examples to show that the switching system is recurrent (resp. transient) even if its subsystems are transient (resp. recurrent). Due to the coexistence of continuous dynamics and discrete events, the dynamics of SDSs are full of uncertainty and challenge.
Since solving SDSs is almost unavailable, numerical scheme or approximation techniques become viable alternatives. The explicit Euler-Maruyama (EM) scheme is popular for approximating diffusion systems and SDSs with global Lipschitz coefficients [2,8,5]. However, the coefficients of many important diffusion systems and SDSs are only locally Lipschitz and superlinear (see, e.g., [9,10,5,3] and the references therein). Hutzenthaler et al. [10,Theorem 2.1] showed that the absolute moments of the EM approximation for a large class of diffusion systems with superlinear growth coefficients diverge to infinity at a finite time point T ∈ (0, ∞). The implicit EM scheme is better than the explicit EM scheme in that its numerical solutions converge strongly to the exact solutions of diffusion and switching diffusion systems with the one-sided Lipschitz drift coefficient and global Lipschitz diffusion coefficient (see Higham et al. [11], Mao and Yuan [2, p.134-153]).
Nevertheless, additional computational efforts are required for its implementation since the solution of an algebraic equation has to be found before each iteration. Due to the advantages of explicit schemes (e.g., simple structure and cheap computational cost), a few modified EM methods have been developed for diffusion systems with nonlinear coefficients including the tamed EM method [12,9,13,14], the tamed Milstein method [15], the stopped EM method [16] and the truncated EM method [17]. These modified EM methods have shown their abilities to approximate the solutions of nonlinear diffusion systems. However, to the best of our knowledge these methods are not developed, even unavailable for a large class of nonlinear SDSs. For instance, Hamilton [18] remarked that the economy may either be in a fast growth or slow growth phase with the regime switching governed by the outcome of a Markov chain. Consider a two-dimensional nonlinear stochastic volatility model switching randomly between a fast growth phase dX(t) = 2.5X(t) 1 − |X(t)| dt + −1 √ 2 √ 2 1 |X(t)| 3/2 dB(t), (1.1) and a slow growth phase dX(t) = (1, 2) T − X(t) dt + 0.2 −0.5 1 0.4 |X(t)|dB(t), (1.2) modulated by a Markov chain r(t). This SDS is one of popular volatility models used for pricing option in Finance (see, e.g., [18,1,19,3] and the references therein), especially, for pricing VIX options. Based on its importance and no closed-form, one of our aims is to construct the explicit scheme available for this kind of SDSs and to study its convergence in the pth moment as well as the rate of convergence.
On the other hand, long-time behaviors of SDSs are also the major concerns in stochastic processes, systems theory, control and optimization (see, e.g., the monographs [2,5] and the references therein). So far, the dynamical properties of SDSs are investigated deeply including stochastic stability (see, e.g., [2,5,20,21]), invariant densities (see, e.g., [22,23]), recurrence and transience (see, e.g., [5,24]) and so on. Although the finite-time convergence is one of the fundamental concerns, how to approximate long-time behaviors of SDSs is significant and challenging. Recently, Higham et al. [25] by using the EM scheme, Mao and Yuan [2, pp.229-249] by using the backward EM (BEM) scheme showed respectively that the numerical solutions preserve the mean-square exponential stability of SDSs with globally Lipschitz continuous coefficients. Mao et al. [26] gave a counter-example that the EM numerical solutions don't share the underlying almost surely exponential stability of SDSs with the nonlinear growth drift term, but the BEM numerical solutions do.
Mao et al. [27], Yuan and Mao [28] and Bao et al. [29] made use of the EM method with a constant stepsize to approximate the underlying invariant measure of SDSs with linear growth coefficients while Yin and Zhu [5, p.159-179] did that by using the EM scheme with the decreasing step sizes.
In the above mentioned works, the diffusion coefficients of SDSs are always required to be globally Lipschitz continuous. For the further development of numerical schemes for SDSs, we refer readers to [30], for example, and the references therein. Accommodating many applications although these systems are more realistically addressing the demands, the nontraditional setup makes the discrete approximations of the long-time behaviors for nonlinear SDSs more difficult. Thus, in order to close the gap, the other aim is to construct an appropriate explicit scheme for a large class of nonlinear SDSs such that the numerical solutions realize the underlying infinite-time dynamical properties, such as the pth moment (p > 0) boundedness, stability as well as approximate the underlying invariant measure.
Motivated by the truncation idea in [17], together with the novel approximation technique, we have constructed a new explicit scheme and get the convergence with 1/2 order rate in finite time interval. Then we go further to improve the scheme according to the structure of the SDSs such that it is easily implementable for approximating the underlying invariant measure (resp. stability).
The schemes proposed in this paper are obviously different from those of [2,13,17,31]. More precisely, the numerical solutions at the grid points are modified before each iteration according to the growth rate of the drift and diffusion coefficients such that the numerical solutions keep the underlying excellent properties of the exact solutions of SDSs. Our contributions are as follows.
• We construct an easily implementable scheme for the SDSs with only local Lipschitz drift and diffusion coefficients and establish finite-time moment convergence results. The rate of convergence is also estimated under slightly stronger conditions.
• Using novel techniques (i.e., combining the Lemma 4.1 with the Perron-Frobenius theorem (see [32, p.6]) to construct appropriate Lyapunov functions depending on the states and analyzing their asymptotic properties), we obtain the criterion on the existence and uniqueness of invariant measures of the SDSs as well as the pth moment (p > 0) and the almost surely exponential stability; see Theorem 4.6 and Corollary 4.5.
• We reconstruct the explicit scheme which numerical solutions admit a unique invariant measure converging to the underlying one in the Wasserstein distance. The restrictions on the coefficients by using the EM scheme (c.f. [28,29,33]) are relaxed. Thus this scheme is more suitable for the measure approximation of nonlinear SDSs.
• Without extra restrictions the numerical solutions of the appropriate explicit scheme stay in step of dynamical properties with the exact solutions.
The rest of the paper is arranged as follows. Section 2 begins with notations and preliminaries on the properties of the exact solutions. Section 3 constructs an explicit scheme, and yields the convergence in pth moment and the rate. Two examples are given to illustrate the availability of this scheme. Section 4 focuses on the analysis of invariant measures. The other explicit scheme is constructed preserving the stability in distribution, which numerical invariant measure approximates the underlying one in the Wasserstein distance. Several numerical experiments are presented to illustrate the results. Section 5 gives some concluding remarks to conclude the paper.

Notations and preliminaries
Throughout this paper, we use the following notations. Let d, m and n denote finite positive integers, |·| denote the Euclidean norm in R n := R n×1 and the trace norm in R n×d . If A is a vector or matrix, its transpose is denoted by A T and its trace norm is denoted by |A| = trace(A T A). For vectors or matrixes A and B with compatible dimensions, AB denotes the usual matrix multiplication. If A is a symmetric matrix, denote by λ max (A) and λ min (A) its largest and smallest eigenvalue, respectively.
For any c = (c(1), . . . , c(m)) (or = (c 1 , . . . , c m )) defineĉ = min 1≤i≤m c(i)(c i ) andč = max 1≤i≤m c(i)(c i ). For any a, b ∈ R, a ∨ b := max{a, b}, and a ∧ b := min{a, b}. If D is a set, its indicator function is denoted by I D , namely I D (x) = 1 if x ∈ D and 0 otherwise. Let C i and C denote two generic positive real constants respectively, whose value may change in different appearances, where C i is dependent on i and C is independent of i.
It is well known that almost every sample path of r(t) is right-continuous step functions with a finite number of simple jumps in any finite interval of R + := [0, +∞)(c.f. [26]).
In this paper, we consider the n-dimensional SDS described by a hybrid stochastic differential equation (HSDE) with an initial value (X(0), r(0)) = (x 0 , ) ∈ R n × S, where the drift and diffusion terms are locally Lipschitz continuous, that is, for any N > 0 there exists a positive constant C N such that, for any x, y ∈ R n with |x| ∨ |y| ≤ N and any i ∈ S, Let C 2 (R n × S; R + ) denote the family of all nonnegative functions V (x, i) on R n × S which are continuously twice differentiable in x. For each V (x, i) ∈ C 2 (R n × S; R + ), define an operator LV from R n × S to R by where In order for the existence of exact regular solutions we impose the following assumption.
Assumption 1 For somep > 0 and each i ∈ S, there exists a symmetric positive-definite matrix Q i ∈ R n×n and a constant α i such that Now we give the regularity of exact solutions as well as the estimation on their pth moment.
Proof. For any i ∈ S and 0 < κ i < 1, Assumption 1 together with the continuity of f and g implies that there exists a positive constant C i such that for any x ∈ R n . For any fixed 0 Direct calculation, together with (2.5), leads to For any (x, i) ∈ R n × S and any p > 0, using the Young inequality, we obtain Therefore it follows from (2.6) and (2.7) that for all p ∈ (0,p], where C is a constant independent of N .

Moment estimate and strong convergence
In this section, we aim to construct an easily implementable explicit scheme and show the strong convergence under Assumption 1. Given a stepsize ∆ > 0 and let t k = k , r k = r(t k ) for k ≥ 0, and one-step transition probability matrix P (∆) = (P ij (∆)) m×m = exp(∆Γ We note that ϕ i is well defined since f (·, i) and g(·, i) are locally Lipschitz continuous. Denote by , ∞) → R + is a strictly increasing continuous function. We also choose a strictly decreasing h : where K is a positive constant independent of the iteration order k and the stepsize . For any given ∈ (0, 1], each i ∈ S, define a truncation mapping π i : R n → R n by where we use the convention x |x| = 0 when x = 0 ∈ R n . Obviously, for any (x, i) ∈ R n × S, Remark 3.1 If there exists a state i ∈ S such that |f (x, i)| ≤ C(1 + |x|), |g(x, i)| 2 ≤ C(1 + |x|) 2 , ∀x ∈ R n , we choose ϕ i (u) ≡ C for any u ≥ 1, and let ϕ −1 i (u) ≡ +∞ for any u ∈ [C, +∞). Then π i (x) = x, and (3.3) hold always.
Next we propose an explicit scheme to approximate the exact solution of SDS (2.1). To avoid the arbitrarily large excursions from Brownian pathes, we truncate the grid point value according to the growth rate of the drift and diffusion functions. Define This method prevents the diffusion term from producing extra-ordinary large value. Additionally, the drift term and diffusion term have the linear property (3.5) To obtain the continuous-time approximations, defineỸ (t), Y (t) andr(t) bỹ For convenience, denote by G t k the σ-algebra generated by {F t k , r k+1 }. Obviously, F t k ⊆ G t k .
The following lemma will play an important role in the proof of the moment boundedness of the numerical solutions.

Lemma 3.2 For any measurable functions
and Proof. First of all, note that Y k and r k are F t k -measurable. Then, by the Markov property, we derive that Thus, the required assertion (3.6) follows. Note also that Y k , r k and r k+1 are G t k -measurable. Using the properties where I d denotes the d × d identity matrix. Thus, we obtain that which implies that (3.7) holds. Moreover, Making use of (3.6) we arrive at (3.8). The proof is complete.
In order to estimate the pth moment of the numerical solution Y (t), we prepare an elementary inequality.
Lemma 3.3 For any given 2k < p ≤ 2(k + 1) (k is a nonnegative integer) the following inequality holds for any u > −1, where a k (u) represents a kth-order polynomial of u which coefficients depend only on p.

Moment estimate
Let us begin to establish the criterion on the pth moment boundedness of the numerical solutions of Scheme (3.4).
for any p ∈ (0,p] and any T > 0, where C is a constant independent of the iteration order k and the stepsize .
Proof. For any integer k ≥ 0, we have and we can see that ζ k > −1. By the virtue of Lemma 3.3, without loss the generality we prove (3.12) only for 0 < p ≤ 2. It follows from (3.11) and (3.13) that 14) It follows from Lemma 3.2 and (3.5) that Thus both of the above inequalities implies Using Lemma 3.2 and (3.5) again yields Using the properties for any A ∈ R 1×d as well as Lemma 3.2, we deduce that By virtue of Lemma 3.2, for any integer l ≥ 1, Making use of the above inequality and (3.5) yields Similarly, we can also prove that for any integer l > 3, E |ζ k | l |F t k ≤ C . Combining (3.14)-(3. 16) and (3.18), using (2.5) and (2.7), we obtain that implies that Repeating this procedure we obtain Taking expectations on both sides, for any integer k satisfying 0 ≤ k ≤ T , then we have Therefore the desired result follows.
In order to establish the strong convergence theory of Scheme (3.4), we give the following lemma.
then for any T > 0, for all p ∈ (0,p], where C is a constant independent of k and . Proof.
On the other hand, for ω ∈ {β < k + 1}, we have β ≤ k and hencẽ Therefore, we derive from (3.4) that for any integer k ≥ 0, By the virtue of Lemma 3.3, without loss the generality we prove the required result only for 0 < p ≤ 2. It follows from (3.11) and (3.21) that By Lemma 3.2 one oberserves that is a continuous local martingale, by the virtue of the Doob martingale stopping time theorem, we know that The above equality together with Lemma 3.2 implies Thus the above inequality and (3.23) imply Using the techniques in the proof of Theorem 3.4, we show that Similarly, we can also show that for any integer l > 3, Combining (3.22) and (3.24)-(3.26), using (2.5) and (2.7), for any integer k ≥ 0, Repeating this procedure we obtain for any integer k satisfying 0 ≤ k ≤ T . Therefore the required assertion follows from that where T represents the integer part of T / . The proof is complete.

Strong convergence
In this subsection, we give the convergence result of Scheme (3.4).
If we can show that the required assertion follows. For this purpose we define the truncated functions with the initial value z(0) = x 0 and r(0) = . For the chosen N , Assumption 1 implies that f N (·, ·) and g N (·, ·) are globally Lipschitz continuous with the Lipschitz constant C N . Therefore, SDS (3.32) has a unique regular solution z(t) satisfying On the other hand, for each ∈ (0, * ], we apply the EM method to (3.32) and denote by u(t) the piecewise constant EM solution (see [2]) which has the property Thus the desired assertion (3.31) follows from (3.34). The proof is complete.

Convergence Rate
In this subsection our aim is to establish the rate of convergence. The rate is optimal similar to the standard results of the explicit EM scheme for SDSs with globally Lipschitz coefficients, see [2, p.115]. To estimate the rate, we need somewhat stronger conditions compared with the convergence alone, which are stated as follows.
Assumption 2 For somep > 2, there exist positive constants L i and l such that Remark 3.7 One observes that if Assumption 2 holds, then One also knows that and by Young's inequality, In order to obtain the rate, we specify h( ) = K −τ for any Making use of Scheme (3.4), we define an auxiliary approximation process bȳ and Y (t) coincide with the discrete solution at the grid points.
Lemma 3.9 If Assumptions 1 and 2 hold, for any p ∈ (0,p/(l + 1)], the process defined by (3.40) has the property where C is a positive constant independent of .
Using the techniques in the proof of Theorem 3.4 and Lemma 3.5 yields the following lemmas.
Lemma 3.10 Under Assumption 1, the process defined by (3.40) has the property that for all p ∈ (0,p], where C is a positive constant independent of .
Choose a small constant ι > 0 such that 1 + ι ≤p − 1, then the application of Young's inequality implies Due to (3.36) and (3.37), by Jensen's inequality and Lemma 3.9, one observes Let j = T / , by (3.38), (3.39), using Young's inequality yields It follows from Theorem 3.4 and the property of Markov chain r(·) that By the virtues of Lemmas 3.9 and 3.12, we yield the optimal rate of strong convergence.
Remark 3.14 If we can find a uniform function ϕ(u) such that (3.1) holds for each i ∈ S, then we can find the unform truncation function π (x). The second equation in (3.4) degenerates to Y k = π (Ỹ k ) independent of r k . Note that the results of Theorem 3.6 and Theorem 3.13 still hold for this special case. One observes that sometimes it is hard to find a uniformly continuous function ϕ(u) ≥ max i∈S ϕ i (u) for u ∈ [1, ∞). However, for the given and h( ), it is easy to findφ −1 (h( )), the minimum of all ϕ −1 i (h( )). Let π (x) = |x| ∧φ −1 (h( )) x/|x|. Clearly, the property (3.5) still holds for Scheme (3.4) with π i (x) ≡ π (x), which implies Theorem 3.6 and Theorem 3.13 still hold. On the other hand, in view of computation, the cost of using the uniform scheme is the same as that of using the non-unform one because the value of r k is fixed before each iteration. More precisely, we illustrate it in Example 3.15.

Numerical examples
In order to illustrate the efficiency of Scheme (3.4) we recall the introductory example and present some simulations.
In order to represent the simulations by Scheme (3.4), we divide it into five steps.
Step 1. Examine the hypothesis. Since that lim sup Assumption 1 holds for any 0 <p ≤ 5/3. By Theorem 2.1, the unique regular solution X(t) exists.
Step 3. MATLAB code. Next we specify the MATLAB code for simulating Y (T ): Step 5. The log-log error plot with M = 1000. The simulation procedure is carried out by steps 3 and 4. The blue solid line depicts log-log error while the red dashed is a reference line of slope 1/2 in Figure 1. On the other hand, as the claim in Remark 3.14, we can also use the uniform truncation mapping    Figure 2).
Thus, the computational cost of the non-uniform scheme is the same as that of the uniform one.
Let us discuss another example to compare our scheme with the implicit EM scheme. dX(t) = a(r(t))X(t)+ b(r(t))X 3 (t) dt + σ(r(t))X(t)dB(t), t ≥ 0, locally Lipschitz continuous. Assumption 1 holds for anyp > 0. Note that there exists a unique regular solution X(t) to SDS (3.53) for any initial data x 0 > 0, ∈ S. In the same way as in [8], we get its closed-form .
and the implicit EM scheme becomes

Approximation of invariant measure
In this section, we are concerned with the existence and uniqueness of invariant measures of the exact and numerical solutions for SDS (2.1). We reconstruct an explicit scheme and show the existence and uniqueness of the numerical invariant measure converging to the underlying exact one in the Wasserstein metric. Our results cover a large kind of nonlinear SDSs with only locally Lipschitz continuous coefficients. For the convenience of invariant measure study we introduce some notations as well as an assumption. We write (X x 0 , t , r t ) in lieu of (X(t), r(t)) to highlight the initial data (X(0), r(0)) = (x 0 , ). For each N > 0, let B N (0) = {x ∈ R n : |x| ≤ N }. Let B(R n ) denote the family of all Borel sets in R n and P(R n × S) denote the family of all probability measures on R n × S. For any p ∈ (0, 1], define a metric on R n × S as follows d p ((x, i), (y, j)) := |x − y| p + I {i =j} , (x, i), (y, j) ∈ R n × S, and the corresponding Wasserstein distance between µ,μ ∈ P(R n × S) by where C(µ,μ) denotes the collection of all probability measures on (R n ×S)×(R n ×S) with marginal measures µ andμ. The Wasserstein distance W p admits a duality representation (see, e.g, [34]), i.e.
whereψ is a continuous function on R n × S and From this section we always assume γ(t) is irreducible, namely, the following linear equation has a unique solution π = (π 1 , . . . , π m ) ∈ R 1×m satisfying π i > 0 for each i ∈ S. This solution is where spec(Γ p,u ) and Re(ν) denote the spectrum of Γ p,u (i.e. the multiset of its eigenvalues) and the real part of ν, respectively. In order to examine the properties of the exact solution of SDS (2.1), we prepare the following lemma.

Boundedness
Since it is related closely to the tightness as well as the ergodicity we go further to investigate the  where C is independent of t and i, each α i is given by Assumption 1, and α := (α 1 , . . . , α m ) T .
Proof. For the given α and any p > 0, diag(α), Γ p,α , η p,α , p * α are defined as Therefore the desired assertion follows from that The proof is complete.
In order to show that the numerical solutions of Scheme (3.4) preserve this asymptotic boundedness perfectly, we further require the function h( ) satisfying for some 0 < θ < 1 2 . Then the asymptotic boundedness of the numerical solution is obtained as follows.
where C is independent of k and .

Invariant measure
In this subsection, we discuss the criterion on the existence and uniqueness of the invariant measure for SDS (2.1). It follows from SDS (2.1) that for any two initial values (x 0 , ), (x 0 , ) ∈ K × S, where K is a compact set in R n . For convenience we impose the following hypothesis.
In order for the unique invariant measure we provide the asymptotic attractivity of SDS (2.1).
As well as we know the stability of the trivial solution is one of the major concerns in many applications. As a corollary, the result on the stability follows from the proof of Lemma 4.4 directly.
For clarity we impose the following assumption.
Proof. In the same way as the proof of Lemma 4.4, for any initial data (x 0 , ) ∈ R n × S, ∞ a.s. Therefore the other required assertion follows. The proof is complete.
Next we give the existence and uniqueness of the invariant measure for the solution (X x 0 , t , r t ) of SDS (2.1). Let P t (x 0 , ; dx × {i}) be the transition probability kernel of the pair X x 0 , t , r t , a time homogeneous Markov process (see, e.g, [2, Theorem 3.27, pp.104-105]). Recall that µ ∈ P(R n × S) Proof. For arbitrary t > 0, define a probability measure Then, for any ε > 0, by Theorem 4.2 and Chebyshev's inequality, there exists a positive integer N > 0 sufficiently large such that is a compact subset of R n . Moreover, one observes that (X x 0 , t , r t ) enjoys the Feller property (see, e.g., [5,Theorem 2.18,p.48]). Thus, the solution (X x 0 , t , r t ) of SDS (2.1) has an invariant measure (see, e.g., [36,Theorem 4.14,p.128]).
We have established the existence of the invariant measure, and next we go a further step to show its uniqueness. For any p ∈ (0, p * α ∧ p * β ∧p) ∩ (0, ρ], (4.15) and the irreducibility of r(·) result in where δ (x,i) stands for the Dirac's measure at the point (x, i). Assume both µ andμ are invariant measures, then we have One then observes that, due to (4.25), W p (μ, µ) → 0 as t → ∞. The desired assertion follows.

The invariant measure of numerical solution
In order to approximate the invariant measure µ of SDS (2.1) we need to construct the appropriate scheme such that the numerical solutions are attractive in ρth moment and admit a unique numerical invariant measure. However, the proposed mapping π i (x) is not suitable for the avoidance of extra wide divergence in distance between two different numerical solutions. Thus we construct a new truncation mappingπ i (x) according to the local Lipschitz continuity of the drift and diffusion coefficients. Then making use of the appropriate truncation mapping we give an explicit scheme.
Finally we show that, it produces a unique numerical invariant measure µ which tends to the invariant measure µ of SDS (2.1) as → 0 in the Wasserstein metric.
For each i ∈ S choose a strictly increasing continuous functionsφ i : R + → R + such that Due to the local Lipschitz continuity of f and g the functionφ i can be well defined as well as its We choose a strictly decreasingh : for some 0 <θ < 1/2 holds, where K is a positive constant independent of k and . For a given stepsize ∈ (0, 1], let us define the truncated mappingπ i : R n → R n bȳ where we let x |x| = 0 when x = 0 ∈ R n . Obviously, for any i ∈ S and x, y ∈ R n , and Remark 4.7 If there exists the state i ∈ S such that we chooseφ i (u) ≡ C for any u ≥ 1, and letφ −1 i (u) ≡ +∞ for any u ∈ [C, +∞). Thenπ i (x) = x, (4.29) and (4.30) hold always.
Then sup which implies Inserting this into (4.42) yields the desired assertion (4.35).
Furthermore we can also use Scheme (4.32) to approximate the stability as follows.
In order for the ergodicity, we show the Markov property of (Z x 0 , k , r k ) k≥0 .
Lemma 4.10 {(Z k , r k )} k≥0 is a time homogeneous Markov chain.
Proof. One observes Since B k and B 0 are identical in probability law, comparing the two fomulas above, we know that (Z k+1 , r k+1 ) and (Z 1 , r 1 ) are identical in probability law under Z k = x, r k = i and x 0 = x, r 0 = i respectively. Thus for any D ∈ B(R n ), j ∈ S, which is the desired homogenous property. For any k ≥ 0, ∈ (0, 1], By (4.32) we know that r k+1 = λ r k k+1 and Z k+1 = θ Z k ,r k k+1,r k+1 . Note that λ i k+1 and θ x,i k+1,j are bounded measurable random functions independent of F t k . Hence, for any D × {j} ∈ B(R n ) × S, using [2, which is the desired Markov property.
Let P ∆ k (x 0 , ; dx × {i}) be the transition probability kernel of the pair Z x 0 , k , r k , a time homogeneous Markov chain. If µ ∈ P(R n × S) satisfies for any k ≥ 0, then µ is called an invariant measure of Z x 0 , k , r k . Moreover, such an invariant measure µ ∈ P(R n × S) is also called a numerical invariant measure. Next we give the existence and uniqueness of the numerical invariant measure for SDS (2.1) using Scheme (4.32). Proof. For arbitrary integer l > 0, define a measure sequence { l } that for any Υ ∈ B(R n ) and i ∈ S. For any p ∈ (0, p * α ) ∩ (0,p] and ∈ (0, * * ], by Theorem 4.3 and Chebyshev's inequality, we derive that { l } is tight, then one can extract a subsequence which converges weakly to an invariant measure. Thus, the numerical solution (Z x 0 , k , r k ) has an invariant measure µ . For any ∈ (0, * * ], we have established the existence of the numerical invariant measure, and now we further show its uniqueness. For any p ∈ (0, p * α ∧ p * β ∧p) ∩ (0, ρ], it follows from (4.19) and (4.35) that (4.45) Assume both µ andμ ∈ P(R n × S) are invariant measures, then we have Due to (4.45), W p (µ ,μ ) → 0 holds as k → ∞. The desired assertion follows.
It follows from Theorem 4.6 that exact solutions of stochastic volatility model with random switching between (1.1) and (1.2) admit a unique invariant measure µ ∈ P(R n ×S). By the virtues of Theorems 4.11 and 4.12 Scheme (4.32) produces a unique numerical invariant measure µ ∈ P(R n × S) converging to µ in the Wasserstein metric. To the best of our knowledge almost all numerical methods in the literatures such as [28,29,33] cannot treat this case.
Secondly we carry out some numerical experiments. Basing on the model structure, we begin to specify the explicit truncated EM scheme (4.32).
Moreover, by Theorems 4.11 and 4.12, the unique numerical invariant measure µ ∈ P(R × S) of the truncated EM scheme exists and converges to µ in the Wasserstein metric.

Concluding Remarks
This paper investigates the approximation methods for the SDSs without globally Lipschitz continuous coefficients. A novelty is to construct two explicit schemes approximating the dynamical properties of SDSs. In finite horizon, by one scheme, we show the boundedness of the numerical solutions, obtain the convergence in the pth moment and estimate the rate of convergence. On the other hand, in infinite horizon we use the other scheme to approximate the underlying invariant measure of exact solutions in the Wasserstein distance. Moreover, this scheme is also suitable to realize the stability of SDSs. Our exploiting schemes perform the dynamical behaviors of exact solutions very well but don't require more restrictions except the structure conditions which guarantee the exact solutions posses some propery such as stability, moment boundedness and ergodicity. Some simulation examples are provided to support the theoretical results and demonstrate the validity of approaches.