Numerical method of highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching

In this paper, we establish a partially truncated Euler–Maruyama scheme for highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching. We investigate the strong convergence rate and almost sure exponential stability of the numerical solutions under the generalized Khasminskii-type condition.


Introduction
Stochastic differential equations play an important role in various fields, such as biology, chemistry, and finance [3,20,27]. In practice, parameters and forms in stochastic systems may change when something unexpected happens. At this point, we can use stochastic differential equations with Markovian switching. Mao and Yuan [24] studied stochastic differential equations with Markovian switching in depth. Many stochastic systems not only depend on the present and past states, but also contain derivatives with delays and the function itself, which can be described by neutral stochastic differential delay equations (NSDDEs) [20]. Kolmanovskii et al. [12] established a fundamental theory for neutral stochastic differential delay equations with Markovian switching (NSDDEwMSs) and discussed some important properties of the solutions.
In many cases the true solutions of the equations cannot be found. So it is very useful to study explicit forms of the numerical solutions. The Euler-Maruyama (EM) method for stochastic differential delay equations with Markovian switching (SDDEwMSs) was investigated in [25] and [37]. Wu and Mao [34] showed the convergence of EM method for neutral stochastic functional differential equations. However, Hutzenthaler et al. [9] showed that pth moments of the EM approximations diverge to infinity for any p ∈ [1, ∞) when the coefficients grow superlinearly. Many implicit methods were established to estimate the solutions of the equations with superlinearly growing coefficients [2,4,8,11,26,30,32,33]. Due to the advantages of explicit numerical solutions, such as less computation, plenty of modified EM methods have been studied to approximate the solutions of superlinear stochastic differential equations. The tamed EM scheme was proposed in [10] to estimate the solutions of stochastic differential equations with one-sided Lipschitz drift coefficient and global Lipschitz diffusion coefficient. Sabanis [28,29] developed tamed EM schemes for nonlinear stochastic differential equations. More detail on the other explicit numerical methods can be found in [1,16,18]. In addition, Mao initialized the truncated EM method in [21] and obtained the convergence rate in [22]. Then Guo et al. [7] discussed the convergence rate of the truncated EM method for stochastic differential delay equations. The truncated EM method for time-changed nonautonomous stochastic differential equations was shown in [19]. To get the asymptotic behaviors easily, Guo et al. [6] proposed the partially truncated EM method. In [38], the partially truncated EM method for stochastic differential delay equations was proposed. Cong et al. [5] used the partially truncated EM method to get the convergence rate and almost sure exponential stability of highly nonlinear SDDEwMSs. Tan and Yuan in [33] showed the convergence rates of the theta-method for nonlinear neutral stochastic differential delay equations driven by Brownian motion and Poisson jumps, but the stability was not analyzed as time goes to infinity. In [39], the convergence of the EM method for NSDDEwMSs was proved, but the convergence rate was not given. To our best knowledge, there are few papers concerning with numerical solutions of highly nonlinear and nonautonomous NSDDEwMSs. Therefore, in this paper, we give the strong convergence rate of the partially truncated EM method for highly nonlinear and nonautonomous NSDDEwMSs.
Moreover, many scholars are interested in the asymptotic behaviors of the stochastic systems [3,5,6,20,24,31]. The almost surely asymptotic stability of NSDDEwMSs was discussed in [23]. Then Li and Mao [15] established LaSalle-type stability theorem for NS-DDEwMSs. Liu et al. [17] showed the mean square polynomial stability of the EM method and the backward EM method for stochastic differential equations. The almost sure exponential stability of EM approximations for stochastic differential delay equations was investigated By means of the semimartingale convergence theorem [36]. The exponential mean square stability of the split-step theta method for NSDDEs was investigated in [40]. Lan and Yuan [14] studied the exponential stability of the exact solutions and θ -EM (1/2 < θ ≤ 1) approximations to NSDDEwMSs. Lan [13] gave the asymptotic mean-square and almost sure exponential stability of the modified truncated EM method for NSDDEs under local Lipschitz condition and nonlinear growth condition. However, there is little literature studying the almost sure exponential stability of the partially truncated EM method for highly nonlinear and nonautonomous NSDDEwMSs. The second goal of this paper is to fill this gap. This paper is organized as follows. We introduce some useful notations and establish the partially truncated EM scheme for NSDDEwMSs in Sect. 2. In Sect. 3, we discuss the strong convergence rate. In Sect. 4, we show the almost sure exponential stability of numerical solutions. Section 5 contains two examples to illustrate that our main result covers a large class of highly nonlinear and nonautonomous NSDDEwMSs.

Mathematical preliminaries
Unless otherwise specified, we use the following notation. If A is a vector or matrix, its transpose is denoted by A T . For x ∈ R n , let |x| denote its Euclidean norm. If A is a ma-trix, denote by |A| = trace(A T A) its trace norm. By A ≤ 0 and A < 0 we mean that A is nonpositive and negative definite, respectively. For real numbers a, b, we denote a ∧ b = min{a, b} and a ∨ b = max{a, b}. Let a be the largest integer that does not exceed a. Let R + = [0, +∞) and τ > 0. By C ([-τ , 0]; R n ) we denote the family of continuous functions ν from [-τ , 0] to R n with the norm ν = sup -τ ≤θ≤0 |ν(θ )|. If H is a set, then I H denotes its indicator function, that is, stand for a generic positive real constant different in different cases. Let (Ω, F, {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous, and F 0 contains all Pnull sets). Let E denote the expectation with respect to P.
. . , B m (t)) T be an m-dimensional Brownian motion defined on the probability space.
Let r(t) (t ≥ 0) be a right-continuous Markov chain on the probability space taking values in a finite state space S = {1, 2, . . . , N} with generator Γ = (γ ij ) N×N given by where Δ > 0, and γ ij is the transition rate from i to j with γ ij > 0 if i = j, whereas γ ii = j =i γ ij . We suppose that the Markov chain r is independent of the Brownian motion B.
As is well known [31], almost every sample path of r is a right-continuous step function with finite number of simple jumps in any finite subinterval of R + , that is, there is a sequence of stopping times 0 = τ 0 < τ 1 < τ 2 < · · · < τ k → ∞ almost surely such that where I is the indicator function defined as before. Hence r is constant on each interval [τ k , τ k+1 ): In this paper, we consider highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching of the form with initial data where r 0 is S-valued F 0 -measurable random variable. Here f : R + × R n × R n × S → R n , g : R + × R n × R n × S → R n×m , and D : R n × S → R n . They are all Borel-measurable functions.
We suppose that the drift and diffusion coefficients can be decomposed as To estimate the partially truncated EM method for (2.1), we need the following lemma [24].
. .} is a discrete Markov chain with the one-step transition probability matrix (2.4) Then we impose two standard necessary hypotheses on the initial data and neutral term.

Assumption 2.2
There exist constants K 1 > 0 and α ∈ (0, 1] such that for all x, y ∈ R n and i ∈ S. By Assumption 2.3 we have |D(x, i)| ≤ K 2 |x| for all x ∈ R n and i ∈ S. Since γ ij is independent of x, the paths of r could be generated before approximating x. The discrete Markovian chain {r Δ k , k = 0, 1, 2, . . .} can be generated as follows: Compute the one-step transition probability matrix P(Δ). Let r Δ 0 = i 0 and generate a random number ξ 1 uniformly distributed in [0, 1]. Define where we set 0 j=1 P i 0 ,j (Δ) = 0 as usual. Then independently generate a new random number ξ 2 uniformly distributed in [0, 1] as well. Define Repeating this procedure, we can obtain a trajectory of {r Δ k , k = 1, 2, . . .}. The procedure can be applied independently to get more trajectories. After generating the discrete Markov chain {r Δ k , k = 0, 1, 2, . . .}, we can now define the partially truncated EM approximate solution for NSDDEwMSs (2.1) with initial data (2.2).
To define the partially truncated EM scheme, we first choose a strictly increasing continuous function ϕ(w) : R + → R + such that ϕ(w) → ∞ as w → ∞ and (2.7) Let ϕ -1 denote the inverse function of ϕ. Hence ϕ -1 is a strictly increasing continuous function from [ϕ(1), ∞) to R + . Then we also choose K 0 ≥ 1 ∨ ϕ(1) and a strictly decreasing function h : For a given step size Δ ∈ (0, 1], define the truncated mapping π Δ from R n to the closed ball {x ∈ R n : |x| ≤ ϕ -1 (h(Δ))} by where we let x |x| = 0 for x = 0. Then we can define the truncated functions Thus we obtain that Moreover, we can easily get that for any x, y ∈ R n , Let us now establish our discrete-time truncated EM numerical solutions to approximate the true solution. For some positive integer M, we take step size Δ = τ /M. It is easy to see that Δ becomes sufficiently small by choosing M sufficiently large. Define t k = kΔ for k = -M, -M + 1, -M + 2, . . . , -1, 0, 1, 2, . . . . Set X Δ (t k ) = ξ (t k ) for k = -M, -M + 1, -M + 2, . . . , -1, 0 and then form for k = 0, 1, 2, . . . , where ΔB k = B(t k+1 ) -B(t k ). To form continuous-time step approximations, define where I is the indicator function. As usual, there are two kinds of continuous-time step approximations. The first one whose sample paths are not continuous is The other one with continuous sample paths is which is continuous in t. Is easy to see that . Namely, they coincide at t k .

Strong convergence rate
In this section, we estimate the strong convergence rate of the partially truncated EM method for (2.1). Now, to achieve this goal, we have to impose the following assumptions on the coefficients. Assumption 3.1 There exist constants K 3 > 0 and β ≥ 0 such that and for all t ∈ [0, T], x, y,x,ȳ ∈ R n , and i ∈ S.
By Assumption 3.1 we get that there exists a constantK 3 > 0 such that and We also derive from Assumption 3.1 that x,ȳ, i) for all t ∈ [0, T], x, y,x,ȳ ∈ R n , and i ∈ S.
Before stating the next assumption, we introduce functionsV i , i = 1, 2, 3, such that for any x, y ∈ R n , Assumption 3.2 There exist constants K 4 > 0 andq > 2 such that By Assumption 3.2 we obtain that for any q ∈ (2,q), The proof is trivial, so we omit it.

Assumption 3.3
There exist constants K 5 > 0 andp >q such that for all t ∈ [0, T], x, y ∈ R n , and i ∈ S.
By Assumption 3.3 we derive that for any p ∈ [2,p), . Assumption 3.4 There exist constants K 6 > 0, K 7 > 0, θ ∈ (0, 1], and σ ∈ (0, 1] such that for all t 1 , t 2 ∈ [0, T], x, y ∈ R n , and i ∈ S, where β is as in Assumption 3.1. The following lemma gives that the p-moment of the true solution is bounded. This lemma can be proved similarly to the proof of Theorem 2.4 presented in [12] by means of the technique used in Theorem 2.1 of [35]. Lemma 3.5 Let Assumptions 3.1 and 3.3 hold. Then neutral stochastic differential delay equations with Markovian switching (2.1) with initial data (2.2) has a unique solution x(t) on t ≥ -τ . In addition, this solution has the property that To get the strong convergence rate, we impose another assumption.
Assumption 3.6 There exist constants K 8 > 0 andp >q such that for all t ∈ [0, T], x, y ∈ R n , and i ∈ S.
Remark 3.7 When D(·, ·) = 0, we can derive that for any functions satisfying Assumption 3.3, We can easily prove that Assumptions 3.3 and 3.6 are satisfied. A detailed proof is presented in Sect. 5.
In the same way as in the proof of (3.19), we have Then Lemma 3.9 gives that We complete the proof.
Then we have

30)
Proof By Itô's formula and Assumption 3.6 we get Note that where (2.8), (2.10), (3.3), Young's inequality, and Lemma 3.9 were used. Then we obtain that where l v * = l v ∨ 2. Using the same technique as in Lemma 3.9 gives that We can get from (2.6) that (3.32) Hence we derive from (3.31) and (3.32) that Then the desired result follows. We complete the proof.
The following lemma can be proved in a similar way as Lemma 3.11 was, so we omit the proof. Assume that q ∈ [2,q) and p > (β + l v + 2)q. Let L > ξ be a real number, and let Δ ∈ (0, 1] be sufficiently small such that ϕ -1 (h(Δ)) ≥ L. Then we have where ρ Δ,L := τ L ∧ τ Δ,L with τ L , τ Δ,L defined as before.
Proof For simplicity, we write ρ Δ, Recalling the definition of F Δ and G Δ , we have By Itô's formula we get Note that Hence By Hölder's inequality, Assumptions 2.2 and 3.2, and Lemmas 3.9 and 3.10 we get As for B 2 , we derive from Assumptions 2.3 and 3.4 that and we have Moreover, let j be the integer part of T/Δ. Then (3.42) where in the last step, we used the fact thatx Δ (s) andx Δ (sτ ) are conditionally independent of I {r(s) =r(t k )} with respect to the σ -algebra generated by r(t k ). Applying the Markov property yields that (3.43) By Lemma 3.9 we have (3.45) In addition, we obtain from Assumptions 2.2 and 3.1 and Lemmas 3.5, 3.9, and 3.10 that Furthermore, let j be the integer part of T/Δ. Then by Assumption 2.3 and Lemma 3.9 we have By (3.43) we get Hence, for any s ∈ [0, T], we derive that (3.47) Inserting (3.47) into (3.46) gives that (3.48) Similarly to B 2 and B 3 , we easily derive that and Using Gronwall's inequality gives that . Thus e Δ (t) = y(t)y Δ (t). Then for any c 3 , c 4 , c 5 > 0, we have Choose c  Hence we derive from (3.47) that Then we have An application of Gronwall's inequality gives that . Then we use the same technique as in Lemma 3.9 to get the convergence rate. Define We find that 2q i+1 < q i and q T/τ +1 = q, i = 1, 2, . . . , T/τ .
Then by Hölder's inequality we obtain that By induction we could get the desired result. We complete the proof.
Remark 4.2 In fact, there are many functions such that D(y, i),F(t, x, y, i), andG(t, x, y, i) satisfying (4.1) and the corresponding F Δ (t, x, y, i) and G Δ (t, x, y, i) satisfying (4.2). The example and proof will be given in Sect. 5.

Example
Example 5.1 Consider a nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching We easily see that  with step sizes 2 -11 , 2 -10 , 2 -9 , 2 -8 , 2 -7 at T = 1. 1000 sample paths were simulated in the numerical experiments. We can observe that the convergence order of partially truncated EM method for (5.1) is approximately 1 4 , which is close to our result. It is easy to see that