The Truncated Theta-EM Method for Nonlinear and Nonautonomous Hybrid Stochastic Differential Delay Equations with Poisson Jumps

In this paper, we study a class of nonlinear and nonautonomous hybrid stochastic differential delay equations with Poisson jumps (HSDDEwPJs). The convergence rate of the truncated theta-EM numerical solutions to HSDDEwPJs is investigated under given conditions. An example is shown to support our theory.


Introduction
Stochastic differential equations have been widely used in many fields and have attracted many scholars [1][2][3]. Sometimes, an emergency may occur in the system, and it is necessary to consider the influence of the emergency. For example, the surprising outbreak of COVID-19 has a huge impact on the world economy, especially on the stock market. erefore, stochastic differential equations with jumps considering continuous and discontinuous random effects have been investigated to analyze these situations [4][5][6][7]. In practical applications, the parameters and forms in the stochastic systems will change when certain emergencies occur. In this case, we could use stochastic differential equations with Markovian switching to describe them [8]. In this paper, we will take the Markovian switching and jumps into consideration; i.e., we shall study hybrid stochastic differential delay equations with Poisson jumps (HSDDEwPJs).
Numerical methods have been extensively studied, due to the fact that many true solutions of plenty of stochastic differential equations could not be obtained. For example, the explicit Euler-Maruyama (EM) schemes are well known for approximating the true solutions [9]. However, when the coefficients grow superlinearly, Hutzenthaler et al. in [10] proved that, for all p ∈ [1, ∞), the pth moment of the EM approximations diverges to infinity. erefore, many implicit methods have been proposed to approximate the solutions of the stochastic differential equations with nonlinear growing coefficients [11][12][13]. In addition, considering that the amount of calculations of the explicit schemes is less, some modified EM methods have been used to approximate the nonlinear stochastic differential equations [14][15][16]. In particular, the truncated EM method was initialized by Mao in [17] with both the drift and diffusion coefficients growing superlinearly. e convergence rate of the truncated EM method was given in [18]. Subsequently, there have been many papers discussing the truncated EM method for stochastic differential equations with superlinear coefficients [19][20][21][22][23][24][25]. In addition, there are many papers which consider the stability of the systems [26][27][28][29][30]. e truncated EM scheme for time-changed nonautonomous stochastic differential equations was shown in [31]. In [32], it was extended to the truncated theta-EM scheme on the basis of truncated EM scheme, and the strong convergence rate of the truncated theta-EM scheme for stochastic delay differential equations under local Lipschitz condition was investigated. e truncated theta-EM method will become the EM method when θ � 0 and degenerate to the backward EM method when θ � 1. Additionally, there are a few results on the numerical solutions for HSDDEwPJs. e convergence of EM approximation solution to the true solution in probability under some weaker conditions was proved in [33]. e EM approximate solutions converge to the true solutions for stochastic differential delay equations with Poisson jumps and Markovian switching under local Lipschitz condition [34]. e convergence of EM method for stochastic differential delay equations with Poisson jumps and Markovian switching in the sense of L 1 -norm under one non-Lipschitz condition was discussed in [35]. e strong convergence between the true solutions and the numerical solutions to stochastic differential delay equations with Poisson jumps and Markovian switching was studied when the drift and diffusion coefficients are Taylor approximations [36]. To the best of our knowledge, there are few papers concerning the numerical solutions of the nonlinear and nonautonomous HSDDEwPJs. us, in this paper, we will give the strong convergence rate of the truncated theta-EM method for nonlinear and nonautonomous HSDDEwPJs.
is paper is organized as follows. We will introduce some necessary notations in Section 2. e rate of convergence in L 2 sense will be discussed in Section 3. Finally, in Section 4, we will give an example to illustrate that our main result could cover a large class of nonlinear and nonautonomous HSDDEwPJs.

Mathematical Preliminaries
roughout this paper, unless otherwise specified, we will use the following notations. If A is a vector or matrix, its transpose is denoted by A T . ∀x ∈ R n , let |x| denote its Euclidean norm. If A is a matrix, its trace norm is denoted by |A| �

��������� � trace(A T A).
A ≤ 0 and A < 0 mean that A is nonpositive and negative definite, respectively. If a, b are real numbers, then a∧b � min a, b { } and a∨b � max a, b { }. Let ⌊a⌋ be the largest integer which does not exceed a. Let R + � [0, +∞) and τ > 0. Let C([− τ, 0]; R n ) be the family of continuous functions ] from [− τ, 0] to R n with the norm ‖]‖ � sup − τ≤θ≤0 |](θ)|. If H is a set, let I H denote its indicator function which means I H (ω) � 1 if ω ∈ H and I H (ω) � 0 if ω ∉ H. Let C be a generic positive real constant which could be 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 while F 0 contains all P-null sets). In addition, let E denote the probability expectation with respect to P. For p > 0, let L p F 0 ([− τ, 0]; R n ) denote the family of all F 0 -measurable and C([− τ, 0]; R n )-value random variables ξ such that E‖ξ‖ p < ∞. Let B(t) � (B 1 (t), . . . , B m (t)) T be an m-dimensional Brownian motion defined on the probability space. Let N(t) denote a scalar Poisson process with the compensated Poisson process N(t) � N(t) − λt, where the parameter λ > 0 is the jump intensity. Moreover, we assume that B(·) and N(·) are independent in this paper.
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 the generator Γ � (c ij ) N×N given by where Δ > 0 and c ij is the transition rate from i to j with c ij > 0 if i ≠ j, while c ii � − j≠i c ij . We suppose that the Markov chain r(·) is independent of B(·) and N(·). As we know in [37], almost every sample path of r(t) is a rightcontinuous step function with a finite number of simple jumps in any finite subinterval of R + . us, there exists a sequence of stopping times 0 � τ 0 < τ 1 < τ 2 < · · · < τ k ⟶ ∞, almost surely such that Hence, r(t) is constant on each interval [τ k , τ k+1 ), In this paper, we consider the nonlinear and nonautonomous hybrid stochastic differential delay equations with Poisson jumps of the form with the initial data Here, f: ey are all Borel-measurable functions.
. .} is a discrete Markov chain with the onestep transition probability matrix en, we impose the standard hypothesis on the initial data.

Assumption 1.
ere exist constants K 1 > 0 and c ∈ (0, 1] such that Since c ij is independent of x, the paths of r could be generated before approximating x. e discrete Markovian chain r Δ k , k � 0, 1, 2, . . . could 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 which is Discrete Dynamics in Nature and Society where we set 0 j�1 P i 0 ,j (Δ) � 0 as usual. en, we generate a new random number ξ 2 independently which is uniformly distributed in [0, 1] as well. Define Repeating this procedure, we could obtain a trajectory of r Δ k , k � 1, 2, . . . . e procedure could 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 truncated theta-EM approximate solution for HSDDEwPJs (4) with the initial data (5).
In order to define the truncated theta-EM scheme, we first choose a strictly decreasing function φ: where K φ(Δ) is a function that depends on φ(Δ). For example, we could choose For a given step size Δ ∈ (0, 1], we give the definition of the truncated mapping where we let (x/|x|) � 0 when x � 0. e truncated functions are defined as Now we give the definition of the discrete-time truncated theta-EM scheme to approximate the true solution of (4). Assume that there exist two positive integers M and M ′ such . . , 0 and then form for It is well known that there exist two kinds of the continuous-time step approximations. e first one is that e other one is that Discrete Dynamics in Nature and Society en, we could observe that . Namely, they coincide at t k . For simplicity, we write

Convergence Rate
To obtain the rate of convergence for the truncated theta-EM method for (4) in L 2 sense, we need to impose the following assumptions on the coefficients.
ere exists a constant K 2 > 0 such that (20) for all t ∈ [0, T], any x, y, x, y ∈ R n , and i ∈ S.
From Assumption 3, we could derive that there exists a constant K 2 > 0 such that where Before presenting the next assumption, we need more notations. Let U be the family of continuous functions for any x, y ∈ R n with |x|∨|y| ≤ b.
for all t ∈ [0, T], any x, y, x, y ∈ R n , and i ∈ S.
e boundedness of the p-moment of the true solution is shown in the following lemma which could be proved by using the standard method. Lemma 2. Let Assumptions 2, 3, and 5 hold. en, for any given initial data (5), there exists a unique solution x(t) to (4) on t ≥ − τ. Moreover, Furthermore, we could obtain the next two lemmas in an analogous way to the proof of Lemmas 2.2 and 2.3 in [32]. 4 Discrete Dynamics in Nature and Society
Proof. Let e Δ (t) � x(t) − y Δ (t) for t ∈ [0, T] and Δ ∈ (0, Δ * ). For simplicity, we rewrite ρ Δ,R � ρ. Recalling the definition of f Δ and g Δ , we have for any 0 ≤ s ≤ t ∧ ρ. By Itô's formula, we could show that Discrete Dynamics in Nature and Society 9 Let q ∈ (2, q), so (62) en, we have 10 Discrete Dynamics in Nature and Society By Assumptions 1 and 4 and Lemma 7, it follows that We derive from Assumption 6 that In addition, let j denote the integer part of T/Δ. us, Discrete Dynamics in Nature and Society where we have utilized the fact that x Δ (s) and x Δ (s − τ) are conditionally independent of I r(s) ≠ r(t k ) { } with reference to the σ-algebra generated by r(t k ).
e application of the Markov property gives From Lemma 6, it follows that Inserting (68) into (65), we get Similar to B 12 , we obtain Combining (63), (64), (69), and (70) together gives By Lemmas 3 and 7, we could show that where the same techniques used in the proofs of B 11 and B 12 have been applied. Similarly, By Assumptions 3 and 7, we obtain that where the skills of estimating B 1 have been used. Inserting (71)-(74) into (61), one can see that us, e use of Gronwall's inequality yields that e proof is completed.

□
Let us now give the rate of convergence in L 2 sense.
By eorem 1, we could get that where c ∈ (0, 1] is defined in Assumption 1. us, the convergence rate of the truncated theta-EM method for (88) is (1 − 4ε)/2∧c. is example shows that our main result could cover a large class of nonlinear and nonautonomous HSDDEwPJs.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.