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

Abstract

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.

1. Introduction

Stochastic differential equations have been widely used in many fields and have attracted many scholars [1–3]. 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. Therefore, stochastic differential equations with jumps considering continuous and discontinuous random effects have been investigated to analyze these situations [4–7]. 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 , the th moment of the EM approximations diverges to infinity. Therefore, many implicit methods have been proposed to approximate the solutions of the stochastic differential equations with nonlinear growing coefficients [11–13]. 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 [14–16]. In particular, the truncated EM method was initialized by Mao in [17] with both the drift and diffusion coefficients growing superlinearly. The 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 [19–25]. In addition, there are many papers which consider the stability of the systems [26–30]. The 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. The truncated theta-EM method will become the EM method when and degenerate to the backward EM method when . Additionally, there are a few results on the numerical solutions for HSDDEwPJs. The convergence of EM approximation solution to the true solution in probability under some weaker conditions was proved in [33]. The EM approximate solutions converge to the true solutions for stochastic differential delay equations with Poisson jumps and Markovian switching under local Lipschitz condition [34]. The convergence of EM method for stochastic differential delay equations with Poisson jumps and Markovian switching in the sense of -norm under one non-Lipschitz condition was discussed in [35]. The 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. Thus, in this paper, we will give the strong convergence rate of the truncated theta-EM method for nonlinear and nonautonomous HSDDEwPJs.

This paper is organized as follows. We will introduce some necessary notations in Section 2. The rate of convergence in 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.

2. Mathematical Preliminaries

Throughout this paper, unless otherwise specified, we will use the following notations. If is a vector or matrix, its transpose is denoted by . , let denote its Euclidean norm. If is a matrix, its trace norm is denoted by . and mean that is nonpositive and negative definite, respectively. If are real numbers, then and . Let be the largest integer which does not exceed . Let and . Let be the family of continuous functions from to with the norm . If is a set, let denote its indicator function which means if and if . Let be a generic positive real constant which could be different in different cases.

Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). In addition, let denote the probability expectation with respect to . For , let denote the family of all -measurable and -value random variables such that . Let be an -dimensional Brownian motion defined on the probability space. Let denote a scalar Poisson process with the compensated Poisson process , where the parameter is the jump intensity. Moreover, we assume that and are independent in this paper.

Let () be a right-continuous Markov chain on the probability space taking values in a finite state space with the generator given bywhere and is the transition rate from to with if , while . We suppose that the Markov chain is independent of and . As we know in [37], almost every sample path of is a right-continuous step function with a finite number of simple jumps in any finite subinterval of . Thus, there exists a sequence of stopping times , almost surely such that

Hence, is constant on each interval ,

In this paper, we consider the nonlinear and nonautonomous hybrid stochastic differential delay equations with Poisson jumps of the formwith the initial data

Here, , , . They are all Borel-measurable functions.

To estimate the truncated theta-EM method for (4), we need the following lemma ([8], Theorem 1.44).

Lemma 1. Given , let for . Then, {} is a discrete Markov chain with the one-step transition probability matrix

Then, we impose the standard hypothesis on the initial data.

Assumption 1. There exist constants and such that

Since is independent of , the paths of could be generated before approximating . The discrete Markovian chain could be generated as follows: Compute the one-step transition probability matrix . Let , and generate a random number which is uniformly distributed in . Definewhere we set as usual. Then, we generate a new random number independently which is uniformly distributed in as well. Define

Repeating this procedure, we could obtain a trajectory of . The procedure could be applied independently to get more trajectories. After generating the discrete Markov chain , 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 : such thatwhere is a function that depends on . For example, we could choosefor some .

For a given step size , we give the definition of the truncated mappingwhere we let when . The 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 and such that . Hence, will become sufficiently small when we choose sufficiently large. Define for . Set for and then formfor , where , . To form the continuous-time scheme, define

It is well known that there exist two kinds of the continuous-time step approximations. The first one is that

The other one is that

Then, we could observe that . Namely, they coincide at . For simplicity, we write

3. Convergence Rate

To obtain the rate of convergence for the truncated theta-EM method for (4) in sense, we need to impose the following assumptions on the coefficients.

Assumption 2. For any , there exists a constant such thatfor all , and any with .

Assumption 3. There exists a constant such thatfor all , any , and .

From Assumption 3, we could derive that there exists a constant such thatwhere .

Before presenting the next assumption, we need more notations. Let be the family of continuous functions such that, for each , there exists a positive constant satisfyingfor any with .

Assumption 4. There exist constants , and , such thatfor all , any , and .

Assumption 5. There exist constants , such thatfor all , any , and .

Assumption 6. There exist constants , and such thatfor all , , and any with .

Assumption 7. There exist constants , such thatfor all , any , and .

The boundedness of the -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. Then, for any given initial data (5), there exists a unique solution to (4) on . 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].

Lemma 3. Let Assumption 2 holds. For any with , we havefor all , , and any .

From Lemma 3, we derive that

In addition, by the monotone operator theory in [38], needs to be satisfied to ensure the existence and uniqueness of for given . We get from (10) that ; thus, we need . Moreover, to get the main result, should be satisfied in the proof of Lemma 6. Denote . Let and in the rest of this paper.

Lemma 4. Let Assumption 5 hold. Then, for any and , we derive thatwhere .

Before the next lemma, define

Lemma 5. Let Assumptions 2 and 3 hold. For any and , we obtain that

Proof. Fix any . By (17), for any and , we get thatwhere .
For any , there always exists an integer such that . By Hölder’s inequality and BDG’s inequality, we obtain thatBy the characteristic function’s argument in [39], for any , we havewhere is a constant independent of . Then, we get from Assumption 3 thatThus,For , the use of Jensen’s inequality yields thatThe proof is completed.

Lemma 6. Let Assumptions 2, 3, and 5 hold; then, we have

Proof. By (17), we get thatwhere .
Using Itô’s formula, we derive thatWith the help of (10), Lemma 4, and Young’s inequality, we could obtainMoreover,By (10), Young’s inequality, and Lemmas 3 and 5, we could show thatIn the same way, with (10) and Lemmas 3 and 5, we haveBy inserting (44) and (45) into (43), it follows thatFrom Assumption 3, one can see thatCombining (41), (42), (46) , and (47) together, we obtainBy the inequality and (29), we getNote that ; hence,Applying Gronwall’s inequality gives the desired result.

Lemma 7. Let Assumptions 2, 3, and 5 hold. For any and , we derive thatHence,

Proof. By Lemmas 5 and 6, we could obtain (51) and (52). Then, the use of a similar technique in Lemma 6 gives the following when :We could get (54) by applying Jensen’s inequality.

By Lemmas 2 and 6 and Chebyshev’s inequality, we could get the following lemma right away.

Lemma 8. Let Assumptions 2, 3, and 5 hold. For any number , define the stopping timeThen, we obtain that

Lemma 9. Let Assumptions 1–7 hold. Let be sufficiently small such that . Then, we getwhere .

Proof. Let for and . For simplicity, we rewrite . Recalling the definition of and , we have