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

Authors: Weifeng Wang; Lei Yan; Shuaibin Gao; Junhao Hu
Journal: Discrete Dynamics in Nature and Society
 (2021)

---

## 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.


---

## Body

## 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 p∈1,∞, the pth 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 θ=0 and degenerate to the backward EM method when θ=1. 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 L1-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 Section2. The rate of convergence in ℒ2 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. IfA is a vector or matrix, its transpose is denoted by AT. ∀x∈ℝn, let x denote its Euclidean norm. If A is a matrix, its trace norm is denoted by A=traceATA. A≤0 and A<0 mean that A is nonpositive and negative definite, respectively. If a,b are real numbers, then a∧b=mina,b and a∨b=maxa,b. Let a be the largest integer which does not exceed a. Let ℝ+=0,+∞ and τ>0. Let C−τ,0;ℝn be the family of continuous functions ν from −τ,0 to ℝn with the norm ν=sup−τ≤θ˜≤0νθ˜. If H is a set, let IH denote its indicator function which means IHω=1 if ω∈H and IHω=0 if ω∉H. Let C be a generic positive real constant which could be different in different cases.LetΩ,ℱ,ℱtt≥0,ℙ be a complete probability space with a filtration ℱtt≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ0 contains all ℙ-null sets). In addition, let E denote the probability expectation with respect to ℙ. For p>0, let Lℱ0p−τ,0;ℝn denote the family of all ℱ0-measurable and C−τ,0;ℝn-value random variables ξ such that Eξp<∞. Let Bt=B1t,…,BmtT be an m-dimensional Brownian motion defined on the probability space. Let Nt denote a scalar Poisson process with the compensated Poisson process N˜t=Nt−λt, where the parameter λ>0 is the jump intensity. Moreover, we assume that B⋅ and N⋅ are independent in this paper.Letrt (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=γijN×N given by(1)ℙrt+Δ=j|rt=i=γijΔ+oΔ,ifi≠j,1+γiiΔ+oΔ,ifi=j,where Δ>0 and γij is the transition rate from i to j with γij>0 if i≠j, while γii=−∑j≠iγij. We suppose that the Markov chain r⋅ is independent of B⋅ and N⋅. As we know in [37], almost every sample path of rt is a right-continuous step function with a finite number of simple jumps in any finite subinterval of ℝ+. Thus, there exists a sequence of stopping times 0=τ0<τ1<τ2<⋯<τk⟶∞, almost surely such that(2)rt=∑k=0∞rτkIτk,τk+1t.Hence,rt is constant on each interval τk,τk+1,(3)rt=rτk,∀t∈τk,τk+1,k=0,1,2,….In this paper, we consider the nonlinear and nonautonomous hybrid stochastic differential delay equations with Poisson jumps of the form(4)dxt=ft,xt,xt−τ,rtdt+gt,xt,xt−τ,rtdBt+ht,xt,xt−τrtdNt,t∈0,T,with the initial data(5)x0=ξ∈Lℱ0p−τ,0;ℝn,r0=r0∈S.Here,f:ℝ+×ℝn×ℝn×S⟶ℝn, g:ℝ+×ℝn×ℝn×S⟶ℝn×m, h:ℝ+×ℝn×ℝn×S⟶ℝn. 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.
GivenΔ>0, let rkΔ=rkΔ for k≥0. Then, {rkΔ,k=0,1,2,…} is a discrete Markov chain with the one-step transition probability matrix(6)ℙΔ=ℙijΔN×N=eΔΓ.Then, we impose the standard hypothesis on the initial data.Assumption 1.
There exist constantsK1>0 and γ∈0,1 such that(7)ξt−ξs≤K1t−sγ,−τ≤t,s≤0.Sinceγij is independent of x, the paths of r could be generated before approximating x. The discrete Markovian chain rkΔ,k=0,1,2,… could be generated as follows: Compute the one-step transition probability matrix ℙΔ. Let r0Δ=i0, and generate a random number ξ1 which is uniformly distributed in 0,1. Define(8)r1Δ=i1,ifi1∈S−Nsuch that∑j=1i1−1ℙi0,jΔ≤ξ1<∑j=1i1ℙi0,jΔ,N,if∑j=1N−1ℙi0,jΔ≤ξ1,where we set ∑j=10ℙi0,jΔ=0 as usual. Then, we generate a new random number ξ2 independently which is uniformly distributed in 0,1 as well. Define(9)r2Δ=i2,ifi2∈S−Nsuch that∑j=1i2−1ℙr1Δ,jΔ≤ξ2<∑j=1i2ℙr1Δ,jΔ,N,if∑j=1N−1ℙr1Δ,jΔ≤ξ2.Repeating this procedure, we could obtain a trajectory ofrkΔ,k=1,2,…. The procedure could be applied independently to get more trajectories. After generating the discrete Markov chain rkΔ,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 functionφ: 0,1⟶0,∞ such that(10)limΔ⟶0φΔ=∞,KφΔΔ1/4≤1,∀Δ∈0,1,where KφΔ is a function that depends on φΔ. For example, we could choose(11)φΔ=Δ−ε,KφΔ=φΔ,for some ε∈0,1/8.For a given step sizeΔ∈0,1, we give the definition of the truncated mapping(12)πΔx=x∧φΔxx,where we let x/x=0 when x=0. The truncated functions are defined as(13)fΔt,x,y,i=ft,πΔx,πΔy,i=ft,x∧φΔxx,y∧φΔyy,i,gΔt,x,y,i=gt,πΔx,πΔy,i=gt,x∧φΔxx,y∧φΔyy,i.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 that Δ=τ/M=T/M′. Hence, Δ will become sufficiently small when we choose M sufficiently large. Define tk=kΔ for k=−M,−M+1,…,0,1,…,M′−1. Set XΔtk=ξtk for k=−M,−M+1,…,0 and then form(14)XΔtk+1=XΔtk+θfΔtk+1,XΔtk+1,XΔtk+1−M,rk+1ΔΔ+1−θfΔtk,XΔtk,XΔtk−M,rkΔΔ+gΔtk,XΔtk,XΔtk−M,rkΔΔBk+htk,XΔtk,XΔtk−M,rkΔΔNk,for k=0,1,2,…,M′−1, where ΔBk=Btk+1−Btk, ΔNk=Ntk+1−Ntk. To form the continuous-time scheme, define(15)μt=∑k=0M′−1tkItk,tk+1t,r¯t=∑k=0M′−1rkΔItk,tk+1t.It is well known that there exist two kinds of the continuous-time step approximations. The first one is that(16)x¯Δt=∑k=0M′−1XΔtkItk,tk+1t.The other one is that(17)xΔt=ξ0−θfΔ0,ξ0,ξ−τ,r0Δ+θfΔt,xΔt,xΔt−τ,rtΔ+∫0tfΔμs,x¯Δs,x¯Δs−τ,r¯sds+∫0tgΔμs,x¯Δs,x¯Δs−τ,r¯sdBs+∫0thμs,x¯Δs,x¯Δs−τ,r¯sdNs.Then, we could observe thatXΔtk=x¯Δtk=xΔtk. Namely, they coincide at tk. For simplicity, we write(18)yΔt=xΔt−θfΔt,xΔt,xΔt−τ,rtΔ,y¯Δt=x¯Δt−θfΔμt,x¯Δt,x¯Δt−τ,r¯tΔ.


## 3. Convergence Rate

To obtain the rate of convergence for the truncated theta-EM method for (4) in ℒ2 sense, we need to impose the following assumptions on the coefficients.Assumption 2.
For anyR≥1, there exists a constant LR>0 such that(19)ft,x,y,i−ft,x¯,y¯,i∨gt,x,y,i−gt,x¯,y¯,i≤LRx−x¯+y−y¯,for all t∈0,T,i∈S, and any x,y,x¯,y¯∈ℝn with x∨y∨x¯∨y¯≤R.Assumption 3.
There exists a constantK2>0 such that(20)ht,x,y,i−ht,x¯,y¯,i≤K2x−x¯+y−y¯,for all t∈0,T, any x,y,x¯,y¯∈ℝn, and i∈S.From Assumption3, we could derive that there exists a constant K¯2>0 such that(21)ht,x,y,i≤K¯21+x+y,where K¯2=3K2+supt∈0,T,i∈Sht,0,0,i.Before presenting the next assumption, we need more notations. LetU be the family of continuous functions U:ℝn×ℝn⟶ℝ+ such that, for each b>0, there exists a positive constant κb satisfying(22)Ux,y≤κbx−y2,for any x,y∈ℝn with x∨y≤b.Assumption 4.
There exist constantsK3>0,q>2, and U∈U, such that(23)x−x¯Tft,x,y,i−ft,x¯,y¯,i+q−12gt,x,y,i−gt,x¯,y¯,i2≤K3x−x¯2+y−y¯2−Ux,x¯+Uy,y¯,for all t∈0,T, any x,y,x¯,y¯∈ℝn, and i∈S.Assumption 5.
There exist constantsK4>0, p>q>2 such that(24)xTft,x,y,i+p−12gt,x,y,i2≤K41+x2+y2,for all t∈0,T, any x,y∈ℝn, and i∈S.Assumption 6.
There exist constantsL¯R>0, ϑ∈0,1 and σ∈0,1 such that(25)ft1,x,y,i−ft2,x,y,i≤L¯Rt1−t2ϑ,gt1,x,y,i−gt2,x,y,i≤L¯Rt1−t2σ,for all t1,t2∈0,T, i∈S, and any x,y∈ℝn with x∨y≤R.Assumption 7.
There exist constantsK5>0, η∈0,1 such that(26)ht1,x,y,i−ht2,x,y,i≤K51+x+yt1−t2η,for all t1,t2∈0,T, any x,y∈ℝn, and i∈S.The boundedness of thep-moment of the true solution is shown in the following lemma which could be proved by using the standard method.Lemma 2.
Let Assumptions2, 3, and 5 hold. Then, for any given initial data (5), there exists a unique solution xt to (4) on t≥−τ. Moreover,(27)sup−τ≤t≤TExtq<∞,∀T>0.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 Assumption2 holds. For any Δ∈0,1 with KφΔ≥1, we have(28)fΔt,x,y,i−fΔt,x¯,y¯,i∨gΔt,x,y,i−gΔt,x¯,y¯,i≤2KφΔx−x¯+y−y¯,x−x¯TfΔt,x,y,i−fΔt,x¯,y¯,i≤5KφΔ2x−x¯2+y−y¯2,for all t∈0,T, i∈S, and any x,y,x¯,y¯∈ℝn.From Lemma3, we derive that(29)fΔt,x,y,i∨gΔt,x,y,i≤2KφΔx+y+supt∈0,T,i∈Sft,0,0,i+gt,0,0,i.In addition, by the monotone operator theory in [38], 5KφΔ2θΔ<1 needs to be satisfied to ensure the existence and uniqueness of XΔtk+1 for given XΔtk. We get from (10) that KφΔ2Δ1/2≤1; thus, we need θ2Δ<1/25. Moreover, to get the main result, θΔ3/4<1/8 should be satisfied in the proof of Lemma 6. Denote Δ∗=1/25θ2∧1/16θ4/3. Let Δ∈0,Δ∗ and θ∈0,1 in the rest of this paper.Lemma 4.
Let Assumption5 hold. Then, for any Δ∈0,Δ∗ and x,y∈ℝn, we derive that(30)xTfΔt,x,y,i+p−12gΔt,x,y,i2≤K¯41+x2+y2,where K¯4=3K41/φ1∨1.Before the next lemma, define(31)κt=tΔΔ,∀−τ≤t≤T.Lemma 5.
Let Assumptions2 and 3 hold. For any Δ∈0,Δ∗ and t∈0,T, we obtain that(32)EyΔt−y¯Δtp¯|ℱκt≤CΔp¯/2KφΔp¯+Δ1+x¯Δsp¯+x¯Δs−τp¯,p¯≥2,E|yΔt−y¯Δt|p¯|ℱκt≤CΔp¯/2KφΔp¯1+x¯Δsp¯+x¯Δs−τp¯,0<p¯<2.Proof.
Fix anyp¯≥2. By (17), for any Δ∈0,Δ∗ and t∈0,T, we get that(33)yΔt=yΔ0+∫0tfΔμs,x¯Δs,x¯Δs−τ,r¯sds+∫0tgΔμs,x¯Δs,x¯Δs−τ,r¯sdBs+∫0thμs,x¯Δs,x¯Δs−τ,r¯sdNs,where yΔ0=ξ0−θfΔ0,ξ0,ξ−τ,r0Δ.
For anyt∈0,T, there always exists an integer k≥0 such that tk≤t<tk+1. By Hölder’s inequality and BDG’s inequality, we obtain that(34)EyΔt−y¯Δtp¯|ℱκt≤CE∫κttfΔμs,x¯Δs,x¯Δs−τ,r¯sdsp¯|ℱκt+CE∫κttgΔμs,x¯Δs,x¯Δs−τ,r¯sdBsp¯|ℱκt+CE∫κtthμs,x¯Δs,x¯Δs−τ,r¯sdNsp¯|ℱκt≤CΔp¯−1E∫κttfΔμs,x¯Δs,x¯Δs−τ,r¯sp¯ds|ℱκt+CΔp¯/2−1E∫κttgΔμs,x¯Δs,x¯Δs−τ,r¯sp¯ds|ℱκt+CE∫κtthμs,x¯Δs,x¯Δs−τ,r¯sdNsp¯|ℱκt≤CΔp¯/2KφΔp¯1+x¯Δsp¯+x¯Δs−τp¯+CE∫κtthμs,x¯Δs,x¯Δs−τ,r¯sdNsp¯|ℱκt.
By the characteristic function’s argument in [39], for any Δ∈0,Δ∗, we have(35)ENt−Nκtp¯≤c0Δ,where c0 is a constant independent of Δ. Then, we get from Assumption 3 that(36)E∫κtthμs,x¯Δs,x¯Δs−τ,r¯sdNsp¯|ℱκt≤Ehκt,xΔκt,xΔκt−τ,rt/ΔΔNt−Nκtp¯|ℱκt≤hκt,xΔκt,xΔκt−τ,rt/ΔΔp¯ENt−Nκtp¯≤CΔ1+x¯Δsp¯+x¯Δs−τp¯.
Thus,(37)EyΔt−y¯Δtp¯|ℱκt≤CΔp¯/2KφΔp¯+Δ1+x¯Δsp¯+x¯Δs−τp¯.
For0<p¯<2, the use of Jensen’s inequality yields that(38)EyΔt−y¯Δtp¯|ℱκt≤EyΔt−y¯Δt2|ℱκtp¯/2≤CΔp¯/2KφΔp¯1+x¯Δsp¯+x¯Δs−τp¯.
The proof is completed.Lemma 6.
Let Assumptions2, 3, and 5 hold; then, we have(39)sup0<Δ<Δ∗sup0≤t≤TExΔtp≤C,∀T>0.Proof.
By (17), we get that(40)yΔt=yΔ0+∫0tfΔμs,x¯Δs,x¯Δs−τ,r¯sds+∫0tgΔμs,x¯Δs,x¯Δs−τ,r¯sdBs+∫0thμs,x¯Δs,x¯Δs−τ,r¯sdNs,where yΔ0=ξ0−θfΔ0,ξ0,ξ−τ,r0Δ.
Using Itô’s formula, we derive that(41)EyΔtp≤EyΔ0p+E∫0tpyΔsp−2yΔTsfΔμs,x¯Δs,x¯Δs−τ,r¯s+p−12gΔμs,x¯Δs,x¯Δs−τ,r¯s2ds+λE∫0txΔs−hμs,x¯Δs,x¯Δs−τ,r¯sp−xΔspds≤EyΔ0p+E∫0tpyΔsp−2x¯ΔTsfΔμs,x¯Δs,x¯Δs−τ,r¯s+p−12gΔμs,x¯Δs,x¯Δs−τ,r¯s2ds+E∫0tpyΔsp−2yΔs−x¯ΔsTfΔμs,x¯Δs,x¯Δs−τ,r¯sds+λE∫0txΔs−hμs,x¯Δs,x¯Δs−τ,r¯sp−xΔspds≕EyΔ0p+A1+A2+A3.
With the help of (10), Lemma 4, and Young’s inequality, we could obtain(42)A1≤pK¯4E∫0tyΔsp−21+x¯Δs2+x¯Δs−τ2ds≤pK¯4E∫0txΔs−θfΔs,xΔs,xΔs−τ,rsΔp−21+x¯Δs2+x¯Δs−τ2ds≤CE∫0t1+xΔsp+KφΔpΔpxΔsp+xΔs−τp+x¯Δsp+x¯Δs−τpds≤C1+∫0tsup0≤l≤sExΔlpds.
Moreover,(43)A2≤CE∫0ty¯Δsp−2yΔs−x¯ΔsfΔμs,x¯Δs,x¯Δs−τ,r¯sds+CE∫0tyΔs−y¯Δsp−2yΔs−x¯ΔsfΔμs,x¯Δs,x¯Δs−τ,r¯sds≕A21+A22.
By (10), Young’s inequality, and Lemmas 3 and 5, we could show that(44)A21≤C∫0tEy¯Δsp−2fΔμs,x¯Δs,x¯Δs−τ,r¯s⋅EyΔs−y¯Δs−θfΔμs,x¯Δs,x¯Δs−τ,r¯sΔ|ℱκsds≤C∫0tEy¯Δsp−2fΔμs,x¯Δs,x¯Δs−τ,r¯s⋅EyΔs−y¯Δs|ℱκsds+C∫0tEy¯Δsp−2fΔμs,x¯Δs,x¯Δs−τ,r¯s2Δds≤C∫0tΔ1/2KφΔ2ExΔs−θfΔs,xΔs,xΔs−τ,rsΔp−2⋅1+x¯Δs2+x¯Δs−τ2ds+C∫0tΔExΔs−θfΔs,xΔs,xΔs−τ,rsΔp−2ds+C∫0tΔKφΔ2ExΔs−θfΔs,xΔs,xΔs−τ,rsΔp−2⋅1+x¯Δs2+x¯Δs−τ2ds≤C1+∫0tsup0≤l≤sExΔlpds.
In the same way, with (10) and Lemmas 3 and 5, we have(45)A22≤C1+∫0tsup0≤l≤sExΔlpds.
By inserting (44) and (45) into (43), it follows that(46)A2≤C1+∫0tsup0≤l≤sExΔlpds.
From Assumption3, one can see that(47)A3≤CE∫0t1+xΔsp+x¯Δsp+x¯Δs−τpds≤C1+∫0tsup0≤l≤sExΔlpds.
Combining (41), (42), (46) , and (47) together, we obtain(48)sup0≤l≤tEyΔlp≤C1+∫0tsup0≤l≤sExΔlpds.
By the inequalityu−vp≥21−pup−vp and (29), we get(49)yΔtp≥21−pxΔtp−θpΔpfΔt,xΔt,xΔt−τ,rtp≥21−pxΔtp−22p−2θpΔpKφΔpxΔtp+xΔt−τp−2p−1θpΔpsupt∈0,T,i∈Sft,0,0,i+gt,0,0,i=21−p−22p−2θpΔpKφΔpxΔtp−22p−2θpΔpKφΔpxΔt−τp−2p−1θpΔpsupt∈0,T,i∈Sft,0,0,i+gt,0,0,i.
Note thatΔ<1/16θ4/3; hence,(50)sup0≤l≤tExΔlp≤C1+∫0tsup0≤l≤sEyΔlpds≤C1+∫0tsup0≤l≤sExΔlpds.
Applying Gronwall’s inequality gives the desired result.Lemma 7.
Let Assumptions2, 3, and 5 hold. For any Δ∈0,Δ∗ and t∈0,T, we derive that(51)EyΔt−y¯Δtp≤CΔp/2KφΔp+Δ,p≥2,(52)EyΔt−y¯Δtp≤CΔp/2KφΔp,0<p<2,(53)ExΔt−x¯Δtp≤CΔp/2KφΔp+Δ,p≥2,(54)ExΔt−x¯Δtp≤CΔp/2KφΔp,0<p<2.
Hence,(55)limΔ⟶0EyΔt−y¯Δtp=limΔ⟶0ExΔt−x¯Δtp=0,∀p>0.Proof.
By Lemmas5 and 6, we could obtain (51) and (52). Then, the use of a similar technique in Lemma 6 gives the following when p≥2:(56)ExΔt−x¯Δtp≤CEyΔt−y¯Δtp≤CΔp/2KφΔp+Δ.
We could get (54) by applying Jensen’s inequality.By Lemmas2 and 6 and Chebyshev’s inequality, we could get the following lemma right away.Lemma 8.
Let Assumptions2, 3, and 5 hold. For any number R≥ξ, define the stopping time(57)τR=inft≥0:xt≥R,τΔ,R=inft≥0:xΔt≥R.
Then, we obtain that(58)ℙτR≤T≤CR2,ℙτΔ,R≤T≤CR2.Lemma 9.
Let Assumptions1–7 hold. Let Δ∈0,Δ∗ be sufficiently small such that φΔ≥R∨ξ. Then, we get(59)ExT∧ρΔ,R−xΔT∧ρΔ,R2≤CKφΔ4Δ∨Δ2γ∧ϑ∧σ∧η,where ρΔ,R:=τR∧τΔ,R.Proof.
LeteΔt=xt−yΔt for t∈0,T and Δ∈0,Δ∗. For simplicity, we rewrite ρΔ,R=ρ. Recalling the definition of fΔ and gΔ, we have(60)fΔμs,x¯Δs,x¯Δs−τ,r¯s=fμs,x¯Δs,x¯Δs−τ,r¯s,gΔμs,x¯Δs,x¯Δs−τ,r¯s=gμs,x¯Δs,x¯Δs−τ,r¯s,for any 0≤s≤t∧ρ. By Itô’s formula, we could show that(61)EeΔt∧ρ2≤Eθ2Δ2fΔ0,ξ0,ξ−τ,r02+E∫0t∧ρ2eΔTsfs,xs,xs−τ,rs−fΔμs,x¯Δs,x¯Δs−τ,r¯s+gs,xs,xs−τ,rs−gΔμs,x¯Δs,x¯Δs−τ,r¯s2ds+λE∫0t∧ρ|eΔs+hs,xs,xs−τ,rs−hμs,x¯Δs,x¯Δs−τ,r¯s|2−eΔs2ds≤CKφΔ2Δ2+E∫0t∧ρ2xs−x¯ΔsT⋅fs,xs,xs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯s+gs,xs,xs−τ,rs−gμs,x¯Δs,x¯Δs−τ,r¯s2ds+E∫0t∧ρ2x¯Δs−xΔsT⋅fs,xs,xs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯sds+E∫0t∧ρ2θΔfΔTs,xΔs,xΔs−τ,rs⋅fs,xs,xs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯sds+λE∫0t∧ρ|eΔs+hs,xs,xs−τ,rs−hμs,x¯Δs,x¯Δs−τ,r¯s|2−eΔs2ds≕CKφΔ2Δ2+B1+B2+B3+B4.
Letq¯∈2,q, so(62)12gs,xs,xs−τ,rs−gμs,x¯Δs,x¯Δs−τ,r¯s2≤q¯−12gs,xs,xs−τ,rs−gs,x¯Δs,x¯Δs−τ,rs2+q¯−12q¯−2gs,x¯Δs,x¯Δs−τ,rs−gμs,x¯Δs,x¯Δs−τ,r¯s2.
Then, we have(63)B1≤2E∫0t∧ρxs−x¯ΔsT⋅fs,xs,xs−τ,rs−fs,x¯Δs,x¯Δs−τ,rs+q¯−12gs,xs,xs−τ,rs−gs,x¯Δs,x¯Δs−τ,rs2ds+2E∫0t∧ρxs−x¯ΔsT⋅fs,x¯Δs,x¯Δs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯sds+E∫0t∧ρq¯−1q¯−2gs,x¯Δs,x¯Δs−τ,rs−gμs,x¯Δs,x¯Δs−τ,r¯s2ds≕B11+B12+B13.
By Assumptions1 and 4 and Lemma 7, it follows that(64)B11≤2E∫0t∧ρK3xs−x¯Δs2+xs−τ−x¯Δs−τ2ds+2E∫0t∧ρ−Uxs,x¯Δs+Uxs−τ,x¯Δs−τds≤4K3E∫0t∧ρxs−xΔs2+xΔs−x¯Δs2+xs−τ−xΔs−τ2+xΔs−τ−x¯Δs−τ2ds+2∫−τ0Uξs,ξκsds≤CE∫0t∧ρxs−xΔs2ds+CE∫0TKφΔ2Δds+C∫−τ0ξs−ξκs2ds≤C∫0tExs∧ρ−xΔs∧ρ2ds+KφΔ2Δ+Δ2γ.
We derive from Assumption6 that(65)B12≤E∫0t∧ρxs−x¯Δs2+fs,x¯Δs,x¯Δs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯s2ds≤2E∫0t∧ρxs−xΔs2+xΔs−x¯Δs2ds+2E∫0t∧ρfs,x¯Δs,x¯Δs−τ,rs−fμs,x¯Δs,x¯Δs−τ,rs2+fμs,x¯Δs,x¯Δs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯s2ds≤C∫0tExs∧ρ−xΔs∧ρ2ds+CKφΔ2Δ+CΔ2ϑ+CE∫0t∧ρfμs,x¯Δs,x¯Δs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯s2ds.
In addition, letj denote the integer part of T/Δ. Thus,(66)E∫0t∧ρfμs,x¯Δs,x¯Δs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯s2ds=∑k=0jE∫tktk+1fμs,x¯Δs,x¯Δs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯s2I0,t∧ρsds≤2∑k=0jE∫tktk+1fμs,x¯Δs,x¯Δs−τ,rs2+fμs,x¯Δs,x¯Δs−τ,r¯s2I0,t∧ρsIrs≠rtkds≤C∑k=0jE∫tktk+1EE1+KφΔ2x¯Δs2+x¯Δs−τ2⋅Irs≠rtk|rtkds,where we have utilized the fact that x¯Δs and x¯Δs−τ are conditionally independent of Irs≠rtk with reference to the σ-algebra generated by rtk. The application of the Markov property gives(67)EIrs≠rtk|rtk=∑i∈SIrtk=iℙrs≠i|rtk=i≤max0≤i≤N−γiiΔ+oΔ∑i∈SIrtk=i≤CΔ+oΔ.
From Lemma6, it follows that(68)E∫0t∧ρfμs,x¯Δs,x¯Δs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯s2ds≤CΔ+oΔ∑k=0j∫tktk+11+KφΔ2Ex¯Δs2+KφΔ2Ex¯Δs−τ2ds≤KφΔ2CΔ+oΔ.
Inserting (68) into (65), we get(69)B12≤C∫0tExs∧ρ−xΔs∧ρ2ds+KφΔ2Δ+Δ2ϑ+oΔ.
Similar toB12, we obtain(70)B13≤CKφΔ2Δ+Δ2σ+oΔ.
Combining (63), (64), (69), and (70) together gives(71)B1≤C∫0tExs∧ρ−xΔs∧ρ2ds+KφΔ2Δ+Δ2γ+Δ2ϑ+Δ2σ+oΔ.
By Lemmas3 and 7, we could show that(72)B2≤4E∫0t∧ρKφΔxΔs−x¯Δs⋅xs−x¯Δs+xs−τ−x¯Δs−τds≤2E∫0t∧ρKφΔ2xΔs−x¯Δs2ds+2E∫0t∧ρ|xs−x¯Δs|+xs−τ−x¯Δs−τds≤C∫0tExs∧ρ−xΔs∧ρ2ds+KφΔ4Δ+KφΔ2Δ+Δ2γ+Δ2ϑ+oΔ,where the same techniques used in the proofs of B11 and B12 have been applied. Similarly,(73)B3≤E∫0t∧ρθ2Δ2fΔs,xΔs,xΔs−τ,rs2ds+E∫0t∧ρfs,xs,xs−τ,rs−fμs,x¯Δs,x¯Δs−τ,r¯s2ds≤CKφΔ2Δ2+KφΔ2Δ+Δ2γ+Δ2ϑ+oΔ.
By Assumptions3 and 7, we obtain that(74)B4≤λE∫0t∧ρeΔs2+2hs,xs,xs−τ,rs−hμs,x¯Δs,x¯Δs−τ,r¯s2ds≤2λE∫0t∧ρxs−xΔs2+θ2Δ2fΔs,xΔs,xΔs−τ,rs2ds+2λE∫0t∧ρhs,xs,xs−τ,rs−hs,x¯Δs,x¯Δs−τ,rs2ds+2λE∫0t∧ρhs,x¯Δs,x¯Δs−τ,rs−hμs,x¯Δs,x¯Δs−τ,rs2ds+2λE∫0t∧ρhμs,x¯Δs,x¯Δs−τ,rs−hμs,x¯Δs,x¯Δs−τ,r¯s2ds≤C∫0tExs∧ρ−xΔs∧ρ2ds+KφΔ2Δ2+KφΔ2Δ+Δ2γ+Δ2η+oΔ,where the skills of estimating B1 have been used. Inserting (71)–(74) into (61), one can see that(75)EeΔt∧ρ2≤C∫0tExs∧ρ−xΔs∧ρ2ds+KφΔ4Δ+Δ2γ+Δ2η+Δ2ϑ+Δ2σ+oΔ.
Thus,(76)Ext∧ρ−xΔt∧ρ2ds≤EeΔt∧ρ2+EθΔfΔt,xΔt,xΔt−τ,rt2≤C∫0tExs∧ρ−xΔs∧ρ2ds+KφΔ4Δ∨Δ2γ∧ϑ∧σ∧η.
The use of Gronwall’s inequality yields that(77)ExT∧ρ−xΔT∧ρ2≤CKφΔ4Δ∨Δ2γ∧ϑ∧σ∧η.
The proof is completed.Let us now give the rate of convergence inℒ2 sense.Theorem 1.
Let Assumptions 1–7 hold. For any sufficiently smallΔ∈0,Δ∗, we assume there exists a positive constant c∗ such that(78)φΔ≥c∗KφΔ4Δ∨Δ2γ∧ϑ∧σ∧η−p/2p−2.
Then, for anyT>0, we have(79)ExT−xΔT2≤CKφΔ4Δ∨Δ2γ∧ϑ∧σ∧η,(80)ExT−x¯ΔT2≤CKφΔ4Δ∨Δ2γ∧ϑ∧σ∧η.Proof.
Letδ>0 be arbitrary. By Young’s inequality, we obtain(81)u2v=δup2/pvp/p−2δ2/p−2p−2/p≤2δpup+p−2pδ2/p−2vp/p−2,∀u,v>0.
Therefore,(82)ExT−xΔT2=ExT−xΔT2Iρ>T+ExT−xΔT2Iρ≤T≤ExT∧ρ−xΔT∧ρ2+2δpExT−xΔTp+p−2pδ2/p−2ℙρ≤T.
By Lemmas2 and 6, we get(83)ExT−xΔTp≤C.
From Lemma8, it follows that(84)ℙρ≤T≤ℙτR≤T+ℙτΔ,R≤T≤CR2.
We can chooseδ=KφΔ4Δ∨Δ2γ∧ϑ∧σ∧η and R=c∗KφΔ4Δ∨Δ2γ∧ϑ∧σ∧η−p/2p−2 such that(85)ExT−xΔT2≤ExT∧ρ−xΔT∧ρ2+CKφΔ4Δ∨Δ2γ∧ϑ∧σ∧η.
With condition (78), we obtain(86)φΔ≥c∗KφΔ4Δ∨Δ2γ∧ϑ∧σ∧η−p/2p−2=R.
From Lemma9 and (85), we derive that(87)ExT−xΔT2≤CKφΔ4Δ∨Δ2γ∧ϑ∧σ∧η.
Then, combining Lemma7 and (79) gives (80). We complete the proof.


## 4. Example

An example is presented to illustrate our theory in this section.Example 1.
Consider the following nonlinear and nonautonomous scalar hybrid stochastic differential delay equation with Poisson jumps:(88)dxt=ft,xt,xt−τ,rtdt+gt,xt,xt−τ,rtdBt+ht,xt,xt−τ,rtdNt,with the initial value x0 which satisfies Assumption 1. Bt is a scalar Brownian motion, and Nt is a scalar Poisson process with intensity λ=0.4. Furthermore, rt is a Markovian chain defined on the state space S=1,2 with generator(89)Γ=−221−1.
Moreover, for anyt∈0,2 and x,y∈ℝ, let(90)ft,x,y,i=−5x3+3xsint+18t2−t1/3y5/4,ifi=1,−20x5+6xcost+18t2−t1/4y5/4,ifi=2,gt,x,y,i=12t2−t1/3x3/2,ifi=1,t2−t1/4x5/2,ifi=2,ht,x,y,i=t2−t1/5x,ifi=1,t2−t1/5y,ifi=2.
Obviously, Assumptions2 and 3 hold with LR=20R4,K2=1. Moreover, when i=1,(91)x−x¯Tft,x,y,1−ft,x¯,y¯,1≤x−x¯2−5x2+xx¯+x¯2+3x−x¯2+18x−x¯y5/4−y¯5/4≤−52x−x¯2x2+x¯2+4x−x¯2+25256y−y¯2y1/4−y¯1/42≤−52x−x¯2x2+x¯2+4x−x¯2+2564y−y¯2+25128y−y¯2y2+y¯2.
Letq=3. In a similar way, we have(92)q−12gt,x,y,1−gt,x¯,y¯,12≤12x3/2−x¯3/22≤98x−x¯2x1/2+x¯1/22≤92x−x¯2+94x−x¯2x2+x¯2.
Thus,(93)x−x¯Tft,x,y,1−ft,x¯,y¯,1+q−12gt,x,y,1−gt,x¯,y¯,12≤9x−x¯2+y−y¯2−14x−x¯2x2+x¯2+14y−y¯2y2+y¯2.
Similarly, fori=2, we have(94)x−x¯Tft,x,y,2−ft,x¯,y¯,2≤x−x¯2−20x4+x3x¯+x2x¯2+xx¯3+x¯4+6x−x¯2+18x−x¯y5/4−y¯5/4≤−10x−x¯2x4+x¯4+7x−x¯2+25256y−y¯2y1/4+y¯1/42≤−10x−x¯2x4+x¯4+7x−x¯2+2564y−y¯2+25128y−y¯2y4+y¯4,q−12gt,x,y,2−gt,x¯,y¯,22≤12x5/2−x¯5/22≤258x−x¯2x1/2+x¯1/22≤252x−x¯2+254x−x¯2x4+x¯4.
Therefore,(95)x−x¯Tft,x,y,2−ft,x¯,y¯,2+q−12gt,x,y,2−gt,x¯,y¯,22≤20x−x¯2+y−y¯2−754x−x¯2x4+x¯4+754y−y¯2y4+y¯4.
Hence, Assumption4 is satisfied for any i∈S. For Assumption 5, by letting p=4, we obtain that(96)xTft,x,y,1+p−12gt,x,y,12≤−5x4+3x2+18xy5/4+38x3≤C1+x2+y2,xTft,x,y,2+p−12gt,x,y,22≤−20x6+6x2+18xy5/4+32x5≤C1+x2+y2.
Thus, Assumption5 is satisfied. Then, it is easy to see that Assumptions 6 and 7 hold with ϑ=σ=1/4,η=1/5.
For anyε∈0,1/4, define φΔ=41/20Δ−ε/4. Therefore, when KφΔ=20φΔ4, we have KφΔΔ1/4=Δ1/4−ε≤1 and limΔ⟶0φΔ=∞. Choose p=4,ε=33/136,c∗=41/20; then, we derive that(97)1204KφΔ4Δ−p/2p−2=1204Δ4ε−1≤1204Δ−ε/4=φΔ.
By Theorem1, we could get that(98)ExT−xΔT2≤CΔ1−4ε∧2γ,where γ∈0,1 is defined in Assumption 1. Thus, the convergence rate of the truncated theta-EM method for (88) is 1−4ε/2∧γ. This example shows that our main result could cover a large class of nonlinear and nonautonomous HSDDEwPJs.



---

*Source: 2882076-2021-06-10.xml*