Analytic Approximation of the Solutions of Stochastic Differential Delay Equations with Poisson Jump and Markovian Switching

Abstract

We are concerned with the stochastic differential delay equations with Poisson jump and Markovian switching (SDDEsPJMSs). Most SDDEsPJMSs cannot be solved explicitly as stochastic differential equations. Therefore, numerical solutions have become an important issue in the study of SDDEsPJMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJMSs when the drift and diffusion coefficients are Taylor approximations.

1. Introduction

Recently there has been an increasing interest in the study of stochastic differential delay equations with Poisson jump and Markovian switching (SDDEsPJMSs). Svīshchuk and Kazmerchuk [1] investigated stability of stochastic differential delay equations with linear Poisson jumps and Markovian switchings. While Luo [2] discussed the comparison principle and several kinds of stability of Itô stochastic differential delay equations with Poisson jump and Markovian switching. Besides, Li and Chang [3] discussed the convergence of the numerical solutions of stochastic differential delay equations with Poisson jump and Markovian switching. In the present paper we will further research this topic and our focus is on the convergence of numerical solution to stochastic differential delay equation with Poisson jump and Markovian switching when the coefficients are Taylor approximations.

Stochastic differential delay equation with Poisson jump and Markovian switching may be considered as extension of stochastic differential delay equation with Poisson jump. Of course, it may also be regarded as an generalization of stochastic differential delay equation with Markovian switching. Similar to stochastic differential delay equations with Poisson jump, explicit solutions can hardly be obtained for the stochastic differential delay equations with Poisson jump and Markovian switching. Thus appropriate numerical approximation schemes such as the Euler (or Euler-Maruyama) are needed if we apply them in practice or to study their properties. There is an extensive literature concerning the approximate schemes for either stochastic differential delay equations with Poisson jump or stochastic differential delay equations with Markovian switching [4–12].

However, the rate of convergence to the true solution by the numerical solution is different for different numerical schemes [13]. Recently, Janković and Ilić [14] have investigated the rate of convergence between the true solution and numerical solution of the stochastic differential equations in the sense of the 𝐿^𝑝-norm when the drift and diffusion coefficients are Taylor approximations, up to arbitrary fixed derivatives. Moreover, Jiang et al. [15] generalized the Taylor method to stochastic differential delay equations with Poisson jump. Since the rate of convergence for such a numerical method is faster than the result obtained in [13], in this paper, we intend to generalize the above method to the SDDEsPJMSs case and consider the strong convergence between the true solution and numerical; solution to SDDEsPJMSs if the drift and diffusion coefficients are Taylor approximations, up to arbitrary fixed derivatives. To the best of our knowledge, so far there seem to be no existing results. Therefore, the aim of this paper is to close this gap.

In Section 2, we introduce necessary notations and approximation scheme. Then comes our main result that the Taylor approximate solutions will converge to the true solutions of SDDEsPJMSs. The proof of this main result is rather technical so we present several lemmas in Section 3 and then complete the proof in Section 4.

2. Approximation Scheme and Hypotheses

Throughout this paper, we 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 ℱ₀ contains all 𝑃-null sets). Let 𝐶([−𝜏,0];𝑅) be the family of continuous function 𝜉 from [−𝜏,0] to 𝑅 with the norm ‖𝜉‖=sup_{−𝜏≤𝑡≤0}|𝜉(𝑡)|. We also denote by 𝐶^𝑏_ℱ0([−𝜏,0];𝑅) the family of all bounded, ℱ₀-measurable, 𝐶([−𝜏,0];𝑅)-valued random variables and denote by 𝐿^𝑝_ℱ𝑡([−𝜏,0];𝑅) the family of all ℱ_𝑡-measurable, 𝐶([−𝜏,0];𝑅)-valued random variables 𝜉={𝜉(𝑡)∶−𝜏≤𝑡≤0} satisfying sup_{−𝜏≤𝑡≤0}𝐸|𝜉(𝑡)|^𝑝<𝜅<∞, where 𝜅 is a positive constant.

Let 𝑤(𝑡),𝑡≥0, be a one-dimensional Brownian motion defined on the probability space and let 𝑁(𝑡),𝑡≥0, be a scalar Poisson process with intensity 𝜆 which is independent of 𝑤(𝑡). 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=(𝛾_𝑖𝑗)_𝑁×𝑁 given by 𝑃𝑟()(𝑡)=𝛾𝑡+𝛿=𝑗∣𝑟=𝑖^𝑖𝑗𝛿+𝑜(𝛿),if𝑖≠𝑗,1+𝛾_𝑖𝑗𝛿+𝑜(𝛿),if𝑖=𝑗, (2.1) where 𝛿>0. Here 𝛾_𝑖𝑗≥0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 while 𝛾_𝑖𝑖=−∑_𝑖≠𝑗𝛾_𝑖𝑗. We assume that the Markov chain 𝑟(⋅) is independent of the Brownian motion 𝑤(⋅) and Poisson process 𝑁(𝑡). It is well known that almost every sample path of 𝑟(⋅) is a right-continuous step function with finite number of simple jumps in any finite subinterval of ℜ₊=[0,∞).

Consider stochastic differential delay equations with Poisson jump and Markovian switching of the form(𝑡)((𝑡)()(𝑡((𝑡)()(𝑡(𝑡)((𝑡)()(𝑡(𝑡),𝑑𝑥=𝑓𝑡,𝑥,𝑥𝑡−𝜏,𝑟))𝑑𝑡+𝑔𝑡,𝑥,𝑥𝑡−𝜏,𝑟))𝑑𝑤+ℎ𝑡,𝑥,𝑥𝑡−𝜏,𝑟))𝑑𝑁 (2.2) on the time interval [0,𝑇]  with initial data 𝜉(𝑡)∈𝐶^𝑏_ℱ0([−𝜏,0];𝑅) being independent of 𝑤(𝑡) and 𝑁(𝑡), and 𝑟(0)=𝑖₀∈𝑆, where 𝑓∶[0,∞)×𝑅×𝑅×𝑆⟶𝑅,𝑔∶[0,∞)×𝑅×𝑅×𝑆⟶𝑅,ℎ∶[0,∞)×𝑅×𝑅×𝑆→𝑅. (2.3)

In the paper, 𝑓_𝑥^(𝑗),𝑔_𝑥^(𝑗), and ℎ_𝑥^(𝑗) denote, respectively, 𝑗th-order partial derivatives of 𝑓,𝑔 and ℎ with respect to 𝑥. Further, to ensure the existence and uniqueness of the solution to (2.2), we impose the following hypotheses.

  (H₁) 𝑓,𝑔, and ℎ satisfy Lipschitz and linear growth condition; that is, there exists a positive constant 𝐿>0 such that 𝑓𝑡,𝑥₁,𝑥₂,𝑖−𝑓𝑡,𝑦₁,𝑦₂,𝑖∨𝑔𝑡,𝑥₁,𝑥₂,𝑖−𝑔𝑡,𝑦₁,𝑦₂∨ℎ,𝑖𝑡,𝑥₁,𝑥₂,𝑖−ℎ𝑡,𝑦₁,𝑦₂,𝑖≤𝐿𝑥₁−𝑦₁²+𝑥₂−𝑦₂²,𝑓𝑡,𝑥₁,𝑥₂,𝑖²∨𝑔𝑡,𝑥₁,𝑥₂,𝑖²∨ℎ𝑡,𝑥₁,𝑥₂,𝑖²≤𝐿²1+𝑥₁²+𝑥₂², (2.4)

for 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝑅 and 𝑖∈𝑆.

  (H₂) There exist constants 𝐾₁>0 and 𝛾∈(0,1] such that, for all −𝜏≤𝑠<𝑡≤0 and 𝜌≥2,𝐸𝜉(𝑡)−𝜉(𝑠)^𝜌≤𝐾₁(𝑡−𝑠)^𝛾. (2.5)

  (H₃) 𝑓,𝑔, and ℎ have Taylor approximations in the second argument, up to 𝑚₁th, 𝑚₂th, and 𝑚₃th derivatives, respectively.

  (H₄) Partial derivatives of the order 𝑚₁+1, 𝑚₂+1, and 𝑚₃+1 of the functions 𝑓, 𝑔, and ℎ, 𝑓^(𝑚1_𝑥⁺¹⁾(𝑡,𝑥,𝑦,𝑖),𝑔^(𝑚2_𝑥⁺¹⁾(𝑡,𝑥,𝑦,𝑖), and ℎ^(𝑚3_𝑥⁺¹⁾(𝑡,𝑥,𝑦,𝑖), are uniformly bounded; that is, there exist positive constants 𝐿₁,𝐿₂, and 𝐿₃ obeying sup_{[]0,𝑇×𝑅×𝑅×𝑆}𝑓^(𝑚1_𝑥⁺¹⁾𝑡,𝑥,𝑦,𝑖≤𝐿₁,sup_{[]0,𝑇×𝑅×𝑅×𝑆}𝑔^(𝑚2_𝑥⁺¹⁾𝑡,𝑥,𝑦,𝑖≤𝐿₂,sup_{[]0,𝑇×𝑅×R×𝑆}ℎ^(𝑚3_𝑥⁺¹⁾𝑡,𝑥,𝑦,𝑖≤𝐿₃. (2.6)

For some sufficiently large integer 𝑀, we define the time step by ℎ=𝜏/𝑀, where 0<ℎ≪1. Then, the approximation solution to (2.2) is computed by 𝑦(𝑡)=𝜉(𝑡) on −𝜏≤𝑡≤0, and for any 𝑡≥0,𝑦(𝑡)=𝜉(0)+^𝑡₀^𝑚1_𝑗=0𝑓_𝑥^(𝑗)𝑠,𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑦(𝑠)𝑗!−𝑧₁(𝑠)^𝑗+𝑑𝑠_{𝑡0𝑚2𝑗=0}𝑔_𝑥^(𝑗)𝑠,𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑦(𝑠)𝑗!−𝑧₁(𝑠)^𝑗𝑑𝑤(𝑠)+^𝑡₀^𝑚3_𝑗=0ℎ_𝑥^(𝑗)𝑠,𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑦(𝑠)𝑗!−𝑧₁(𝑠)^𝑗𝑑𝑁(𝑠), (2.7) where, for 𝑡_𝑘=𝑘ℎ with integer 𝑘≥0,𝑧₁(𝑡)=^∞_𝑘=0𝑦𝑡_𝑘𝐼_{[𝑡𝑘,𝑡𝑘+1)}(𝑡),𝑧₂(𝑡)=^∞_𝑘=0𝑦𝑡_𝑘−𝜏𝐼_{[𝑡𝑘,𝑡𝑘+1)}(𝑡),𝑟(𝑡)=^∞_𝑘=0𝑟𝑡_𝑘𝐼_{[𝑡𝑘,𝑡𝑘+1)}(𝑡). (2.8)

In order to show the strong convergence of the numerical solutions and the exact solutions to (2.2), we will need the following assumption.

  (H₅) There exists a positive constant 𝐾₂ such that 𝐸sup_{0≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝∨𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)^{(𝑚+1)2𝑝}≤𝐾₂, (2.9) for any 𝑝>0, where 𝑚=max{𝑚₁,𝑚₂,𝑚₃}.

Remark 2.1. (H₁) shows that the exact solutions to (2.2) admit finite moments; see [15]. If 𝑚₁=𝑚₂=𝑚₃=0, then (H₁) shows also that the numerical solutions and the exact solutions admit finite moments (see, [1, 3]).
We can now state our main result of this paper.

Theorem 2.2. Under assumptions(H₁)–(H₅), for any  𝑝≥2, then lim_ℎ→0𝐸sup_{0≤𝑡≤𝑇}𝑥(𝑡)−𝑦(𝑡)^𝑝=0. (2.10)

The proof of this theorem is rather technical. We will present a number of useful lemmas in Section 3 and then complete the proof in Section 4.

3. Lemmas

Throughout our analysis, 𝐶_𝑖,𝑖=1,2,… denote generic constants, independent of ℎ. In order to prove the main theorem, the following lemmas are useful.

Lemma 3.1. If assumptions(H₁),(H₃),(H₄), and (H₅)hold, then, for2≤𝜌≤(𝑚+1)𝑝, 𝐸𝑦(𝑡)−𝑧₁(𝑡)^𝜌≤𝐶ℎ^𝜌/2,𝑡≥0, (3.1) where 𝐶 is a positive constant independent of ℎ.

Proof. For notation simplicity reason, let us denote that 𝐴𝑡,𝑦(𝑡),𝑧₁(𝑡),𝑧₂(𝑡),𝑟(𝑡)=^𝑚1_𝑗=0𝑓_𝑥^(𝑗)𝑡,𝑧₁(𝑡),𝑧₂(𝑡),𝑟(𝑡)𝑦(𝑡)𝑗!−𝑧₁(𝑡)^𝑗,𝐵𝑡,𝑦(𝑡),𝑧₁(𝑡),𝑧₂(𝑡),𝑟(𝑡)=^𝑚2_𝑗=0𝑔_𝑥^(𝑗)𝑡,𝑧₁(𝑡),𝑧₂(𝑡),𝑟(𝑡)𝑦(𝑡)𝑗!−𝑧₁(𝑡)^𝑗,𝐶𝑡,𝑦(𝑡),𝑧₁(𝑡),𝑧₂(𝑡),𝑟(𝑡)=^𝑚3_𝑗=0ℎ_𝑥^(𝑗)𝑡,𝑧₁(𝑡),𝑧₂(𝑡),𝑟(𝑡)𝑦(𝑡)𝑗!−𝑧₁(𝑡)^𝑗. (3.2) Obviously, for any 𝑡≥0, there exists an integer 𝑘≥0 such that 𝑡∈[𝑡_𝑘,𝑡_𝑘+1]. Then, by (2.7) and (2.8) we obtain 𝑦(𝑡)−𝑧₁(𝑡)=𝑦(𝑡)−𝑦𝑡_𝑘=^𝑡_𝑡𝑘𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑠+^𝑡_𝑡𝑘𝐵𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)(𝑠)+𝑑𝑤_𝑡𝑡𝑘𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑁(𝑠). (3.3) Hence, we have 𝐸𝑦(𝑡)−𝑧₁(𝑡)^𝜌≤3^𝜌−1𝐸^𝑡_𝑡𝑘𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑠^𝜌+𝐸^𝑡_𝑡𝑘𝐵𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑤(𝑠)^𝜌+𝐸^𝑡_𝑡𝑘𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑁(𝑠)^𝜌. (3.4) By the Hölder inequality, we have 𝐸^𝑡_𝑡𝑘𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑠^𝜌≤𝑡−𝑡_𝑘^𝜌−1𝑡_𝑡𝑘𝐸𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝜌𝑑𝑠. (3.5) By the Burkholder-Davis-Gundy inequality and the Hölder inequality, we yield, for some positive constant 𝐶_𝜌, 𝐸^𝑡_𝑡𝑘𝐵𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑤(𝑠)^𝜌≤𝐶_𝜌𝑡−𝑡_𝑘^{𝜌/2−1𝑡}_𝑡𝑘𝐸𝐵𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝜌𝑑𝑠. (3.6) For the jump integral, we convert to the compensated Poisson process 𝑁(𝑡)∶=𝑁(𝑡)−𝜆𝑡, which is a martingale with 𝐸^𝑡2_𝑡1𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑁(𝑠)²=𝜆^𝑡2_𝑡1𝐸𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)²𝑑𝑠; (3.7) see, for example, [1, 4, 12, 16]. By the Burkholder-Davis-Gundy inequality and the Hölder inequality, for some positive constant 𝐶_𝜆,𝜌, we then obtain 𝐸^𝑡_𝑡𝑘𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑁(𝑠)^𝜌=𝐸^𝑡_𝑡𝑘𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑁(𝑠)+𝜆^𝑡_𝑡𝑘𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑠^𝜌≤2^𝜌−1𝐸^𝑡_𝑡𝑘𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑁(𝑠)^𝜌+2^𝜌−1𝐸𝜆^𝑡_𝑡𝑘𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑠^𝜌≤2^𝜌−1𝐶_𝜆,𝜌𝑡−𝑡_𝑘^{𝜌/2−1𝑡}_𝑡𝑘𝐸𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝜌𝑑𝑠+2^𝜌−1𝜆^𝜌𝑡−𝑡_𝑘^𝜌−1𝑡_𝑡𝑘𝐸𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝜌𝑑𝑠. (3.8) Now, by the mean value theorem there is a 𝜃∈(0,1) such that 𝐽₁(𝑡)=^𝑡_𝑡𝑘𝐸𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝜌=𝑑𝑠_𝑡𝑡𝑘𝐸𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠−𝑓)^{(𝑚1𝑥+1)}𝑠,𝑧₁(𝑠)+𝜃𝑦(𝑠)−𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑚₁!×𝑦(𝑠)+1−𝑧₁(𝑠)^𝑚1+1𝜌𝑑𝑠. (3.9) This, together with (H₁),(H₃)–(H₅),yields that 𝐽₁(𝑡)≤2^𝜌−1𝑡_𝑡𝑘𝐸𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)^2𝜌/2+𝐿^𝜌₁𝑚₁+1!^𝜌𝐸𝑦(𝑠)−𝑧₁(𝑠)^{(𝑚1+1)𝜌}𝑑𝑠≤2^𝜌−1𝑡_𝑡𝑘3^𝜌/2𝐿^𝜌1+𝐸𝑦(𝑠)^𝜌+𝐸𝑧₂(𝑠)^𝜌+𝐿^𝜌₁2^{(𝑚1+1)𝜌}𝑚₁+1!^𝜌×𝐸𝑦(𝑠)^{(𝑚1+1)𝜌}+𝐸𝑧₁(𝑠)^{(𝑚1+1)𝜌}𝑑𝑠≤2^𝜌−1𝑡_𝑡𝑘3^𝜌/2𝐿^𝜌1+2𝐾₂+𝐿+𝜅^𝜌12^{(𝑚1+1)𝜌+1}𝑚₁+1!^𝜌𝐾₂𝑑𝑠=𝐶₁𝑡−𝑡_𝑘. (3.10) Similarly, we can also show that there are two positive constant 𝐶₂ and 𝐶₃ for which 𝐽₂(𝑡)=^𝑡_𝑡𝑘𝐸𝐵𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝜌𝑑𝑠≤𝐶₂𝑡−𝑡_𝑘,𝐽₃(𝑡)=^𝑡_𝑡𝑘𝐸𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝜌𝑑𝑠≤𝐶₃𝑡−𝑡_𝑘. (3.11) Next, since the boundedness of 𝐽₁(𝑡), 𝐽₂(𝑡), and 𝐽₃(𝑡), we then have some positive constant 𝐶, independent of ℎ, such that 𝐸𝑦(𝑡)−𝑧₁(𝑡)^𝜌≤𝐶ℎ^𝜌/2. (3.12) The desired assertion is complete.

Lemma 3.2. If assumptions (H₁)-(H₅) hold, then, for 2≤𝜌≤(𝑚+1)𝑝, 𝐸𝑦(𝑡−𝜏)−𝑧₂(𝑡)^𝜌≤𝐶ℎ^𝛾,𝑡≥0, (3.13) where 𝛾∈(0,1] and 𝐶 is a positive constant independent of ℎ.

Proof. Clearly, for any 𝑡≥0, there exists an integer 𝑘≥0 such that 𝑡∈[𝑡_𝑘,𝑡_𝑘+1). In what follows, we split the following three cases to complete the proof.
Case 1. If −𝜏≤𝑡_𝑘−𝜏≤𝑡−𝜏≤0, we then have, from (H₂), 𝐸𝑦(𝑡−𝜏)−𝑧₂(𝑡)^𝜌=𝐸𝑦(𝑡−𝜏)−𝑦𝑡_𝑘−𝜏^𝜌=𝐸𝜉(𝑡−𝜏)−𝜉𝑡_𝑘−𝜏^𝜌≤𝐾₁ℎ^𝛾. (3.14)Case 2. If 0≤𝑡_𝑘−𝜏≤𝑡−𝜏, then, by Lemma 3.1, we have 𝐸𝑦(𝑡−𝜏)−𝑧₂(𝑡)^𝜌≤𝐶₄ℎ^𝜌/2. (3.15)Case 3. If −𝜏≤𝑡_𝑘−𝜏≤0≤𝑡−𝜏, note that 𝐸𝑦(𝑡−𝜏)−𝑧₂(𝑡)^𝜌≤2^𝜌−1𝐸𝑦(𝑡−𝜏)−𝜉(0)^𝜌+2^𝜌−1𝐸𝑦𝑡_𝑘−𝜏−𝜉(0)^𝜌. (3.16) Then, by the above two cases, it follows easily that 𝐸𝑦(𝑡−𝜏)−𝑧₂(𝑡)^𝜌≤𝐶₅ℎ^𝜌/2+ℎ^𝛾. (3.17) Now, combining the three cases altogether, for 2≤𝜌≤(𝑚+1)𝑃 and 𝛾∈(0,1], 𝐸𝑦(𝑡−𝜏)−𝑧₂(𝑡)^𝜌≤𝐶ℎ^𝛾; (3.18) where 𝐶 is a positive constant independent of ℎ. The proof is complete.

Lemma 3.3. If assumptions (H₁) and (H₅) hold, then, for 𝑝≥2, ^𝑇₀𝐸𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠≤𝐶₁ℎ,_𝑇0𝐸𝑔𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−𝑔𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠≤𝐶₂ℎ,_𝑇0𝐸ℎ𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−ℎ𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠≤𝐶₃ℎ, (3.19)(3.20)(3.21) where 𝐶₁,𝐶₂, and 𝐶₃ are positive constants dependent on max_{0≤𝑖≤𝑁}(−𝛾_𝑖𝑖), but independent of ℎ.

Proof. Let 𝑛=[𝑇/ℎ] be the integer part of 𝑇/ℎ. Then 𝐸^𝑇₀|𝑓(𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠))−𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝=𝑑𝑠_𝑛𝑘=0𝐸^𝑡𝑘+1_𝑡𝑘𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟𝑡_𝑘^𝑝𝑑𝑠, (3.22) with 𝑡_𝑛+1 being 𝑇. By (H₁) and (H₅), we derive 𝐸^𝑡𝑘+1_𝑡𝑘𝑓(𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−𝑓𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟𝑡_𝑘^𝑝𝑑𝑠≤2^𝑝−1𝐸^𝑡𝑘+1_𝑡𝑘𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝+𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟𝑡_𝑘^𝑝𝐼_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}𝑑𝑠≤2^𝑝𝐿^𝑝𝐸^𝑡𝑘+1_𝑡𝑘1+𝑦(𝑠)²+𝑧₂(𝑠)^2𝑝/2𝐼_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}𝑑𝑠≤2^𝑝𝐿^𝑝3^{𝑝/2−1𝑡𝑘+1}_𝑡𝑘𝐸𝐸1+𝑦(𝑠)^𝑝+𝑧₂(𝑠)^𝑝𝐼_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}∣𝑟𝑡_𝑘𝑑𝑠≤2^𝑝𝐿^𝑝3^{𝑝/2−1𝑡𝑘+1}_𝑡𝑘𝐸𝐸1+2𝐾₂+𝜅∣𝑟𝑡_𝑘𝐸𝐼_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}∣𝑟𝑡_𝑘𝑑𝑠. (3.23) Now, by the Markov property [6], we have 𝐸𝐼_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}∣𝑟𝑡_𝑘=_𝑖∈𝑆𝐼_{{𝑟(𝑡𝑘)=𝑖}}𝑃𝑟(𝑠)≠𝑖∣𝑟𝑡_𝑘=𝑖=_𝑖∈𝑆𝐼_{{𝑟(𝑡𝑘)=𝑖}𝑗≠𝑖}𝛾_𝑖𝑗𝑠−𝑡_𝑘+𝑜𝑠−𝑡_𝑘≤_𝑖∈𝑆𝐼_{{𝑟(𝑡𝑘)=𝑖}}max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑗▵+𝑜(▵). (3.24) So, (3.19) is complete. Similarly, we can show (3.20) and (3.21).

4. Proof of Theorem 2.2

Let us now begin to prove our main result Theorem 2.2. Clearly,𝐸sup_{0≤𝑡≤𝑇}𝑥(𝑡)−𝑦(𝑡)^𝑝=𝐸sup_{0≤𝑡≤𝑇}^𝑡₀𝑓(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)+𝑑𝑠_𝑡0𝑔(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐵𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)(𝑠)+𝑑𝑤_𝑡0[ℎ(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐶(𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠))]𝑑𝑁(𝑠)^𝑝≤3^𝑝−1𝐸sup_{0≤𝑡≤𝑇}^𝑡₀𝑓(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑠^𝑝+𝐸sup_{0≤𝑡≤𝑇}^𝑡₀𝑔(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐵𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑤(𝑠)^𝑝+𝐸sup_{0≤𝑡≤𝑇}^𝑡₀ℎ(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑁(𝑠)^𝑝. (4.1) So, for any 𝑡₁≤𝑇, by the Hölder inequality,𝐸sup_{0≤𝑡≤𝑡1}^𝑡₀𝑓(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(s),𝑟(𝑠)𝑑𝑠^𝑝≤𝑇^{𝑝−1𝑡1}₀𝐸𝑓(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠. (4.2) By the Burkholder-Davis-Gundy inequality, for some positive constant 𝐶_𝑝, we have𝐸sup_{0≤𝑡≤𝑡1}^𝑡₀𝑔(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐵𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑤(𝑠)^𝑝≤𝐶_𝑝𝑇^{𝑝/2−1𝑡1}₀𝐸𝑔(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐵𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠. (4.3) By Doob’s martingale inequality and the compensated Poisson integral, for some positive constant 𝐶_{𝜆,𝑝,𝑇}, we yield𝐸sup_{0≤𝑡≤𝑡1}^𝑡₀ℎ(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)𝑑𝑁(𝑠)^𝑝≤𝑝𝑝−1^𝑝2^𝑝−1𝐶_{𝜆,𝑝,𝑇}^𝑡1₀𝐸ℎ(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠+2^𝑝−1𝜆^𝑝𝑇^{𝑝−1𝑡1}₀𝐸ℎ(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠. (4.4) Now, by virtue of the assumption (H₁), we obtain that 𝐽₄(𝑡)=^𝑡1₀𝐸𝑓(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝=𝑑𝑠_𝑡10𝐸𝑓(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠)(𝑠)(𝑠),𝑟+𝑓𝑠,𝑦,𝑧₂(𝑠),𝑟(𝑠)−𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)+𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠≤3^{𝑝−1𝑡1}₀𝐸𝑓(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝+𝑑𝑠_𝑡10𝐸𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝+𝑑𝑠_𝑡10𝐸𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑s. (4.5)

Moreover, by (H₁) and Lemma 3.2, we have^𝑡1₀𝐸𝑓(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠≤𝐿^𝑝𝑡1₀𝐸𝑥(𝑠)−𝑦(𝑠)+𝑥(𝑠−𝜏)−𝑧₂(𝑠)^𝑝𝑑𝑠≤𝐿^𝑝2^{2𝑝−2𝑡1}₀𝐸𝑥(𝑠)−𝑦(𝑠)^𝑝+𝐸𝑥(𝑠−𝜏)−𝑦(𝑠−𝜏)^𝑝+𝐸𝑦(𝑠−𝜏)−𝑧₂(𝑠)^𝑝𝑑𝑠≤𝐿^𝑝2^{2𝑝−2𝑡1}₀𝐸𝑥(𝑠)−𝑦(𝑠)^𝑝+𝐸𝑥(𝑠−𝜏)−𝑦(𝑠−𝜏)^𝑝𝑑𝑠+𝐿^𝑝2^2𝑝−2𝑇𝐶ℎ^𝛾, (4.6) and by Lemma 3.1 and (H₄), there exists a 𝜃₁∈(0,1) such that^𝑡1₀𝐸𝑓𝑠,𝑦(𝑠),𝑧₂(𝑠),𝑟(𝑠)−𝐴𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝=𝑑𝑠_𝑡10𝐸𝑓^(𝑚1_𝑥⁺¹⁾𝑠,𝑧₁(𝑠)+𝜃₁𝑦(𝑠)−𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑚₁+1!^𝑝𝑦(𝑠)−𝑧₁(𝑠)^{(𝑚1+1)𝑝}≤𝐿𝑑𝑠_𝑝1𝑇𝐶𝑚₁+1!^𝑝ℎ^{(𝑚1+1)𝑝/2}. (4.7) Hence, by (4.6) and (4.7), together with Lemma 3.3, we yield𝐽₄(𝑡)≤3^𝑝−1𝐶₁𝐿ℎ+^𝑝1𝑇𝐶𝑚₁+1!^𝑝ℎ^{(𝑚1+1)𝑝/2}+𝐿^𝑝2^2𝑝−2𝑇𝐶ℎ^𝛾+𝐿^𝑝2^{2𝑝−2𝑡1}₀𝐸𝑥(𝑠)−𝑦(𝑠)^𝑝+𝐸𝑥(𝑠−𝜏)−𝑦(𝑠−𝜏)^𝑝𝑑𝑠. (4.8) Similarly, we have𝐽₅(𝑡)=^𝑡1₀𝐸𝑔(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐵𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠≤3^𝑝−1𝐶₂𝐿ℎ+^𝑝2𝑇𝐶𝑚₂+1!^𝑝ℎ^{(𝑚2+1)𝑝/2}+𝐿^𝑝2^2𝑝−2𝑇𝐶ℎ^𝛾+𝐿^𝑝2^{2𝑝−2𝑡1}₀𝐸𝑥(𝑠)−𝑦(𝑠)^𝑝+𝐸𝑥(𝑠−𝜏)−𝑦(𝑠−𝜏)^𝑝,𝐽𝑑𝑠₆(𝑡)=^𝑡1₀𝐸ℎ(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))−𝐶𝑠,𝑦(𝑠),𝑧₁(𝑠),𝑧₂(𝑠),𝑟(𝑠)^𝑝𝑑𝑠≤3^𝑝−1𝐶₃𝐿ℎ+^𝑝3𝑇𝐶𝑚₃+1!^𝑝ℎ^{(𝑚3+1)𝑝/2}+𝐿^𝑝2^2𝑝−2𝑇𝐶ℎ^𝛾+𝐿^𝑝2^{2𝑝−2𝑡1}₀𝐸𝑥(𝑠)−𝑦(𝑠)^𝑝+𝐸𝑥(𝑠−𝜏)−𝑦(𝑠−𝜏)^𝑝𝑑𝑠. (4.9)(4.10) Now we use 𝐾 to denote a generic positive constant that may change between occurrences. Hence, by (4.2)–(4.4) and (4.8)–(4.10), we yield that 𝐸sup_{0≤𝑡≤𝑡1}𝑥(𝑡)−𝑦(𝑡)^𝑝≤𝐾ℎ+𝐾ℎ^{(𝑚+1)𝑝/2}+𝐾ℎ^𝛾+𝐾^𝑡1₀𝐸𝑥(𝑠)−𝑦(𝑠)^𝑝+𝐸𝑥(𝑠−𝜏)−𝑦(𝑠−𝜏)^𝑝𝑑𝑠≤𝐾ℎ+𝐾ℎ^{(𝑚+1)𝑝/2}+𝐾ℎ^𝛾+2𝐾^𝑡1₀𝐸sup_{0≤𝑟≤𝑠}𝑥(𝑠)−𝑦(𝑠)^𝑝𝑑𝑠. (4.11)

The continuous Gronwall inequality then gives 𝐸sup_{0≤𝑡≤𝑇}𝑥(𝑡)−𝑦(𝑡)^𝑝≤𝐾ℎ+𝐾ℎ^{(𝑚+1)𝑝/2}+𝐾ℎ^𝛾𝑒^2𝐾𝑇. (4.12)

This proof is therefore complete.

Remark 4.1. If 𝑚₁=𝑚₂=𝑚₃=0, then the approximate scheme (2.7) is reduced to Euler-Maruyama method for stochastic differential delay equations with Poisson Jump and Markovian switching, which has been discussed in [3].

Remark 4.2. As is well known, Taylor approximation is effectively applicable in engineering if the equations can be solved explicitly. If not, since polynomials are very useful analytic functions, the approximation in the paper can be useful in other applications of stochastic Taylor expansion, especially in the construction of various time discrete approximations of Itô processes by using Itô-Taylor expansion such as Euler-Maruyama approximation and Milstein approximation, which has order 1/2 and 1, respectively [3, 8, 13]. All these show that numerical methods based on Taylor expansions of higher degrees could be improved by combining them with analytic approximations presented in this paper.

Acknowledgments

The project reported here was supported by the Foundation of Wuhan Polytechnic University and Zhongnan University of Economics and Law for Zhenxing.