# Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion

Authors: Lin Hu; Siqing Gan
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs) driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean-square convergence theory is illustrated by the stochasticθ-methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.


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## Body

## 1. Introduction

In stochastic numerical analysis, the order of convergence plays a crucial role in the design of numerical algorithms. Unlike in the deterministic modelling situation, there exist in the stochastic environment different types of convergence. Both in the literature and in practice, most attention is focused on two major types of convergence, that is, strong convergence and weak convergence. There is a rich literature on this subject; we here only mention [1–4] and the references therein.For stochastic differential equations (SDEs), Milstein [1] presented a fundamental convergence theorem which established the order of mean-square convergence of explicit one-step methods. The conditions of this theorem use both properties of mean and mean-square deviation of one-step approximation. The theorem showed that under certain conditions global error estimates for a method can be inferred from estimates on its local error. Buckwar [4] extended the convergence theory in [1] to stochastic functional differential equations. Recently, Zhang and Gan [5] extend the theory to neutral stochastic differential delay equations (NSDDEs). Therefore, the convergence theory in [1] and its generalization have received some attention in the case of nonjump SDEs. However, in the jump-SDE context, which is becoming increasingly important in mathematical finance [6–8], to our best knowledge, no corresponding convergence theory of numerical methods for NSDDEs with jumps has been presented in the literature. Motivated by the work of Zhang and Gan [5], our aim is to establish a relationship between the consistent order and the convergence order of the methods for the NSDDEs with jumps.In this paper, a class of drift-implicit one-step schemes are proposed for NSDDEs driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the theory about mean-square convergence is illustrated by the stochasticθ-methods and the balanced implicit methods. It is derived from Theorem 3.5 that the order of the mean square convergence of both of them for NSDDEs with jumps is 1⁄2. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.


## 2. Neutral Stochastic Delay Differential Equations with Jumps

Let(Ω,ℱ,{ℱt}t≥0,ℙ) be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all the ℙ-null sets). Let W(t):=(W1(t),…,Wb(t))T be a b-dimensional Wiener process, and N(t) is a scalar Poisson process with intensity λ (λ>0), both defined on the space (Ω,ℱ,{ℱt}t⩾0,ℙ). And |·| is used to denote both the norm in ℝd and the trace norm (F-norm) in ℝd×b. Also, C([t1,t2];ℝd) is used to represent the family of continuous mappings ψ from [t1,t2] to ℝd. Finally, Lℱtp([t1,t2];ℝd) is used to denote a family of ℱt-measurable, C([t1,t2];ℝd)-valued random variables ψ={ψ(u):t1≤u≤t2} such that ∥ψ∥𝔼p:=supt1≤u≤t2E|ψ(u)|p<∞. 𝔼 denote mathematical expectation with respect to ℙ.Consider the neutral stochastic delay differential equations (NSDDEs) with Poisson-driven jumps(2.1)d[x(t)-G(x(t-τ))]=f(x(t-),x(t--τ))dt+g(x(t-),x(t--τ))dW(t)+u(x(t-),x(t--τ))dN(t),t∈[0,T],
with initial data
(2.2)x(t)=ψ(t),t∈[-τ,0],
where ψ(t)∈Lℱ02([-τ,0];ℝd). Here, τ>0 is a constant, x(t-) denotes lims→t-x(s), f:ℝd×ℝd→ℝd, g:ℝd×ℝd→ℝd×b, u:ℝd×ℝd→ℝd, and G:ℝd→ℝd.By the definition of Itô-interpreted stochastic differential equations, the integral version of (2.1) is expressed as follows:(2.3)x(t)=G(x(t-τ))+x(0)-G(x(-τ))+∫0tf(x(s-),x(s--τ))ds+∫0tg(x(s-),x(s--τ))dW(s)+∫0tu(x(s-),x(s--τ))dN(s),t>0.Definition 2.1 (see [9]).
Anℝd-valued stochastic process x(t) on -τ≤t≤T is called a solution to (2.1) with initial data (2.2) if it has the following properties:(i)
{x(t)}0≤t≤T is continuous and ℱt-adapted;(ii)
{f(x(t),x(t-τ))}∈ℒ1([0,T];ℝd), {g(x(t),x(t-τ))}∈ℒ2([0,T];ℝd×b),{u(x(t),x(t-τ))}∈ℒ1([0,T];ℝd);(iii)
x(t)=ψ(t), -τ≤t≤0, and (2.3) holds for every t∈[0,T] with probability 1, where ℒp([0,T];ℝd) denotes the family of Borel measurable functions ϖ:[0,T]→ℝd such that ∫0T|ϖ(t)|pdt<∞ a.s.
A solution{x(t)} is said to be unique if any other solution {x¯(t)} is indistinguishable from {x(t)}, that is,
(2.4)P{x(t)=x¯(t)∀t∈[0T]}=1.In order to guarantee the existence and uniqueness of the solution, we impose the following hypothesis.Assumption 1 (global Lipschitz condition).
There exists a positive constantK such that for all x1,x2,y1,y2∈ℝd,
(2.5)|v(x1,y1)-v(x2,y2)|2⩽K(|x1-x2|2+|y1-y2|2),v=f,goru.Assumption 2 (linear growth condition).
There exists a positive constantL such that for all x,y∈ℝd,
(2.6)|v(x,y)|2⩽L(1+|x|2+|y|2),v=f,goru.Assumption 3.
There is a constantη∈(0,1) such that for all x1,x2∈ℝd(2.7)|G(x1)-G(x2)|≤η|x1-x2|.Remark 2.2.
In this paper, we always assume thatη≠0. Otherwise, the system (2.1) reduces to the stochastic delay differential equations with jumps.We useJ1,J2,… to denote the constants which are independent of the stepsize h.Lemma 2.3.
If Assumptions1–3 hold, then (2.1) has a unique strong solution x(t) on t≥-τ, and the solution of (2.1) satisfies
(2.8)E(sup⁡-τ≤t≤T|x(t)|2)≤J1,
where J1 is a positive constant which depends on constants T, η, L, and initial function ψ(t).It is not hard to prove Lemma2.3 in a similar way as the proof of Theorems 6.2.2 and 6.4.5 in [9].Lemma 2.4.
Let Assumptions2 and 3 hold. Assume that the initial function ψ(t) is uniformly Lipschitz L2-continuous, that is, there is a positive constant H¯ such that
(2.9)E|ψ(u1)-ψ(u2)|2≤H¯(u1-u2),if-τ≤u2<u1≤0,
then
(2.10)E|x(t1)-x(t2)|2≤Ĥ(t1-t2),
for all 0≤t1<t2≤T with t1-sτ∈[-τ,0], t2-sτ∈[-τ,0], where the constant Ĥ depends on constants H¯, T, initial function ψ(t), and positive integer s.Lemma2.4 is a modified version of [10, Lemma 2.1]. In a similar way, it is not hard to obtain the estimate (2.10).In this paper, we will use the following inequality. For anya,c>0 and 0<α<1, we have
(2.11)(a+c)2≤a2α+c21-α.


## 3. Implicit One-Step Schemes

Define a mesh with uniform steph which satisfies τ=mh for an integer number m (for convenience, we assume that m⩾2), and suppose that q is a positive integer with q=T/h. Let tn=nh, n=0,1,…,q. The drift-implicit one-step methods for the simulation of the solution x(t) of (2.1) are defined as follows:
(3.1)Yn+1=G(Yn+1-m)-G(Yn-m)+Yn+Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)+Φg(h,Yn,Yn-m,ΔWn,ΔNn)+Φu(h,Yn,Yn-m,ΔWn,ΔNn),n=0,1,…,q-1,
where Yn is an approximation of the exact solution x(tn),  and Φv (v=f, g, and u) are increment functions. ΔWn=W(tn+1)-W(tn), ΔNn=N(tn+1)-N(tn), and ΔWn is independent of ΔNn. Yn-m=ψ(tn-τ) when n-m⩽0.Remark 3.1.
Now, we discuss the solvability of (3.1). If increment function Φf does not depend on Yn+1, it is not difficult to find that the approximations Yn can be computed iteratively. If Φf depends on Yn+1, in order to guarantee the existence and uniqueness of a solution, the general approach is to assume Lipschitz continuity of Φf with respect to Yn+1 with the Lipschitz constant less than 1, and then to apply Banach's contraction mapping principle [4].We denote byY¯(tn+1) the value that is obtained when the exact solution values are inserted into the right-hand side of (3.1), that is,
(3.2)Y¯(tn+1)=G(x(tn+1-τ))-G(x(tn-τ))+x(tn)+Φf(h,x(tn),x(tn+1),x(tn-τ),x(tn+1-τ),ΔWn,ΔNn)+Φg(h,x(tn),x(tn-τ),ΔWn,ΔNn)+Φu(h,x(tn),x(tn-τ),ΔWn,ΔNn),n=0,1,…,q-1.We introduce the following definitions, which are presented in the literature, see [1, 3], for example.Definition 3.2.
Thelocal  error of method (3.1) is the sequence of random variables
(3.3)δn+1:=x(tn+1)-Y¯(tn+1),n=0,1,…,q-1.
The global  error of method (3.1) is the sequence of random variables
(3.4)εn:=x(tn)-Yn,n=0,1,…,q.Definition 3.3.
The numerical method (3.1) is said to be consistent with orderp1in the mean and with orderp2in the mean square sense if the following estimates hold with p2⩾1/2 and p1⩾p2+1/2:
(3.5)max⁡0≤n≤q-1‖E(δn+1∣Ftn)‖L2≤H0hp1,ash⟶0,max⁡0≤n≤q-1‖δn+1‖L2≤H1hp2,ash⟶0,
where the constants H0 and H1 do not depend on h but may depend on T and the initial data. Here, ∥z∥L2:=(𝔼|z|2)1/2.Definition 3.4.
The numerical method (3.1) is said to be convergent with orderpin the mean square sense, on the mesh points, if the following estimate holds:
(3.6)max⁡1≤n≤q‖εn‖L2≤H2hp,ash⟶0,
where the constant H2 does not depend on h but may depend on T and the initial data.In order to obtain the main result, the following properties of the increment functions are required. There exist positive constantsLf such that for xi,yi∈ℝd (i=1,2,3,4),(3.7)|Φf(h,x1,x2,x3,x4,ΔWn,ΔNn)-Φf(h,y1,y2,y3,y4,ΔWn,ΔNn)|≤hLf∑i=14|xi-yi|,
and there exist the positive constants Lg,Lu,Lu¯ such that for all ℱtn-measurable random variables x1,x3,y1,y3∈ℝd,
(3.8)E|Φg(h,x1,x3,ΔWn,ΔNn)-Φg(h,y1,y3,ΔWn,ΔNn)|2≤hLg[E|x1-y1|2+E|x3-y3|2],(3.9)E|Φu(h,x1,x3,ΔWn,ΔNn)-Φu(h,y1,y3,ΔWn,ΔNn)|2≤hLu[E|x1-y1|2+E|x3-y3|2],(3.10)E(Φg(h,x1,x3,ΔWn,ΔNn)-Φg(h,y1,y3,ΔWn,ΔNn)∣Ftn)=0,(3.11)|E[Φu(h,x1,x3,ΔWn,ΔNn)-Φu(h,y1,y3,ΔWn,ΔNn)∣Ftn]|≤hLu¯[|x1-y1|+|x3-y3|].Now, we state our result on the convergence of the one-step method (3.1).Theorem 3.5.
Suppose that Assumption3 and the conditions (3.7)–(3.11) hold. Assume that the one-step method (3.1) is consistent with order p1 in the mean and order p2 in the mean square sense, then the method (3.1) is convergent with order p=p2-1/2 in the mean square sense.Proof.
By (3.4), we have
(3.12)εn+1=ε̂n+1+[G(x(tn+1-τ))-G(Yn+1-m)],
where ε̂n+1 is defined as follows:
(3.13)ε̂n+1=x(tn+1)-G(x(tn+1-τ))-[Yn+1-G(Yn+1-m)].
Squaring and taking expectation on both sides of (3.12), using Assumption 3 and (2.11), we have
(3.14)E|εn+1|2≤11-ηE|ε̂n+1|2+1ηE|G(x(tn+1-τ))-G(Yn+1-m)|2≤11-ηE|ε̂n+1|2+ηE|x(tn+1-τ)-Yn+1-m|2=11-ηE|ε̂n+1|2+ηE|εn+1-m|2.
It follows from (3.1), (3.2), and (3.13) that
(3.15)ε̂n+1=x(tn+1)-G(x(tn+1-τ))-Y¯(tn+1)+Y¯(tn+1)-[Yn+1-G(Yn+1-m)]=x(tn+1)-Y¯(tn+1)+x(tn)-G(x(tn-τ))+Φf(h,x(tn),x(tn+1),x(tn-τ),x(tn+1-τ),ΔWn,ΔNn)+Φg(h,x(tn),x(tn-τ),ΔWn,ΔNn)+Φu(h,x(tn),x(tn-τ),ΔWn,ΔNn)-Yn+G(Yn-m)-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)-Φg(h,Yn,Yn-m,ΔWn,ΔNn)-Φu(h,Yn,Yn-m,ΔWn,ΔNn)=δn+1+ε̂n+Rn,
where
(3.16)Rn=Φf(h,x(tn),x(tn+1),x(tn-τ),x(tn+1-τ),ΔWn,ΔNn)-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)+Φg(h,x(tn),x(tn-τ),ΔWn,ΔNn)-Φg(h,Yn,Yn-m,ΔWn,ΔNn)+Φu(h,x(tn),x(tn-τ),ΔWn,ΔNn)-Φu(h,Yn,Yn-m,ΔWn,ΔNn).
Squaring and taking expectation on both sides of (3.15) yields
(3.17)E|ε̂n+1|2≤E|ε̂n|2+2E|δn+1|2+2E|Rn|2+2E〈δn+1,ε̂n〉+2E〈ε̂n,Rn〉.
We will now estimate the separate terms in (3.17) individually. Without loss of generality, we can assume that 0<h<1. We notice that the method (3.1) is consistent with order p2 in the mean square sense; thus, there exists a constant J2 such that
(3.18)E|δn+1|2≤J2h2p2.
By (3.7)–(3.9), we obtain
(3.19)E|Rn|2≤3E|Φf(h,x(tn),x(tn+1),x(tn-τ),x(tn+1-τ),ΔWn,ΔNn)-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)|2+3E|Φg(h,x(tn),x(tn-τ),ΔWn,ΔNn)-Φg(h,Yn,Yn-m,ΔWn,ΔNn)|2+3E|Φu(h,x(tn),x(tn-τ),ΔWn,ΔNn)-Φu(h,Yn,Yn-m,ΔWn,ΔNn)|2≤3h2Lf2E(|x(tn)-Yn|+|x(tn+1)-Yn+1|+|x(tn-τ)-Yn-m|+|x(tn+1-τ)-Yn+1-m|)2+3hLg(E|x(tn)-Yn|2+E|x(tn-τ)-Yn-m|2)+3hLu(E|x(tn)-Yn|2+E|x(tn-τ)-Yn-m|2)≤(12h2Lf2+3hLg+3hLu)(E|x(tn)-Yn|2+E|x(tn-τ)-Yn-m|2)+12h2Lf2(E|x(tn+1)-Yn+1|2+E|x(tn+1-τ)-Yn+1-m|2)=(12h2Lf2+3hLg+3hLu)(E|εn|2+E|εn-m|2)+12h2Lf2(E|εn+1|2+E|εn+1-m|2),
which, by the inequality (3.14), yields
(3.20)E|Rn|2≤(12h2Lf2+3hLg+3hLu)(11-ηE|ε̂n|2+ηE|εn-m|2)+(12h2Lf2+3hLg+3hLu)E|εn-m|2+12h2Lf2(11-ηE|ε̂n+1|2+ηE|εn+1-m|2)+12h2Lf2E|εn+1-m|2≤(12h2Lf2+3hLg+3hLu)(E|ε̂n|2+E|εn-m|2)max⁡{11-η,1+η}+12h2Lf2(E|ε̂n+1|2+E|εn+1-m|2)max⁡{11-η,1+η}≤(12TLf2+3Lg+3Lu)h(E|ε̂n|2+E|εn-m|2+E|ε̂n+1|2+E|εn+1-m|2)max⁡{11-η,1+η}=J3h(E|ε̂n|2+E|εn-m|2+E|ε̂n+1|2+E|εn+1-m|2),
where J3=(12TLf2+3Lg+3Lu)max{1/1-η,1+η}.
Since method (3.1) is consistent with order p1 in the mean square, there exists a constant J4 such that
(3.21)(E|E(δn+1∣Ftn)|2)1/2≤J4hp1.
Noticing that ε̂n is ℱtn-measurable and by (3.21), we arrive at
(3.22)2E(〈ε̂n,δn+1〉)=2E[E(〈ε̂n,δn+1〉∣Ftn)]≤2E|E(〈ε̂n,δn+1〉∣Ftn)|=2E|〈ε̂n,E(δn+1∣Ftn)〉|≤2(hE|ε̂n|2)1/2(h-1E|E(δn+1∣Ftn)|2)1/2≤hE|ε̂n|2+h-1E|E(δn+1∣Ftn)|2≤hE|ε̂n|2+(J4)2h2p1-1=hE|ε̂n|2+J5h2p1-1,
where J5=(J4)2. Applying the inequality |𝔼(x∣ℱ)|2≤𝔼(|x|2∣ℱ), (3.7), (3.10), and (3.11) yields
(3.23)|E(Rn∣Ftn)|2=|E[-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)|2Φf(h,x(tn),x(tn+1),x(tn-τ),x(tn+1-τ),ΔWn,ΔNn)-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)+Φg(h,x(tn),x(tn-τ),ΔWn,ΔNn)-Φg(h,Yn,Yn-m,ΔWn,ΔNn)+Φu(h,x(tn),x(tn-τ),ΔWn,ΔNn)-Φu(h,Yn,Yn-m,ΔWn,ΔNn)∣Ftn-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)|2]|2=|E[Φf(h,x(tn),x(tn+1),x(tn-τ),x(tn+1-τ),ΔWn,ΔNn)-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)∣Ftn]+E[Φu(h,x(tn),x(tn-τ),ΔWn,ΔNn)-Φu(h,Yn,Yn-m,ΔWn,ΔNn)∣Ftn]|2≤2E[|-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)|2Φf(h,x(tn),x(tn+1),x(tn-τ),x(tn+1-τ),ΔWn,ΔNn)-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)|2∣Ftn]+2|E[-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)|2Φu(h,x(tn),x(tn-τ),ΔWn,ΔNn)-Φu(h,Yn,Yn-m,ΔWn,ΔNn)∣Ftn]|2≤2h2(Lf)2E[-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)|2(|x(tn)-Yn|+|x(tn+1)-Yn+1|+|x(tn-τ)-Yn-m|+|x(tn+1-τ)-Yn+1-m|)2∣Ftn-Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)|2]+2h2(Lu¯)2[|x(tn)-Yn|+|x(tn-τ)-Yn-m|]2≤8h2(Lf)2[|εn|2+E(|εn+1|2∣Ftn)+|εn-m|2+|εn+1-m|2]+4h2(Lu¯)2[|εn|2+|εn-m|2]=[8h2(Lf)2+4h2(Lu¯)2]|εn|2+8h2(Lf)2E(|εn+1|2∣Ftn)+[8h2(Lf)2+4h2(Lu¯)2]|εn-m|2+8h2(Lf)2|εn+1-m|2.
Here, the fact used has been that x(tn), x(tn-τ), x(tn+1-τ), Yn, Yn-m, Yn+1-m are ℱtn-measurable. Using (3.14) and (3.23), we arrive at
(3.24)2E(〈ε̂n,Rn〉)=2E[E(〈ε̂n,Rn〉∣Ftn)]=2[E〈ε̂n,E(Rn∣Ftn)〉]≤2E[|ε̂n|⋅|E(Rn∣Ftn)|]≤2(hE|ε̂n|2)1/2(h-1E|E(Rn∣Ftn)|2)1/2≤hE|ε̂n|2+h-1E|E(Rn∣Ftn)|2≤hE|ε̂n|2+[8h(Lf)2+4h(Lu¯)2]E|εn|2+8h(Lf)2E|εn+1|2+[8h(Lf)2+4h(Lu¯)2]E|εn-m|2+8h(Lf)2E|εn+1-m|2≤hE|ε̂n|2+[8h(Lf)2+4h(Lu¯)2][11-ηE|ε̂n|2+ηE|εn-m|2]+8h(Lf)2[11-ηE|ε̂n+1|2+ηE|εn+1-m|2]+[8h(Lf)2+4h(Lu¯)2]E|εn-m|2+8h(Lf)2E|εn+1-m|2=[1+(8(Lf)2+4(Lu¯)2)11-η]hE|ε̂n|2+8(Lf)211-ηhE|ε̂n+1|2+(8(Lf)2+4(Lu¯)2)(1+η)hE|εn-m|2+8(Lf)2(1+η)hE|εn+1-m|2≤[1+(8(Lf)2+4(Lu¯)2)max⁡{11-η,1+η}]h,[E|ε̂n|2+E|ε̂n+1|2+E|εn+1-m|2+E|εn-m|2]=J6h[E|ε̂n|2+E|ε̂n+1|2+E|εn+1-m|2+E|εn-m|2],
where J6=1+(8(Lf)2+4(Lu¯)2)max{1/(1-η),1+η}. Inserting (3.18), (3.20), (3.22), and (3.24) into (3.17) yields
(3.25)E|ε̂n+1|2≤E|ε̂n|2+2J2h2p2+2J3h(E|ε̂n|2+E|εn-m|2+E|ε̂n+1|2+E|εn+1-m|2)+hE|ε̂n|2+J5h2p1-1+J6h(E|ε̂n|2+E|ε̂n+1|2+E|εn+1-m|2+E|εn-m|2)=(2J3+J6)hE|ε̂n+1|2+(1+(1+2J3+J6)h)E|ε̂n|2+(2J3+J6)h(E|εn+1-m|2+E|εn-m|2)+(2J2+J5)h2p2.
The following proof is analogous to that of Theorem 3.1 in [5]; thus, it is not hard to derive the convergence result. The proof is completed.Remark 3.6.
Notice that ifu(·)=0 in (2.1), then (2.1) reduces to the NSDDEs without jumps, our Theorem 3.5 coincides with Theorem 3.1 in [5], that is to say, Theorem 3.5 generalizes Theorem 3.1 in [5] to the case of NSDDEs with jumps.


## 4. The Examples

Theorem3.5 presents the convergence result about the general implicit one-step methods for NSDDEs with jumps. In this section, we discuss the applicability of the theory presented in the previous section. We will give the convergence orders of the stochastic θ-methods and the balanced implicit methods.Example 4.1.
Consider the stochasticθ-methods for system (2.1),
(4.1)Yn+1=G(Yn+1-m)+Yn-G(Yn-m)+[(1-θ)f(Yn,Yn-m)+θf(Yn+1,Yn+1-m)]h+g(Yn,Yn-m)ΔWn+u(Yn,Yn-m)ΔNn,n=0,1,…,q-1,
where m=τ/h, 0≤θ≤1.Lemma 4.2.
Let Assumption1 hold, then the stochastic θ-methods (4.1) are consistent with order p1=3/2 in the mean and order p2=1 in the mean square sense.Proof.
Combining (2.1), (3.2) with (4.1) yields
(4.2)δn+1=x(tn+1)-Y¯(tn+1)=∫tntn+1[f(x(s),x(s-τ))-f(x(tn),x(tn-τ))]ds+∫tntn+1[g(x(s),x(s-τ))-g(x(tn),x(tn-τ))]dW(s)+∫tntn+1[u(x(s),x(s-τ))-u(x(tn),x(tn-τ))]dN(s)-θh[f(x(tn+1),x(tn+1-τ))-f(x(tn),x(tn-τ))].
Noticing the compensated Poisson process, Ñ(t)=N(t)-λt, which satisfies
(4.3)E(∫abu(s)dÑ(s)∣Fa)=0.
Using Assumption 1, (4.3), Hölder inequality, Lemma 2.4, the properties of conditional expectation, and Jenson’s inequality: |𝔼(x∣ℱ)|2≤𝔼(|x|2∣ℱ), we compute that
(4.4)E|E(δn+1∣Ftn)|2≤5E|E(∫tntn+1[f(x(s),x(s-τ))-f(x(tn),x(tn-τ))]ds∣Ftn)|2+5E|E(∫tntn+1[g(x(s),x(s-τ))-g(x(tn),x(tn-τ))]dW(s)∣Ftn)|2+5E|E(∫tntn+1[u(x(s),x(s-τ))-u(x(tn),x(tn-τ))]dÑ(s)∣Ftn)|2+5λ2E|E(∫tntn+1[u(x(s),x(s-τ))-u(x(tn),x(tn-τ))]ds∣Ftn)|2+5h2E|E(f(x(tn+1),x(tn+1-τ))-f(x(tn),x(tn-τ))∣Ftn)|2≤5E[E(|∫tntn+1f(x(s),x(s-τ))-f(x(tn),x(tn-τ))ds|2∣Ftn)]+5λ2E[E(|∫tntn+1u(x(s),x(s-τ))-u(x(tn),x(tn-τ))ds|2∣Ftn)]+5h2E[E(|f(x(tn+1),x(tn+1-τ))-f(x(tn),x(tn-τ))|2∣Ftn)]≤5(1+λ2)Kh∫tntn+1E(|x(s)-x(tn)|2+|x(s-τ)-x(tn-τ)|2)ds+5h2KE(|x(tn+1)-x(tn)|2+|x(tn+1-τ)-x(tn-τ)|2)≤[10(1+λ2)KĤ+10KĤ]h3,(4.5)E|δn+1|2≤4hE∫tntn+1|f(x(s),x(s-τ))-f(x(tn),x(tn-τ))|2ds+4E∫tntn+1|g(x(s),x(s-τ))-g(x(tn),x(tn-τ))|2ds+8λ∫tntn+1E|u(x(s),x(s-τ))-u(x(tn),x(tn-τ))|2ds+8λ2h∫tntn+1E|u(x(s),x(s-τ))-u(x(tn),x(tn-τ))|2ds+4h2E|f(x(tn+1),x(tn+1-τ))-f(x(tn),x(tn-τ))|2≤(4h+4+8λ+8λ2h)K∫tntn+1E(|x(s)-x(tn)|2+|x(s-τ)-x(tn-τ)|2)ds+4Kh2E(|x(tn+1)-x(tn)|2+|x(tn+1-τ)-x(tn-τ)|2)≤8KĤ(1+2T+2λ+2λ2T)h2,
where the compensated Poisson process Ñ(t)=N(t)-λt satisfies
(4.6)E|∫t1t2u(x(s))dÑ(s)|2=λ∫t1t2E|u(x(s))|2ds.
Hence, the stochastic θ-methods (4.1) are consistent with order 3/2 in the mean and order 1 in the mean square sense. The proof is completed.Theorem 4.3.
Let Assumption1 hold, then the stochastic θ-methods (4.1) are convergent with order p=1/2 in the mean square sense.Proof.
Lemma4.2 shows that the stochastic θ-methods (4.1) are consistent with order p1=3/2 in the mean and order p2=1 in the mean square sense. In order to prove that the stochastic θ-methods (4.1) are convergent with order p=1/2 in the mean square sense, by Theorem 3.5, we only need to verify the conditions (3.7)–(3.11).
From (4.1), we find that
(4.7)Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)=[(1-θ)f(Yn,Yn-m)+θf(Yn+1,Yn+1-m)]h,Φg(h,Yn,Yn-m,ΔWn,ΔNn)=g(Yn,Yn-m)ΔWn,Φu(h,Yn,Yn-m,ΔWn,ΔNn)=u(Yn,Yn-m)ΔNn.
For the random variables xi,yi∈ℝd (i=1,2,3,4), by Assumption 1, we have
(4.8)|Φf(h,x1,x2,x3,x4,ΔWn,ΔNn)-Φf(h,y1,y2,y3,y4,ΔWn,ΔNn)|=|[(1-θ)f(x1,x3)+θf(x2,x4)]h-[(1-θ)f(y1,y3)+θf(y2,y4)]h|≤Kh∑i=14|xi-yi|.
Noticing that 𝔼|ΔWn|2=bh, 𝔼|ΔNn|2=λh(1+λh), and the random variables x1,x3,y1,y3 are ℱtn-measurable, we derive that
(4.9)E|Φg(h,x1,x3,ΔWn,ΔNn)-Φg(h,y1,y3,ΔWn,ΔNn)|2≤E|g(x1,x3)ΔWn-g(y1,y3)ΔWn|2≤bKhE|x1-y1|2+E|x3-y3|2,(4.10)E|Φu(h,x1,x3,ΔWn,ΔNn)-Φu(h,y1,y3,ΔWn,ΔNn)|2≤E|u(x1,x3)ΔNn-u(y1,y3)ΔNn|2≤λ(1+λT)Kh[E|x1-y1|2+E|x3-y3|2].
From (4.8)–(4.10), we see that the increment functions of the stochastic θ-methods (4.1) satisfy the inequalities (3.7)–(3.9) with Lf=K, Lg=bK, and Lu=λ(1+λT)K. Noting that ΔWn,ΔNn are independent of ℱtn and x1,x3,y1, and y3 are ℱtn-measurable, then using |𝔼(x∣ℱ)|≤𝔼(|x|∣ℱ), we find that
(4.11)E[Φg(h,x1,x3,ΔWn,ΔNn)-Φg(h,y1,y3,ΔWn,ΔNn)∣Ftn]=E[g(x1,x3)ΔWn-g(y1,y3)ΔWn∣Ftn]=[g(x1,x3)-g(y1,y3)]E(ΔWn∣Ftn)=0,|E[Φu(h,x1,x3,ΔWn,ΔNn)-Φu(h,y1,y3,ΔWn,ΔNn)∣Ftn]|=|E[u(x1,x3)ΔNn-u(y1,y3)ΔNn∣Ftn]|=|[u(x1,x3)-u(y1,y3)]|⋅E(ΔNn∣Ftn)≤λKh[|x1-y1|+|x3-y3|].
Here, the fact used has been that 𝔼ΔWn=0 and 𝔼ΔNn=λh. From above, we find that the increment functions of the stochastic θ-methods (4.1) satisfy the estimations (3.10) and (3.11) with Lu¯=λK. Therefore, the conditions (3.7)–(3.11) are satisfied. By Lemma 4.2 and Theorem 3.5, it is not difficult to find that the stochastic θ-methods (4.1) are convergent with order p=1/2 in the mean square sense. The proof is completed.Remark 4.4.
For the case ofG(·)=0, (2.1) reduces to the stochastic delay differential equations with jumps
(4.12)dx(t)=f(x(t-),x(t--τ))dt+g(x(t-),x(t--τ))dW(t)+u(x(t-),x(t--τ))dN(t),0<t<T,x(t)=ψ(t),-τ≤t≤0,
where ψ∈Lℱ02([-τ,0];ℝd).
Theorem4.3 implies that the stochastic θ-methods for (4.12) are convergent with order p=1/2, which coincides with Theorem 3.2 in [11].Example 4.5.
Consider the balanced implicit methods for system (2.1),
(4.13)Yn+1=G(Yn+1-m)+Yn-G(Yn-m)+f(Yn,Yn-m)h+g(Yn,Yn-m)ΔWn+u(Yn,Yn-m)ΔNn+C(Yn,Yn-m)[Yn-G(Yn-m)-Yn+1+G(Yn+1-m)].
Here, the d×d matrix C(Yn,Yn-m) is given by
(4.14)C(Yn,Yn-m)=C0(Yn,Yn-m)h+C1(Yn,Yn-m)|ΔWn|=C0nh+C1n|ΔWn|,
where the C0n=C0(Yn,Yn-m), C1n=C1(Yn,Yn-m) are, in general matrix, called control functions which are often chosen as constants.Furthermore, the control functions must satisfy some conditions.Assumption 4.
TheC0(x,y) and C1(x,y) represent bounded d×d-matrix-valued functions for x,y∈ℝd. For any real numbers α0∈[0,α¯],α1≥0, where α¯≥h for all step sizes h considered and x,y∈ℝd, the matrix M(x,y)=I+α0C0(x,y)+α1C1(x,y) has an inverse and satisfies the condition
(4.15)|(M(x,y))-1|≤H<∞.
Here, I is the unit matrix, and H is a positive constant. Notice that if C0n,C1n satisfy Assumption 4, the methods (4.13) are well defined and can be rewritten as
(4.16)Yn+1=G(Yn+1-m)+Yn-G(Yn-m)+[I+C(Yn,Yn-m)]-1×(f(Yn,Yn-m)h+g(Yn,Yn-m)ΔWn+u(Yn,Yn-m)ΔNn).Lemma 4.6 (see [12]).
IfF(y,ω) is independent of ℱ,y∈ℝd, ω∈Ω, 𝔼(F(y,ω))=φ(y), and ς is ℱ-measurable, then 𝔼(F(ς,ω)∣ℱ)=φ(ς).For simplicity, from now on, we suppose thatC0n,C1n in (4.14) are constants, that is to say, C0n=C0, C1n=C1.Lemma 4.7.
Let Assumptions1, 2, and 4 hold, then the balanced implicit methods (4.13) are consistent with order p1=3/2 in the mean and order p2=1 in the mean square sense.Proof.
Without loss of generality, we can assume that0<h<1. From Lemma 4.2, we know that the Euler-Maruyama method is consistent with order 3/2 in the mean, thus using (4.4), we have
(4.17)E[|E[x(tn+1)-Y¯B(tn+1)∣Ftn]|2]=E{|E[x(tn+1)-Y¯E(tn+1)∣Ftn]+E[Y¯E(tn+1)-Y¯B(tn+1)∣Ftn]|2}≤2E[|E[x(tn+1)-Y¯E(tn+1)∣Ftn]|2]+2E[|E[Y¯E(tn+1)-Y¯B(tn+1)∣Ftn]|2]≤[20(1+λ2)KĤ+20KĤ]h3+2E[|E[Y¯E(tn+1)-Y¯B(tn+1)∣Ftn]|2],
where Y¯E(tn+1) and Y¯B(tn+1) are defined as follows:
(4.18)Y¯E(tn+1)=G(x(tn+1-τ))+x(tn)-G(x(tn-τ))+f(x(tn),x(tn-τ))h+g(x(tn),x(tn-τ))ΔWn+u(x(tn),x(tn-τ))ΔNn,Y¯B(tn+1)=G(x(tn+1-τ))+x(tn)-G(x(tn-τ))+[I+Cn]-1×(f(x(tn),x(tn-τ))h+g(x(tn),x(tn-τ))ΔWn+u(x(tn),x(tn-τ))ΔNn),
where Cn=C0h+C1|ΔWn|. By (4.18), we have
(4.19)Y¯E(tn+1)-Y¯B(tn+1)=(I-[I+Cn]-1)⋅(f(x(tn),x(tn-τ))h+g(x(tn),x(tn-τ))ΔWn+u(x(tn),x(tn-τ))ΔNn)=[I+Cn]-1⋅((I+Cn)-I)⋅(f(x(tn),x(tn-τ))h+g(x(tn),x(tn-τ))ΔWn+u(x(tn),x(tn-τ))ΔNn)=[I+Cn]-1⋅Cn⋅(f(x(tn),x(tn-τ))h+g(x(tn),x(tn-τ))ΔWn+u(x(tn),x(tn-τ))ΔNn).
Noticing that x(tn),x(tn-τ) are ℱtn-measurable, ΔWn is independent of ℱtn, and using Lemma 4.6, we find that
(4.20)E[(I+Cn)-1⋅Cn⋅g(x(tn),x(tn-τ))ΔWn∣Ftn]=0.
We notice that C0 and C1 are constants; thus, there exists a positive constant B, such that |Ci|≤B (i=0,1). Since ΔWn,ΔNn are independent of ℱtn, by Assumption 4, (4.19), (4.20), |𝔼(x∣ℱ)|≤𝔼(|x|∣ℱ), 𝔼|ΔWn|≤bh, and 𝔼ΔNn=λh, we obtain
(4.21)|E[Y¯E(tn+1)-Y¯B(tn+1)∣Ftn]|≤|f(x(tn),x(tn-τ))|h⋅E(|(I+Cn)-1|⋅|C0h+C1|ΔWn||∣Ftn)+|u(x(tn),x(tn-τ))|⋅E(|(I+Cn)-1|⋅|C0h+C1|ΔWn||⋅ΔNn∣Ftn)≤|f(x(tn),x(tn-τ))|Hh⋅E(|C0h+C1|ΔWn||∣Ftn)+|u(x(tn),x(tn-τ))|H⋅E(|C0h+C1|ΔWn||⋅ΔNn∣Ftn)≤(|f(x(tn),x(tn-τ))|+|u(x(tn),x(tn-τ))|λ)Hh3/2B(h+b).
It follows from Assumption 2, Lemma 2.3, and (4.21) that
(4.22)E[|E[Y¯E(tn+1)-Y¯B(tn+1)∣Ftn]|2]≤2H2B2(T+b)2h3[E|f(x(tn),x(tn-τ))|2+E|u(x(tn),x(tn-τ))|2λ2]≤2LH2B2(1+λ2)(T+b)2(1+2J1)h3.
Inserting (4.22) into (4.17) yields
(4.23)E[|E[x(tn+1)-Y¯B(tn+1)∣Ftn]|2]≤(20(1+λ2)KĤ+20KĤ+4LH2B2(1+λ2)(T+b)2(1+2J1))h3,
which implies that the balanced implicit methods (4.13) are consistent with order 3/2 in the mean. In the following, we will show that the balanced implicit methods (4.13) are consistent with order 1 in the mean square sense. Using Assumptions 2 and 4, Lemma 2.3, |Ci|≤B (i=0,1), 𝔼|ΔWn|2=bh, 𝔼|ΔWn|4≤3b2h2, 𝔼ΔNn2=λh(1+λh), and (4.19), we compute that
(4.24)E[|Y¯E(tn+1)-Y¯B(tn+1)|2]≤3h2H2E(|C0h+C1|ΔWn||2⋅|f(x(tn),x(tn-τ))|2)+3H2E(|C0h+C1|ΔWn||2⋅|g(x(tn),x(tn-τ))|2|ΔWn|2)+3H2E(|C0h+C1|ΔWn||2⋅|u(x(tn),x(tn-τ))|2ΔNn2)≤3h2H2E((2|C0|2h2+2|C1|2|ΔWn|2)⋅|f(x(tn),x(tn-τ))|2)+3H2E((2|C0|2h2|ΔWn|2+2|C1|2|ΔWn|4)⋅|g(x(tn),x(tn-τ))|2)+3H2E((2|C0|2h2+2|C1|2|ΔWn|2)⋅|u(x(tn),x(tn-τ))|2ΔNn2)≤3h2H2(2B2h2+2B2bh)E[|f(x(tn),x(tn-τ))|2]+3H2(2B2h2bh+2B2⋅3b2h2)E[|g(x(tn),x(tn-τ))|2]+3H2λh(1+λh)(2B2h2+2B2bh)E[|u(x(tn),x(tn-τ))|2]≤6H2B2L[T2+2bT+3b2+λ(1+λT)(T+b)](1+2J1)h2.
Theorem 4.3 implies that the Euler-Maruyama method is convergent with order p=1/2. Thus, by (4.5) and (4.24), we have
(4.25)E[|x(tn+1)-Y¯B(tn+1)|2]≤2E[|x(tn+1)-Y¯E(tn+1)|2]+2E[|Y¯E(tn+1)-Y¯B(tn+1)|2]≤[16KĤ(1+2T+2λ+2λ2T)+12H2B2L×[T2+2bT+3b2+16KĤ(1+2T+2λ+2λ2T)λ(1+λT)(T+b)](1+2J1)]h2.
The proof is completed.Theorem 4.8.
Let Assumptions1–4 hold, then the balanced implicit methods (4.16) are convergent with order p=1/2 in the mean square sense.Proof.
By (4.16), the increment functions Φf, Φg, and Φu of the balanced implicit methods (4.16) are given as follows:
(4.26)Φf(h,Yn,Yn+1,Yn-m,Yn+1-m,ΔWn,ΔNn)=[I+C(Yn,Yn-m)]-1f(Yn,Yn-m)h,(4.27)Φg(h,Yn,Yn-m,ΔWn,ΔNn)=[I+C(Yn,Yn-m)]-1g(Yn,Yn-m)ΔWn,Φu(h,Yn,Yn-m,ΔWn,ΔNn)=[I+C(Yn,Yn-m)]-1u(Yn,Yn-m)ΔNn.
For xi,yi∈ℝd (i=1,2,3,4), by Assumptions 1–4, (4.26), we arrive at
(4.28)|Φf(h,x1,x2,x3,x4,ΔWn,ΔNn)-Φf(h,y1,y2,y3,y4,ΔWn,ΔNn)|=|[I+C0h+C1|ΔWn|]-1f(x1,x3)h-[I+C0h+C1|ΔWn|]-1f(y1,y3)h|≤HKh(|x1-y1|+|x3-y3|).
Noticing that 𝔼|ΔWn|2=bh, 𝔼ΔNn2=λh(1+λh), the random variables x1, x3, y1, and y3 are ℱtn-measurable and using Assumptions 1–4, (4.27), we obtain
(4.29)E|Φg(h,x1,x3,ΔWn,ΔNn)-Φg(h,y1,y3,ΔWn,ΔNn)|2=E|[I+Cn)]-1g(x1,x3)ΔWn-[I+Cn]-1g(y1,y3)ΔWn|2≤H2bKh[E|x1-y1|2+E|x3-y3|2],(4.30)E|Φu(h,x1,x3,ΔWn,ΔNn)-Φu(h,y1,y3,ΔWn,ΔNn)|2=E|[I+Cn]-1u(x1,x3)ΔNn-[I+Cn]-1u(y1,y3)ΔNn|2≤H2Kλ(1+λT)h[E|x1-y1|2+E|x3-y3|2].
Since ΔWn,ΔNn are independent of ℱtn and the random variables x1, x3, y1, and y3 are ℱtn-measurable; thus, by the inequality |𝔼[x∣ℱ]|≤𝔼[|x|∣ℱ], 𝔼ΔNn=λh, and Lemma 4.6, we compute that
(4.31)E[Φg(h,x1,x3,ΔWn,ΔNn)-Φg(h,y1,y3,ΔWn,ΔNn)∣Ftn]=E[(I+Cn)-1g(x1,x3)ΔWn∣Ftn]-E[(I+Cn)-1g(y1,y3)ΔWn∣Ftn]=0,(4.32)|E[Φu(h,x1,x3,ΔWn,ΔNn)-Φu(h,y1,y3,ΔWn,ΔNn)∣Ftn]|≤E|[I+Cn]-1u(x1,x3)ΔNn-[I+Cn]-1u(y1,y3)ΔNn∣Ftn|≤|u(x1,x3)-u(y1,y3)|⋅E[|(I+Cn)-1|⋅ΔNn∣Ftn]≤HKλh[|x1-y1|+|x3-y3|].
From (4.28)–(4.32), we see that the increment functions of the balanced implicit methods (4.13) satisfy the conditions (3.7)–(3.11) with Lf=HK, Lg=H2Kb, Lu=H2Kλ(1+λT), and Lu¯=HKλ. A combination of Lemma 4.7 and Theorem 3.5 leads to the conclusion that the balanced implicit methods (4.13) are convergent with order p=1/2 in the mean square sense. The proof is completed.Remark 4.9.
For the case ofG(·)=0 and τ=0, (2.1) reduces to the stochastic differential equations with jumps
(4.33)dx(t)=f(t,x(t-))dt+g(t,x(t-))dW(t)+u(t,x(t-))dN(t),0<t<T,x(0-)=x0.
From Theorem 4.8, it is not difficult to find that the balanced implicit methods for (4.33) are convergent with order p=1/2 in the mean square sense, which coincides with Theorem 2.1 in [13].


## 5. Numerical Experiments

In this section, several numerical examples are given to illustrate our theoretical results in the previous sections. Consider the nonlinear equation(5.1)d[(x(t)-x(t-1))]=[-x(t)+x(t-1)1+x2(t-1)]dt+[sin⁡(x(t))+cos⁡(x(t-1))]dW(t)+12x(t)dN(t),t∈[0,T],with initial data(5.2)x(t)≡1,t∈[-1,0].To show the convergence of theθ-methods (4.1) and the balanced implicit methods (4.16), we choose θ=0.5, C0=1, and C1=0.5. In all the numerical experiments, we identify the numerical solution using very small stepsize h=Δt as the exact solution and compare this with the numerical approximations using h=4Δt,8Δt,16Δt,32Δt,64Δt for Δt=2-14 over 2000 different discretized Brownian paths. The mean-square errors εr=ε21+rΔt,r=1,2,3,4,5, all measured at time T=1, are estimated by trajectory averaging, that is, εr≅1/2000∑j=12000|xr,T(ωj)-Yr,q(ωj)|2. We plot our approximation to εr against h on a log-log scale in Figure 1. For reference, a dashed line of slope one-half is added in two graphs. In Figure 1., we show the convergence of the θ-method (4.1) in the left picture and the balanced implicit methods (4.16) in the right picture, respectively.The convergence of the numerical methods. (a) Stochasticθ-method (4.1); (b) the balanced implicit methods (4.16).
(a)(b)We see that the slopes of the two curves appear to match well in two pictures in Figure1, which is consistent with the strong order of one-half implied in Theorem 4.3 and Theorem 4.8.


## 6. Conclusion

In this paper, we consider a family of implicit one-step methods for the NSDDEs with jumps. A relationship between the consistent order and the convergence order is established. A general framework for mean-square convergence of the methods is provided. The applicability of the mean-square convergence theory is illustrated by the stochasticθ-methods and the balanced implicit methods. We have generalized the existing results. The examples presented in Section 4 show that the main result in this paper can be applied not only to semi-implicit methods but also to full implicit methods.



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