Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 675781 | 17 pages | https://doi.org/10.1155/2012/675781

Numerical Solutions of Stochastic Differential Equations Driven by Poisson Random Measure with Non-Lipschitz Coefficients

Academic Editor: Said Abbasbandy
Received15 Jan 2012
Revised20 Mar 2012
Accepted22 Mar 2012
Published10 Jul 2012

Abstract

The numerical methods in the current known literature require the stochastic differential equations (SDEs) driven by Poisson random measure satisfying the global Lipschitz condition and the linear growth condition. In this paper, Euler's method is introduced for SDEs driven by Poisson random measure with non-Lipschitz coefficients which cover more classes of such equations than before. The main aim is to investigate the convergence of the Euler method in probability to such equations with non-Lipschitz coefficients. Numerical example is given to demonstrate our results.

1. Introduction

In finance market and other areas, it is meaningful and significant to model the impact of event-driven uncertainty. Events such as corporate defaults, operational failures, market crashes, or central bank announcements require the introduction of stochastic differential equations (SDEs) driven by Poisson random measure (see [1, 2]), since such equations were initiated in [3].

Actually, we can only obtain the explicit solutions of a small class of SDEs driven by Poisson random measure and so numerical methods are necessary. In general, numerical methods can be divided into strong approximations and weak approximations. Strong approximations focus on pathwise approximations while weak approximations (see [4, 5]) are fit for problems such as derivative pricing.

We give an overview of the results on the strong approximations of stochastic differential equations (SDEs) driven by Poisson random measure in the existing literature. In [6], a convergence result for strong approximations of any given order π›Ύβˆˆ{0.5,1,1.5,…} was presented. Moreover, N. Bruti-Liberatiand E. Platen (see [7]) obtain the jump-adapted order 1.5 scheme, and they also give the derivative-free or implicit jump-adapted schemes with desired order of strong convergence. And for the specific case of pure jump SDEs, they (see [8]) establish the strong convergence of Taylor’s methods under weaker conditions than the currently known. In [5, 7], the drift-implicit schemes which achieve strong order π›Ύβˆˆ{0.5,1} are given. Recently, Mordecki et al. [9] improved adaptive time stepping algorithms based on a jump augmented Monte Carlo Euler-Maruyama method, which achieve a prescribed precision. M. Wei [10] demonstrates the convergence of numerical solutions for variable delay differential equations driven by Poisson random measure. In [11], the developed Runge-Kutta methods are presented to improve the accuracy behaviour of problems with small noise to SDEs with Poisson random measure.

Clearly, the results above require that the SDEs driven by Poisson random measure satisfy the global Lipschitz condition and the linear growth condition. However, there are many equations which do not satisfy above conditions, and we can see such equations in Section 5 in our paper. Our main contribution is to present Euler’s method for these equations with non-Lipschitz coefficients. Here non-Lipschitz coefficients are interpreted in [12], that is to say, the drift coefficients and the diffusion coefficients satisfy the local Lipschitz conditions, the jump coefficients satisfy the global Lipschitz conditions, and the one-sided linear growth condition is considered. Our work is motivated by [12] in which the existence of global solutions for these equations with non-Lipschitz coefficients is proved, while there is no numerical method is presented in our known literature. And our aim in this paper is to close this gap.

Our work is organized as follows. In Section 2, the property of SDEs driven by Poisson random measure with non-Lipschitz coefficients is given. In Section 3, Euler method is analyzed for such equations. In Section 4, we present the convergence in probability of the Euler method. In Section 5, an example is presented.

2. The SDEs Driven by Poisson Random Measure with Non-Lipschitz Coefficients

Throughout this paper, unless specified, we use the following notations. Let 𝑒1βˆ¨π‘’2=max{𝑒1,𝑒2} and 𝑒1βˆ§π‘’2=min{𝑒1,𝑒2}. Let |β‹…| and βŸ¨β‹…,β‹…βŸ© be the Euclidean norm and the inner product of vectors in 𝐑𝑑,π‘‘βˆˆπ. If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴𝑇. If 𝐴 is a matrix, its trace norm is denoted by √|𝐴|=trace(𝐴𝑇𝐴). Let 𝐿2β„±0(Ξ©;πšπ‘‘) denote the family of 𝐑𝑑-valued β„±0-measurable random variables πœ‰ with 𝐄|πœ‰|2<∞. [𝑧] denotes the largest integer which is less than or equal to 𝑧 in 𝐑. Iπ’œ denotes the indicator function of a set π’œ.

The following 𝑑-dimensional SDE driven by Poisson random measure is considered in our paper: ξ€œπ‘‘π‘₯(𝑑)=π‘Ž(π‘₯(π‘‘βˆ’))𝑑𝑑+𝑏(π‘₯(π‘‘βˆ’))π‘‘π‘Š(𝑑)+πœ€π‘(π‘₯(π‘‘βˆ’),𝑣)Μƒπ‘πœ™(𝑑𝑣×𝑑𝑑),(2.1) for 𝑑>0 with initial condition π‘₯(0βˆ’)=π‘₯(0)=π‘₯0∈𝐿2β„±0(Ξ©;𝐑𝑑), where π‘₯(π‘‘βˆ’) denotes limπ‘ β†’π‘‘βˆ’π‘₯(𝑠) and Μƒπ‘πœ™(𝑑𝑣×𝑑𝑑)∢=π‘πœ™(𝑑𝑣×𝑑𝑑)βˆ’πœ™(𝑑𝑣)𝑑𝑑.

The drift coefficient π‘ŽβˆΆπ‘π‘‘β†’π‘π‘‘, the diffusion coefficient π‘βˆΆπ‘π‘‘β†’π‘π‘‘Γ—π‘š, and the jump coefficient π‘βˆΆπ‘π‘‘Γ—πœ€β†’π‘π‘‘ are assumed to be Borel measurable functions.

The randomness of (2.1) is generated by the following (see [9]). An π‘š-dimensional Wiener process π‘Š={π‘Š(𝑑)=(π‘Š1(𝑑),…,π‘Šπ‘š(𝑑))𝑇} with independent scalar components is defined on a filtered probability space (Ξ©π‘Š,β„±π‘Š,(β„±π‘Šπ‘‘)𝑑β‰₯0,ππ‘Š). A Poisson random measure π‘πœ™(πœ”,𝑑𝑣×𝑑𝑑) is on Ξ©π½Γ—πœ€Γ—[0,∞), where πœ€βŠ†π‘π‘Ÿβ§΅{0} with π‘Ÿβˆˆπ, and its deterministic compensated measure πœ™(𝑑𝑣)𝑑𝑑=πœ†π‘“(𝑣)𝑑𝑣𝑑𝑑, that is, 𝐄(π‘πœ™(𝑑𝑣×𝑑𝑑))=πœ™(𝑑𝑣)𝑑𝑑. 𝑓(𝑣) is a probability density, and we require finite intensity πœ†=πœ™(πœ€)<∞. The Poisson random measure is defined on a filtered probability space (Ω𝐽,ℱ𝐽,(ℱ𝐽𝑑)𝑑β‰₯0,𝐏𝐽). The Wiener process and the Poisson random measure are mutually independent. The process π‘₯(𝑑) is thus defined on a product space (Ξ©,β„±,(ℱ𝑑)𝑑β‰₯0,𝐏), where Ξ©=Ξ©π‘ŠΓ—Ξ©π½,β„±=β„±π‘ŠΓ—β„±π½,(ℱ𝑑)𝑑β‰₯0=(β„±π‘Šπ‘‘)𝑑β‰₯0Γ—(ℱ𝐽𝑑)𝑑β‰₯0,𝐏=ππ‘ŠΓ—ππ½ and β„±0 contains all 𝐏-null sets.

Now, the condition of non-Lipschitz coefficients is given by the following assumptions.

Assumption 2.1. For each integer π‘˜β‰₯1, there exists a positive constant πΆπ‘˜, dependent on π‘˜, such that ||||π‘Ž(π‘₯)βˆ’π‘Ž(𝑦)2∨||||𝑏(π‘₯)βˆ’π‘(𝑦)2β‰€πΆπ‘˜||||π‘₯βˆ’π‘¦2,(2.2) for π‘₯,π‘¦βˆˆπ‘π‘‘ with |π‘₯|∨|𝑦|β‰€π‘˜,π‘˜βˆˆπ. And there exists a positive constant 𝐢 such that ξ€œπœ€||||𝑐(π‘₯,𝑣)βˆ’π‘(𝑦,𝑣)2||||πœ™(𝑑𝑣)≀𝐢π‘₯βˆ’π‘¦2,(2.3) for π‘₯,π‘¦βˆˆπ‘π‘‘.

Assumption 2.2. There exists a positive constant 𝐿 such that ||||2⟨π‘₯,π‘Ž(π‘₯)⟩+𝑏(π‘₯)2+ξ€œπœ€||||𝑐(π‘₯,𝑣)2ξ€·πœ™(𝑑𝑣)≀𝐿1+|π‘₯|2ξ€Έ,(2.4) for π‘₯βˆˆπ‘π‘‘.

A unique global solution of (2.1) exists under Assumptions 2.1 and 2.2, see [12].

Assumption 2.3. Consider ||||π‘Ž(0)2+||||𝑏(0)2+ξ€œπœ€||||𝑐(0,𝑣)2ξ‚ξ‚πœ™(𝑑𝑣)≀𝐿,𝐿>0.(2.5)

Actually, Assumptions 2.1 and 2.3 imply the linear growth conditions ||||π‘Ž(π‘₯)2∨||||𝑏(π‘₯)2β‰€ξ‚‹πΆπ‘˜ξ€·1+|π‘₯|2ξ€Έ,(2.6) for π‘₯βˆˆπ‘π‘‘ with |π‘₯|β‰€π‘˜ and ξ‚‹πΆπ‘˜>0, and ξ€œπœ€||||𝑐(π‘₯,𝑣)2ξ‚πΆξ€·πœ™(𝑑𝑣)≀1+|π‘₯|2ξ€Έ,(2.7) for π‘₯βˆˆπ‘π‘‘ and 𝐢>0.

The following result shows that the solution of (2.1) keeps in a compact set with a large probability.

Lemma 2.4. Under Assumptions 2.1 and 2.2, for any pair of πœ–βˆˆ(0,1) and 𝑇>0, there exists a sufficiently large integer π‘˜βˆ—, dependent on πœ– and 𝑇, such that πξ€·πœπ‘˜ξ€Έβ‰€π‘‡β‰€πœ–,βˆ€π‘˜β‰₯π‘˜βˆ—,(2.8) where πœπ‘˜=inf{𝑑β‰₯0∢|π‘₯(𝑑)|β‰₯π‘˜} for π‘˜β‰₯1.

Proof. Using ItΓ΄'s formula (see [1]) to |π‘₯(𝑑)|2, for 𝑑β‰₯0, we have ||||π‘₯(𝑑)2=||π‘₯0||2+ξ€œπ‘‘0ξ‚€||||⟨2π‘₯(π‘ βˆ’),π‘Ž(π‘₯(π‘ βˆ’))⟩+𝑏(π‘₯(π‘ βˆ’))2+ξ€œπ‘‘s𝑑0ξ€œπœ€ξ‚€||||π‘₯(π‘ βˆ’)+𝑐(π‘₯(π‘ βˆ’),𝑣)2βˆ’||||π‘₯(π‘ βˆ’)2+ξ€œβˆ’βŸ¨2π‘₯(π‘ βˆ’),𝑐(π‘₯(π‘ βˆ’),𝑣)βŸ©πœ™(𝑑𝑣)𝑑𝑠𝑑0ξ€œβŸ¨2π‘₯(π‘ βˆ’),𝑏(π‘₯(π‘ βˆ’))βŸ©π‘‘π‘Š(𝑠)+𝑑0ξ€œπœ€ξ‚€||||π‘₯(π‘ βˆ’)+𝑐(π‘₯(π‘ βˆ’),𝑣)2βˆ’||||π‘₯(π‘ βˆ’)2ξ‚Μƒπ‘πœ™(𝑑𝑣×𝑑𝑠),(2.9) which gives 𝐄||π‘₯ξ€·π‘‘βˆ§πœπ‘˜ξ€Έ||2||π‘₯=𝐄0||2ξ€œ+π„π‘‘βˆ§πœπ‘˜0ξ‚€||||⟨2π‘₯(π‘ βˆ’),π‘Ž(π‘₯(π‘ βˆ’))⟩+𝑏(π‘₯(π‘ βˆ’))2ξ‚ξ€œπ‘‘π‘ +π„π‘‘βˆ§πœπ‘˜0ξ€œπœ€||||𝑐(π‘₯(π‘ βˆ’),𝑣)2||π‘₯πœ™(𝑑𝑣)𝑑𝑠=𝐄0||2ξ€œ+π„π‘‘βˆ§πœπ‘˜0ξ‚΅||||⟨2π‘₯(π‘ βˆ’),π‘Ž(π‘₯(π‘ βˆ’))⟩+𝑏(π‘₯(π‘ βˆ’))2+ξ€œπœ€||||𝑐(π‘₯(π‘ βˆ’),𝑣)2ξ‚Άπœ™(𝑑𝑣)𝑑𝑠,(2.10) for π‘‘βˆˆ[0,𝑇]. By Assumption 2.2, we thus have 𝐄||π‘₯ξ€·π‘‘βˆ§πœπ‘˜ξ€Έ||2||π‘₯≀𝐄0||2ξ€œ+π„π‘‘βˆ§πœπ‘˜0𝐿||||1+π‘₯(π‘ βˆ’)2||π‘₯𝑑𝑠≀𝐄0||2ξ€œ+𝐿𝑇+𝐿𝑑0𝐄||π‘₯ξ€·π‘ βˆ§πœπ‘˜βˆ’ξ€Έ||2𝑑𝑠,(2.11) for π‘‘βˆˆ[0,𝑇]. Consequently by using the Gronwall inequality (see [13]), we obtain 𝐄||π‘₯ξ€·π‘‘βˆ§πœπ‘˜ξ€Έ||2≀𝐄||π‘₯0||2𝑒+𝐿𝑇𝐿𝑇,(2.12) for π‘‘βˆˆ[0,𝑇]. We therefore get 𝐄||π‘₯0||2𝑒+𝐿𝑇𝐿𝑇||π‘₯ξ€·β‰₯π„π‘‡βˆ§πœπ‘˜ξ€Έ||2ξ‚€||π‘₯ξ€·πœβ‰₯π„π‘˜ξ€Έ||2𝐼{πœπ‘˜β‰€π‘‡}β‰₯π‘˜2πξ€·πœπ‘˜ξ€Έβ‰€π‘‡,(2.13) which means πξ€·πœπ‘˜ξ€Έβ‰€π‘’β‰€π‘‡πΏπ‘‡π‘˜2𝐄||π‘₯0||2.+𝐿𝑇(2.14) So for any πœ–βˆˆ(0,1), we can choose π‘˜βˆ—=βŽ‘βŽ’βŽ’βŽ£ξƒŽπ‘’πΏπ‘‡π„||π‘₯0||2+πΏπ‘‡π‘’πΏπ‘‡πœ–βŽ€βŽ₯βŽ₯⎦+1,(2.15) such that πξ€·πœπ‘˜ξ€Έβ‰€π‘‡β‰€πœ–,βˆ€π‘˜β‰₯π‘˜βˆ—.(2.16) Hence, we have the result (2.8).

3. The Euler Method

In this section, we introduce the Euler method to (2.1) under Assumptions 2.1, 2.2, and 2.3. Subsequently, we give two lemmas to analyze the Euler method over a finite time interval [0,𝑇], where 𝑇 is a positive number.

Given a step size Ξ”π‘‘βˆˆ(0,1), the Euler method applied to (2.1) computes approximation π‘‹π‘›β‰ˆπ‘₯(𝑑𝑛), where 𝑑𝑛=𝑛Δ𝑑,𝑛=0,1,…, by setting 𝑋0=π‘₯0 and forming 𝑋𝑛+1=𝑋𝑛𝑋+π‘Žπ‘›ξ€Έξ€·π‘‹Ξ”π‘‘+π‘π‘›ξ€ΈΞ”π‘Šπ‘›+ξ€œπ‘‘π‘›+1π‘‘π‘›ξ€œπœ€π‘ξ€·π‘‹π‘›ξ€Έ,π‘£Μƒπ‘πœ™(𝑑𝑣×𝑑𝑑),(3.1) where Ξ”π‘Šπ‘›=π‘Š(𝑑𝑛+1)βˆ’π‘Š(𝑑𝑛).

The continuous-time Euler method is then defined by 𝑋(𝑑)∢=𝑋0+ξ€œπ‘‘0ξ€œπ‘Ž(𝑍(𝑠))𝑑𝑠+𝑑0ξ€œπ‘(𝑍(𝑠))π‘‘π‘Š(𝑠)+𝑑0ξ€œπœ€π‘(𝑍(𝑠),𝑣)Μƒπ‘πœ™(𝑑𝑣×𝑑𝑠),(3.2) where 𝑍(𝑑)=𝑋𝑛 for π‘‘βˆˆ[𝑑𝑛,𝑑𝑛+1),𝑛=0,1,….

Actually, we can see in [8], π‘πœ™={π‘πœ™(𝑑)∢=π‘πœ™(πœ€Γ—[0,𝑑])} is a stochastic process that counts the number of jumps until some given time. The Poisson random measure π‘πœ™(𝑑𝑣×𝑑𝑑) generates a sequence of pairs {(πœ„π‘–,πœ‰π‘–),π‘–βˆˆ{1,2,…,π‘πœ™(𝑇)}} for a given finite positive constant 𝑇 if πœ†<∞. Here {πœ„π‘–βˆΆΞ©β†’π‘+,π‘–βˆˆ{1,2,…,π‘πœ™(𝑇)}} is a sequence of increasing nonnegative random variables representing the jump times of a standard Poisson process with intensity πœ†, and {πœ‰π‘–βˆΆΞ©β†’πœ€,π‘–βˆˆ{1,2,…,π‘πœ™(𝑇)} is a sequence of independent identically distributed random variables, where πœ‰π‘– is distributed according to πœ™(𝑑𝑣)/πœ™(πœ€). Then (3.1) can equivalently be the following form: 𝑋𝑛+1=𝑋𝑛+ξ‚΅π‘Žξ€·π‘‹π‘›ξ€Έβˆ’ξ€œπœ€π‘ξ€·π‘‹π‘›ξ€Έξ‚Άξ€·π‘‹,π‘£πœ™(𝑑𝑣)Δ𝑑+π‘π‘›ξ€ΈΞ”π‘Šπ‘›+π‘πœ™(𝑑𝑛+1)𝑖=π‘πœ™(𝑑𝑛)+1𝑐𝑋𝑛,πœ‰π‘–ξ€Έ.(3.3)

The following lemma shows the close relation between the continuous-time Euler method (3.2) and its step function 𝑍(𝑑).

Lemma 3.1. Under Assumptions 2.1 and 2.3, for any 𝑇>0, there exists a positive constant 𝐾1(π‘˜), dependent on π‘˜ and independent of Δ𝑑, such that for all Ξ”π‘‘βˆˆ(0,1) the continuous-time Euler method (3.2) satisfies 𝐄||||𝑋(𝑑)βˆ’π‘(𝑑)2≀𝐾1(π‘˜)Δ𝑑,(3.4) for 0β‰€π‘‘β‰€π‘‡βˆ§πœπ‘˜βˆ§πœŒπ‘˜, where πœŒπ‘˜=inf{𝑑β‰₯0∢|𝑋(𝑑)|β‰₯π‘˜} for π‘˜β‰₯1 and πœπ‘˜ is defined in Lemma 2.4.

Proof. For 0β‰€π‘‘β‰€π‘‡βˆ§πœπ‘˜βˆ§πœŒπ‘˜, there is an integer 𝑛 such that π‘‘βˆˆ[𝑑𝑛,𝑑𝑛+1). So it follows from (3.2) that 𝑋(𝑑)βˆ’π‘(𝑑)=𝑋𝑛+ξ€œπ‘‘π‘‘π‘›ξ€œπ‘Ž(𝑍(𝑠))𝑑𝑠+π‘‘π‘‘π‘›ξ€œπ‘(𝑍(𝑠))π‘‘π‘Š(𝑠)+π‘‘π‘‘π‘›ξ€œπœ€π‘(𝑍(𝑠),𝑣)Μƒπ‘πœ™(𝑑𝑣×𝑑𝑠)βˆ’π‘‹π‘›.(3.5) Thus, by taking expectations and using the Cauchy-Schwarz inequality and the martingale properties of π‘‘π‘Š(𝑑) and Μƒπ‘πœ™(𝑑𝑣×𝑑𝑑), we have 𝐄||||𝑋(𝑑)βˆ’π‘(𝑑)2||||ξ€œβ‰€3𝐄𝑑𝑑𝑛||||π‘Ž(𝑍(𝑠))𝑑𝑠2||||ξ€œ+3𝐄𝑑𝑑𝑛||||𝑏(𝑍(𝑠))π‘‘π‘Š(𝑠)2||||ξ€œ+3π„π‘‘π‘‘π‘›ξ€œπœ€π‘(𝑍(𝑠),𝑣)Μƒπ‘πœ™||||(𝑑𝑣×𝑑𝑠)2ξ‚΅ξ€œβ‰€3𝐄𝑑𝑑𝑛12ξ€œπ‘‘π‘ π‘‘π‘‘π‘›||||π‘Ž(𝑍(𝑠))2ξ‚Άξ€œπ‘‘π‘ +3𝐄𝑑𝑑𝑛||||𝑏(𝑍(𝑠))2ξ€œπ‘‘π‘ +3π„π‘‘π‘‘π‘›ξ€œπœ€||||𝑐(𝑍(𝑠),𝑣)2ξ€œπœ™(𝑑𝑣)𝑑𝑠≀3Δ𝑑𝐄𝑑𝑑𝑛||||π‘Ž(𝑍(𝑠))2ξ€œπ‘‘π‘ +3𝐄𝑑𝑑𝑛||||𝑏(𝑍(𝑠))2ξ€œπ‘‘π‘ +3π„π‘‘π‘‘π‘›ξ€œπœ€||||𝑐(𝑍(𝑠),𝑣)2πœ™(𝑑𝑣)𝑑𝑠,(3.6) where the inequality |𝑒1+𝑒2+𝑒3|2≀3|𝑒1|2+3|𝑒2|2+3|𝑒3|2 for 𝑒1,𝑒2,𝑒3βˆˆπ‘π‘‘ is used. Therefore, by applying Assumptions 2.1 and 2.3, we get π„ξ€œπ‘‘π‘‘π‘›||||π‘Ž(𝑍(𝑠))2ξ‚‹πΆπ‘‘π‘ β‰€π‘˜π„ξ€œπ‘‘π‘‘π‘›ξ‚€||||1+𝑍(𝑠)2ξ‚ξ‚‹πΆπ‘‘π‘ β‰€π‘˜ξ‚πΆΞ”π‘‘+π‘˜π‘˜2π„ξ€œΞ”π‘‘,𝑑𝑑𝑛||||𝑏(𝑍(𝑠))2ξ‚‹πΆπ‘‘π‘ β‰€π‘˜ξ‚πΆΞ”π‘‘+π‘˜π‘˜2π„ξ€œΞ”π‘‘,π‘‘π‘‘π‘›ξ€œπœ€||||𝑐(𝑍(𝑠),𝑣)2ξ‚ξ‚πœ™(𝑑𝑣)𝑑𝑠≀𝐢Δ𝑑+πΆπ‘˜2Δ𝑑,(3.7) which lead to 𝐄||||𝑋(𝑑)βˆ’π‘(𝑑)2ξ‚€3ξ‚‹πΆβ‰€Ξ”π‘‘π‘˜Ξ”π‘‘+3π‘˜2ξ‚‹πΆπ‘˜ξ‚‹πΆΞ”π‘‘+3π‘˜+3π‘˜2ξ‚‹πΆπ‘˜ξ‚+3𝐢+3π‘˜2𝐢,(3.8) for π‘‘βˆˆ[0,π‘‡βˆ§πœπ‘˜βˆ§πœŒπ‘˜]. Therefore, we obtain the result (3.4) by choosing 𝐾1𝐢(π‘˜)=6π‘˜+6π‘˜2ξ‚‹πΆπ‘˜ξ‚+3𝐢+3π‘˜2𝐢.(3.9)

In accord with Lemma 2.4, we give the following lemma which demonstrates that the solution of continuous-time Euler method (3.2) remains in a compact set with a large probability.

Lemma 3.2. Under Assumptions 2.1, 2.2, and 2.3, for any pair of πœ–βˆˆ(0,1) and 𝑇>0, there exist a sufficiently large π‘˜βˆ— and a sufficiently small Ξ”π‘‘βˆ—1 such that πξ€·πœŒπ‘˜βˆ—ξ€Έβ‰€π‘‡β‰€πœ–,βˆ€Ξ”π‘‘β‰€Ξ”π‘‘βˆ—1,(3.10) where πœŒπ‘˜βˆ— is defined in Lemma 3.1.

Proof . Applying generalized ItΓ΄'s formula (see [1]) to |𝑋(𝑑)|2, for 𝑑β‰₯0, yields ||||𝑋(𝑑)2=||𝑋0||2+ξ€œπ‘‘02+||||𝑋(𝑠),π‘Ž(𝑍(𝑠))𝑏(𝑍(𝑠))2+ξ€œπ‘‘π‘ π‘‘0ξ€œπœ€ξ‚€||||𝑋(𝑠)+𝑐(𝑍(𝑠),𝑣)2βˆ’||||𝑋(𝑠)2βˆ’ξ‚¬2+ξ€œπ‘‹(𝑠),𝑐(𝑍(𝑠),𝑣)ξ‚­ξ‚πœ™(𝑑𝑣)𝑑𝑠𝑑02ξ‚­ξ€œπ‘‹(𝑠),𝑏(𝑍(𝑠))π‘‘π‘Š(𝑠)+𝑑0ξ€œπœ€ξ‚€||||𝑋(𝑠)+𝑐(𝑍(𝑠),𝑣)2βˆ’||||𝑋(𝑠)2ξ‚Μƒπ‘πœ™(𝑑𝑣×𝑑𝑠).(3.11) By taking expectations, we thus have 𝐄||π‘‹ξ€·π‘‘βˆ§πœŒπ‘˜ξ€Έ||2||𝑋=𝐄0||2ξ€œ+π„π‘‘βˆ§πœŒπ‘˜02ξ‚­+||||𝑋(𝑠),π‘Ž(𝑍(𝑠))𝑏(𝑍(𝑠))2+ξ€œπœ€||||𝑐(𝑍(𝑠),𝑣)2ξ‚Ά||π‘‹πœ™(𝑑𝑣)𝑑𝑠=𝐄0||2ξ€œ+Eπ‘‘βˆ§πœŒπ‘˜02𝑋(𝑠),π‘Ž+|||𝑏𝑋(𝑠)|||𝑋(𝑠)2+ξ€œπœ€|||𝑐𝑋|||(𝑠),𝑣2πœ™ξ‚Άξ€œ(𝑑𝑣)𝑑𝑠+π„π‘‘βˆ§πœŒπ‘˜02𝑋(𝑠),π‘Ž(𝑍(𝑠))βˆ’π‘Žπ‘‹ξ€œ(𝑠)𝑑𝑠+π„π‘‘βˆ§πœŒπ‘˜0ξ‚΅||||𝑏(𝑍(𝑠))2βˆ’|||𝑏|||𝑋(𝑠)2ξ‚Άξ€œπ‘‘π‘ +π„π‘‘βˆ§πœŒπ‘˜0ξ€œπœ€ξ‚΅||||𝑐(𝑍(𝑠),𝑣)2βˆ’|||𝑐|||𝑋(𝑠),𝑣2ξ‚Άπœ™(𝑑𝑣)𝑑𝑠.(3.12) For π‘‘βˆˆ[0,𝑇]. Now, by using the inequalities βŸ¨π‘’1,𝑒2βŸ©β‰€|𝑒1||𝑒2| for 𝑒1,𝑒2βˆˆπ‘π‘‘, (2.2) in Assumption 2.1, Fubini's theorem, Cauchy-Schwarz’s inequality, and Lemma 3.1, we get π„ξ€œπ‘‘βˆ§πœŒπ‘˜02𝑋(𝑠),π‘Ž(𝑍(𝑠))βˆ’π‘Žξ€œπ‘‹(𝑠)𝑑𝑠≀2π„π‘‘βˆ§πœŒπ‘˜0|||||||𝑋(𝑠)π‘Ž(𝑍(𝑠))βˆ’π‘Žξ‚|||βˆšπ‘‹(𝑠)𝑑𝑠≀2π‘˜πΆπ‘˜ξ€œπ‘‘0𝐄||π‘ξ€·π‘ βˆ§πœŒπ‘˜ξ€Έβˆ’π‘‹ξ€·π‘ βˆ§πœŒπ‘˜ξ€Έ||βˆšπ‘‘π‘ β‰€2π‘˜πΆπ‘˜ξ€œπ‘‘0𝐄||π‘ξ€·π‘ βˆ§πœŒπ‘˜ξ€Έβˆ’π‘‹ξ€·π‘ βˆ§πœŒπ‘˜ξ€Έ||21/2βˆšπ‘‘π‘ β‰€2π‘˜π‘‡πΆπ‘˜πΎ1(π‘˜)Δ𝑑.(3.13) And, similarly as above, we have π„ξ€œπ‘‘βˆ§πœŒπ‘˜0ξ‚΅||||𝑏(𝑍(𝑠))2βˆ’|||𝑏|||𝑋(𝑠)2ξ‚Άξ€œπ‘‘π‘ β‰€π„π‘‘βˆ§πœŒπ‘˜0ξ‚€||||+|||𝑏𝑏(𝑍(𝑠))|||×||||βˆ’|||𝑏𝑋(𝑠)𝑏(𝑍(𝑠))|||𝑋(𝑠)𝑑𝑠≀2ξ‚‹πΆπ‘˜ξ€·1+π‘˜2ξ€Έπ„ξ€œπ‘‘βˆ§πœŒπ‘˜0|||𝑏(𝑍(𝑠))βˆ’π‘ξ‚|||𝑋(𝑠)𝑑𝑠≀2πΆπ‘˜ξ‚‹πΆπ‘˜ξ€·1+π‘˜2ξ€Έξ€œπ‘‘0𝐄||π‘ξ€·π‘ βˆ§πœŒπ‘˜ξ€Έβˆ’π‘‹ξ€·π‘ βˆ§πœŒπ‘˜ξ€Έ||𝑑𝑠≀2π‘‡πΆπ‘˜ξ‚‹πΆπ‘˜πΎ1ξ€·(π‘˜)1+π‘˜2Δ𝑑.(3.14) Moreover, in the same way, we obtain π„ξ€œπ‘‘βˆ§πœŒπ‘˜0ξ€œπœ€ξ‚΅||||𝑐(𝑍(𝑠),𝑣)2βˆ’|||𝑐|||𝑋(𝑠),𝑣2ξ‚Άξ€œπœ™(𝑑𝑣)𝑑𝑠=π„π‘‘βˆ§πœŒπ‘˜0ξ€œπœ€ξ‚΅|||𝑐(𝑍(𝑠),𝑣)βˆ’π‘ξ‚ξ‚€π‘‹(𝑠),𝑣+𝑐|||𝑋(𝑠),𝑣2βˆ’|||𝑐|||𝑋(𝑠),𝑣2ξ‚Άξ€œπœ™(𝑑𝑣)π‘‘π‘ β‰€π„π‘‘βˆ§πœŒπ‘˜0ξ€œπœ€ξ‚΅2|||𝑐(𝑍(𝑠),𝑣)βˆ’π‘ξ‚|||𝑋(𝑠),𝑣2+|||𝑐|||𝑋(𝑠),𝑣2ξ‚Άξ€œπœ™(𝑑𝑣)𝑑𝑠≀2πΆπ„π‘‘βˆ§πœŒπ‘˜0||𝑍(𝑠)βˆ’||𝑋(𝑠)2ξ‚ξ€œπ‘‘π‘ +πΆπ„π‘‘βˆ§πœŒπ‘˜0ξ‚€||1+||𝑋(𝑠)2ξ‚ξ€œπ‘‘π‘ β‰€2𝐢𝑑0𝐄||π‘ξ€·π‘ βˆ§πœŒπ‘˜ξ€Έβˆ’π‘‹ξ€·π‘ βˆ§πœŒπ‘˜ξ€Έ||2ξ‚ξ€œπ‘‘π‘ +πΆπ„π‘‘βˆ§πœŒπ‘˜0ξ‚€||1+||𝑋(𝑠)2𝑑𝑠≀2𝐢𝑇𝐾1ξ‚ξ‚ξ€œ(π‘˜)Δ𝑑+𝐢𝑇+πΆπ„π‘‘βˆ§πœŒπ‘˜0||||𝑋(𝑠)2𝑑𝑠,(3.15) where the inequality |𝑒1+𝑒2|2≀2|𝑒1|2+2|𝑒2|2 for 𝑒1,𝑒2βˆˆπ‘π‘‘, (2.3) in Assumptions 2.1 and 2.3, Fubini's theorem, and Lemma 3.1 are used. Subsequently, substituting (3.13), (3.14), and (3.15) into (3.12) together with Assumption 2.2 leads to 𝐄||π‘‹ξ€·π‘‘βˆ§πœŒπ‘˜ξ€Έ||2||𝑋≀𝐄0||2ξ€œ+πΏπ„π‘‘βˆ§πœŒπ‘˜0ξ‚€||1+||𝑋(𝑠)2ξ‚ξ‚ξ€œπ‘‘π‘ +πΆπ„π‘‘βˆ§πœŒπ‘˜0||||𝑋(𝑠)2βˆšπ‘‘π‘ +2π‘˜π‘‡πΆπ‘˜πΎ1(π‘˜)Δ𝑑+2π‘‡πΆπ‘˜ξ‚‹πΆπ‘˜πΎ1ξ€·(π‘˜)1+π‘˜2Δ𝑑+2𝐢𝑇𝐾1ξ‚β‰€ξ‚€ξ‚πΆξ‚ξ€œ(π‘˜)Δ𝑑+𝐢𝑇𝐿+𝑑0𝐄||π‘‹ξ€·π‘ βˆ§πœŒπ‘˜ξ€Έ||2||𝑋𝑑𝑠+𝐄0||2+ξ‚΅βˆš+𝐿𝑇+𝐢𝑇2π‘˜π‘‡πΆπ‘˜πΎ1(π‘˜)+2π‘‡πΆπ‘˜ξ‚‹πΆπ‘˜πΎ1ξ€·(π‘˜)1+π‘˜2ξ€Έξ‚ΆβˆšΞ”π‘‘+2𝐢𝑇𝐾1(π‘˜)Δ𝑑,(3.16) for 0≀𝑑≀𝑇. Therefore, by the Gronwall inequality (see [13]), for 0≀𝑑≀𝑇, we get 𝐄||π‘‹ξ€·π‘‘βˆ§πœŒπ‘˜ξ€Έ||2≀𝛼1𝛼4+𝛼4𝛼2√(π‘˜)Δ𝑑+𝛼4𝛼3(π‘˜)Δ𝑑,(3.17) where 𝛼1||𝑋=𝐄0||2𝛼+𝐿𝑇+𝐢𝑇,2√(π‘˜)=2π‘˜π‘‡πΆπ‘˜πΎ1(π‘˜)+2π‘‡πΆπ‘˜ξ‚‹πΆπ‘˜πΎ1ξ€·(π‘˜)1+π‘˜2ξ€Έ,𝛼3(π‘˜)=2𝐢𝑇𝐾1𝛼(π‘˜),4.=exp𝐿𝑇+𝐢𝑇(3.18) We thus obtain that π‘˜2πξ€·πœŒπ‘˜ξ€Έξ‚€||β‰€π‘‡β‰€π„π‘‹ξ€·πœŒπ‘˜ξ€Έ||2𝐼{πœŒπ‘˜β‰€π‘‡}||β‰€π„π‘‹ξ€·π‘‡βˆ§πœŒπ‘˜ξ€Έ||2≀𝛼1𝛼4+𝛼4𝛼2√(π‘˜)Δ𝑑+𝛼4𝛼3(π‘˜)Δ𝑑.(3.19) So for any πœ–βˆˆ(0,1), we can choose sufficiently large integer π‘˜=π‘˜βˆ— such that 𝛼1𝛼4π‘˜βˆ—2β‰€πœ–2,(3.20) and choose sufficiently small Ξ”π‘‘βˆ—1∈(0,1) such that 𝛼4𝛼2ξ€·π‘˜βˆ—ξ€ΈβˆšΞ”π‘‘βˆ—1+𝛼4𝛼3ξ€·π‘˜βˆ—ξ€ΈΞ”π‘‘βˆ—1π‘˜βˆ—2β‰€πœ–2.(3.21) Hence, we have πξ€·πœŒπ‘˜βˆ—ξ€Έβ‰€π‘‡β‰€πœ–,βˆ€Ξ”π‘‘β‰€Ξ”π‘‘βˆ—1.(3.22)

4. Convergence in Probability

In this section, we present two convergence theorems of the Euler method to the SDE with Poisson random measure (2.1) over a finite time interval [0,𝑇].

At the beginning, we give a lemma based on Lemma 3.1.

Lemma 4.1. Under Assumptions 2.1 and 2.3, for any 𝑇>0, there exists a positive constant 𝐾2(π‘˜), dependent on π‘˜ and independent of Δ𝑑, such that for all Ξ”π‘‘βˆˆ(0,1) the solution of (2.1) and the continuous-time Euler method (3.2) satisfy 𝐄sup0≀𝑑≀𝑇||π‘₯ξ€·π‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έβˆ’π‘‹ξ€·π‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2≀𝐾2(π‘˜)Δ𝑑,(4.1) where πœπ‘˜ and πœŒπ‘˜ are defined in Lemmas 2.4 and 3.1, respectively.

Proof. From (2.1) and (3.2), for any 0≀𝑑′≀𝑇, we have 𝐄sup0≀𝑑≀𝑑′||π‘₯ξ€·π‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έβˆ’π‘‹ξ€·π‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2ξƒͺ≀3𝐄sup0≀𝑑≀𝑑′||||ξ€œπ‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||||(π‘Ž(π‘₯(π‘ βˆ’))βˆ’π‘Ž(𝑍(𝑠)))𝑑𝑠2ξƒͺ+3𝐄sup0≀𝑑≀𝑑′||||ξ€œπ‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||||(𝑏(π‘₯(π‘ βˆ’))βˆ’π‘(𝑍(𝑠)))π‘‘π‘Š(𝑠)2ξƒͺ+3𝐄sup0≀𝑑≀𝑑′||||ξ€œπ‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜0ξ€œπœ€(𝑐(π‘₯(π‘ βˆ’),𝑣)βˆ’π‘(𝑍(𝑠),𝑣))Μƒπ‘πœ™(||||𝑑𝑣×𝑑𝑠)2ξƒͺ,(4.2) where the inequality |𝑒1+𝑒2+𝑒3|2≀3|𝑒1|2+3|𝑒2|2+3|𝑒3|2 for 𝑒1,𝑒2,𝑒3βˆˆπ‘π‘‘ is used. Therefore, by using the Cauchy-Schwarz inequality, (2.2) in Assumption 2.1, Lemma 3.1 and Fubini's theorem, we obtain 𝐄sup0≀𝑑≀𝑑′||||ξ€œπ‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||||(π‘Ž(π‘₯(π‘ βˆ’))βˆ’π‘Ž(𝑍(𝑠)))𝑑𝑠2ξƒͺ≀𝐄sup0β‰€π‘‘β‰€π‘‘β€²ξ€œπ‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜012ξ€œπ‘‘π‘ π‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||||π‘Ž(π‘₯(π‘ βˆ’))βˆ’π‘Ž(𝑍(𝑠))2ξƒͺξƒ©ξ€œπ‘‘π‘ β‰€π‘‡π„π‘‘β€²βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||||π‘Ž(π‘₯(π‘ βˆ’))βˆ’π‘Ž(𝑍(𝑠))2ξƒͺ𝑑𝑠≀2π‘‡πΆπ‘˜π„ξƒ©ξ€œπ‘‘β€²βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||||𝑋(𝑠)βˆ’π‘(𝑠)2ξƒͺ𝑑𝑠+2π‘‡πΆπ‘˜π„ξƒ©ξ€œπ‘‘β€²βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||π‘₯(π‘ βˆ’)βˆ’||𝑋(𝑠)2ξƒͺ𝑑𝑠≀2π‘‡πΆπ‘˜ξ€œπ‘‘β€²0𝐄||π‘‹ξ€·π‘ βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έξ€·βˆ’π‘π‘ βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2𝑑𝑠+2π‘‡πΆπ‘˜ξ€œπ‘‘β€²0𝐄||π‘₯ξ€·π‘ βˆ§πœπ‘˜βˆ§πœŒπ‘˜βˆ’ξ€Έβˆ’π‘‹ξ€·π‘ βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2𝑑𝑠≀2𝑇2πΆπ‘˜πΎ1(π‘˜)Δ𝑑+2π‘‡πΆπ‘˜ξ€œπ‘‘β€²0𝐄sup0≀𝑒≀𝑠||π‘₯ξ€·π‘’βˆ§πœπ‘˜βˆ§πœŒπ‘˜βˆ’ξ€Έβˆ’π‘‹ξ€·π‘’βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2𝑑𝑠.(4.3) Moreover, by using the martingale properties of π‘‘π‘Š(𝑑) and Μƒπ‘πœ™(𝑑𝑣×𝑑𝑑), Assumption 2.1, Lemma 3.1, and Fubini's theorem, we have 𝐄sup0≀𝑑≀𝑑′||||ξ€œπ‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||||(𝑏(π‘₯(π‘ βˆ’))βˆ’π‘(𝑍(𝑠)))π‘‘π‘Š(𝑠)2ξƒͺξ€œβ‰€4π„π‘‘β€²βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||||𝑏(π‘₯(π‘ βˆ’))βˆ’π‘(𝑍(𝑠))2𝑑𝑠≀8πΆπ‘˜π„ξ€œπ‘‘β€²βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||||𝑋(𝑠)βˆ’π‘(𝑠)2𝑑𝑠+8πΆπ‘˜π„ξ€œπ‘‘β€²βˆ§πœπ‘˜βˆ§πœŒπ‘˜0||π‘₯(π‘ βˆ’)βˆ’||𝑋(𝑠)2𝑑𝑠≀8πΆπ‘˜ξ€œπ‘‘β€²0𝐄||π‘‹ξ€·π‘ βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έξ€·βˆ’π‘π‘ βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2𝑑𝑠+8πΆπ‘˜ξ€œπ‘‘β€²0𝐄||π‘₯ξ€·π‘ βˆ§πœπ‘˜βˆ§πœŒπ‘˜βˆ’ξ€Έβˆ’π‘‹ξ€·π‘ βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2𝑑𝑠≀8π‘‡πΆπ‘˜πΎ1(π‘˜)Δ𝑑+8πΆπ‘˜ξ€œπ‘‘β€²0𝐄sup0≀𝑒≀𝑠||π‘₯ξ€·π‘’βˆ§πœπ‘˜βˆ§πœŒπ‘˜βˆ’ξ€Έβˆ’π‘‹ξ€·π‘’βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2𝐄𝑑𝑠,sup0≀𝑑≀𝑑′||||ξ€œπ‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜0ξ€œπœ€(𝑐(π‘₯(π‘ βˆ’),𝑣)βˆ’π‘(𝑍(π‘ βˆ’),𝑣))Μƒπ‘πœ™||||(𝑑𝑣×𝑑𝑠)2ξƒͺ||||ξ€œβ‰€4π„π‘‘β€²βˆ§πœπ‘˜βˆ§πœŒπ‘˜0ξ€œπœ€(𝑐(π‘₯(π‘ βˆ’),𝑣)βˆ’π‘(𝑍(π‘ βˆ’),𝑣))Μƒπ‘πœ™||||(𝑑𝑣×𝑑𝑠)2ξ€œ=4π„π‘‘β€²βˆ§πœπ‘˜βˆ§πœŒπ‘˜0ξ€œπœ€||||𝑐(π‘₯(π‘ βˆ’),𝑣)βˆ’π‘(𝑍(π‘ βˆ’),𝑣)2πœ™(𝑑𝑣)𝑑𝑠≀8𝑇𝐢𝐾1(ξ€œπ‘˜)Δ𝑑+8𝐢𝑑′0𝐄sup0≀𝑒≀𝑠||π‘₯ξ€·π‘’βˆ§πœπ‘˜βˆ§πœŒπ‘˜βˆ’ξ€Έβˆ’π‘‹ξ€·π‘’βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2𝑑𝑠.(4.4) Hence, by substituting (4.3) and (4.4) into (4.2), we get 𝐄sup0≀𝑑≀𝑑′||π‘₯ξ€·π‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έβˆ’π‘‹ξ€·π‘‘βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2ξƒͺ≀Δ𝑑6𝑇2πΆπ‘˜πΎ1(π‘˜)+24π‘‡πΆπ‘˜πΎ1(π‘˜)+24𝑇𝐢𝐾1(ξ€Έ+ξ€·π‘˜)6π‘‡πΆπ‘˜+24πΆπ‘˜ξ€ΈΓ—ξ€œ+24𝐢𝑑′0𝐄sup0≀𝑒≀𝑠||π‘₯ξ€·π‘’βˆ§πœπ‘˜βˆ§πœŒπ‘˜βˆ’ξ€Έβˆ’π‘‹ξ€·π‘’βˆ§πœπ‘˜βˆ§πœŒπ‘˜ξ€Έ||2𝑑𝑠.(4.5) So using the Gronwall inequality (see [13]), we have the result (4.1) by choosing 𝐾2ξ€·(k)=6𝑇2πΆπ‘˜πΎ1(π‘˜)+24π‘‡πΆπ‘˜πΎ1(π‘˜)+24𝑇𝐢𝐾1ξ€Έξ€·(π‘˜)exp6𝑇2πΆπ‘˜+24π‘‡πΆπ‘˜ξ€Έ.+24𝑇𝐢(4.6)

Now, let's state our theorem which demonstrates the convergence in probability of the continuous-time Euler method (3.2).

Theorem 4.2. Under Assumptions 2.1, 2.2, and 2.3, for sufficiently small πœ–,𝜍∈(0,1), there is a Ξ”π‘‘βˆ— such that for all Δ𝑑<Ξ”π‘‘βˆ—πξ‚΅sup0≀𝑑≀𝑇||π‘₯(𝑑)βˆ’||𝑋(𝑑)2ξ‚Άβ‰₯πœβ‰€πœ–,(4.7) for any 𝑇>0.

Proof. For sufficiently small πœ–,𝜍∈(0,1), we define ξ‚»Ξ©=πœ”βˆΆsup0≀𝑑≀𝑇||π‘₯(𝑑)βˆ’||𝑋(𝑑)2ξ‚Ό.β‰₯𝜍(4.8) According to Lemmas 2.4 and 3.2, there exists a pair of π‘˜β‹† and Δ𝑑⋆1 such that πξ€·πœπ‘˜β‹†ξ€Έβ‰€πœ–β‰€π‘‡3,πξ€·πœŒπ‘˜β‹†ξ€Έβ‰€πœ–β‰€π‘‡3,βˆ€Ξ”π‘‘β‰€Ξ”π‘‘β‹†1.(4.9) We thus have πξ‚€Ξ©ξ‚ξ‚€β‰€πξ€½πœΞ©βˆ©π‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Ύξ‚ξ€·πœ>𝑇+ππ‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Έξ‚€β‰€π‘‡β‰€πξ€½πœΞ©βˆ©π‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Ύξ‚ξ€·πœ>𝑇+ππ‘˜β‹†ξ€Έξ€·πœŒβ‰€π‘‡+ππ‘˜β‹†ξ€Έξ‚€β‰€π‘‡β‰€πξ€½πœΞ©βˆ©π‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Ύξ‚+>𝑇2πœ–3,(4.10) for Δ𝑑≀Δ𝑑⋆1. Moreover, according to Lemma 4.1, we have ξ‚€πœπξ€½πœΞ©βˆ©π‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Ύξ‚ξ‚΅πΌ>𝑇≀𝐄{πœπ‘˜β‹†βˆ§πœŒπ‘˜β‹†>𝑇}sup0≀𝑑≀𝑇||π‘₯(𝑑)βˆ’||𝑋(𝑑)2≀𝐄sup0≀𝑑≀𝑇||π‘₯ξ€·π‘‘βˆ§πœπ‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Έβˆ’π‘‹ξ€·π‘‘βˆ§πœπ‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Έ||2≀𝐾2ξ€·π‘˜β‹†ξ€ΈΞ”π‘‘,(4.11) which leads to πξ‚€ξ€½πœΞ©βˆ©π‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Ύξ‚β‰€πœ–>𝑇3,(4.12) for Δ𝑑≀Δ𝑑⋆2. Therefore, from the inequalities above, we obtain πξ‚€Ξ©ξ‚β‰€πœ–,(4.13) for Δ𝑑≀Δ𝑑⋆, where Δ𝑑⋆=min{Δ𝑑⋆1,Δ𝑑⋆2}.

We remark that the continuous-time Euler solution 𝑋(𝑑) (3.2) cannot be computed, since it requires knowledge of the entire Brownian motion and Poisson random measure paths, not just only their Δ𝑑-increments. Therefore, the last theorem shows the convergence in probability of the discrete Euler solution (3.1).

Theorem 4.3. Under Assumptions 2.1, 2.2, and 2.3, for sufficiently small πœ–,𝜍∈(0,1), there is a Ξ”π‘‘βˆ— such that for all Δ𝑑<Ξ”π‘‘βˆ—πξ‚€||||π‘₯(𝑑)βˆ’π‘(𝑑)2β‰₯𝜍,0β‰€π‘‘β‰€π‘‡β‰€πœ–,(4.14) for any 𝑇>0.

Proof. For sufficiently small πœ–,𝜍∈(0,1), we define ||||Ξ©=πœ”βˆΆπ‘₯(𝑑)βˆ’π‘(𝑑)2β‰₯𝜍,0≀𝑑≀𝑇.(4.15) A similar analysis as Theorem 4.2 gives πξ‚€ξ‚Ξ©ξ‚ξ‚€ξ‚ξ€½πœβ‰€πΞ©βˆ©π‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Ύξ‚+>𝑇2πœ–3.(4.16) Recalling that ξ‚€ξ‚ξ€½πœπœπΞ©βˆ©π‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Ύξ‚ξ‚€||||>𝑇≀𝐄π‘₯(𝑇)βˆ’π‘(𝑇)2𝐼{πœπ‘˜β‹†βˆ§πœŒπ‘˜β‹†>𝑇}||π‘₯ξ€·β‰€π„π‘‡βˆ§πœπ‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Έξ€·βˆ’π‘π‘‡βˆ§πœπ‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Έ||2≀2𝐄sup0≀𝑑≀𝑇||π‘₯ξ€·π‘‘βˆ§πœπ‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Έβˆ’π‘‹ξ€·π‘‘βˆ§πœπ‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Έ||2ξ‚Ά||+2π„π‘‹ξ€·π‘‡βˆ§πœπ‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Έξ€·βˆ’π‘π‘‡βˆ§πœπ‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Έ||2≀2𝐾1ξ€·π‘˜β‹†ξ€ΈΞ”π‘‘+2𝐾2ξ€·π‘˜β‹†ξ€ΈΞ”π‘‘,(4.17) and using Lemmas 3.1 and 4.1, we get that πξ‚€ξ‚ξ€½πœΞ©βˆ©π‘˜β‹†βˆ§πœŒπ‘˜β‹†ξ€Ύξ‚β‰€πœ–>𝑇3,(4.18) for sufficiently small Δ𝑑. Consequently, the inequalities above show that πξ‚€ξ‚Ξ©ξ‚β‰€πœ–,(4.19) for all sufficiently small Δ𝑑.
So we complete the result (4.14).

5. Numerical Example

In this section, a numerical example is analyzed under Assumptions 2.1, 2.2, and 2.3 which cover more classes of SDEs driven by Poisson random measure.

Now, we consider the following equation: ξ€œπ‘‘π‘₯(𝑑)=π‘Ž(π‘₯(π‘‘βˆ’))𝑑𝑑+𝑏(π‘₯(π‘‘βˆ’))π‘‘π‘Š(𝑑)+πœ€π‘(π‘₯(π‘‘βˆ’),𝑣)Μƒπ‘πœ™(𝑑𝑣×𝑑𝑑),𝑑>0,(5.1) with π‘₯(0)=π‘₯(0βˆ’)=0, where 𝑑=π‘š=π‘Ÿ=1. The coefficients of this equation have the form 1π‘Ž(π‘₯)=2ξ€·π‘₯βˆ’π‘₯3ξ€Έ,𝑏(π‘₯)=π‘₯2,𝑐(π‘₯,𝑣)=π‘₯𝑣.(5.2) The compensated measure of the Poisson random measure π‘πœ™(𝑑𝑣×𝑑𝑑) is given by πœ™(𝑑𝑣)𝑑𝑑=πœ†π‘“(𝑣)𝑑𝑣𝑑𝑑, where πœ†=5 and 1𝑓(𝑣)=βˆšξ‚΅βˆ’2πœ‹π‘£exp(ln𝑣)22ξ‚Ά,0≀𝑣<∞(5.3) is the density function of a lognormal random variable.

Clearly, the equation cannot satisfy the global Lipschitz conditions and the linear growth conditions. On the other hand, we have ||||2⟨π‘₯,π‘Ž(π‘₯)⟩+𝑏(π‘₯)2+ξ€œπœ€||||𝑐(π‘₯,𝑣)2ξ€·πœ™(𝑑𝑣)=π‘₯π‘₯βˆ’π‘₯3ξ€Έ+π‘₯4+ξ€œπœ€π‘₯2𝑣2πœ†1βˆšξ‚΅βˆ’2πœ‹π‘£exp(ln𝑣)22≀𝑑𝑣1+5𝑒2ξ€Έξ€·1+π‘₯2ξ€Έ,(5.4) that is to say, Assumptions 2.1, 2.2, and 2.3 in Section 2 are satisfied. Therefore, Albeverio et al. [12] guarantee that (5.1) has a unique global solution on [0,∞).

Given the stepsize Δ𝑑, we can have the Euler method 𝑋𝑛+1=𝑋𝑛+12ξ€·π‘‹π‘›βˆ’π‘‹3𝑛Δ𝑑+𝑋2π‘›Ξ”π‘Šπ‘›+π‘‹π‘›ξ€œπ‘‘π‘›+1π‘‘π‘›ξ€œπœ€π‘£Μƒπ‘πœ™(𝑑𝑣×𝑑𝑑),(5.5) with 𝑋0=0.

And in Matlab experiment, each discretized trajectory is actually given in detail by the following.

Algorithm  Simulate π‘‹βˆ’π‘›+1∢=𝑋𝑛+(1/2)(π‘‹π‘›βˆ’π‘‹3π‘›βˆšβˆ’10𝑒𝑋𝑛)Δ𝑑+𝑋2π‘›Ξ”π‘Šπ‘›; Simulate variable π‘πœ™(𝑑𝑛+1)βˆ’π‘πœ™(𝑑𝑛), where π‘πœ™(𝑑𝑛) is from Poisson distribution with parameter πœ†π‘‘π‘›; Simulate π‘πœ™(𝑑𝑛+1)βˆ’π‘πœ™(𝑑𝑛) independent random variables πœ„π‘– uniformly distributed on the interval [π‘πœ™(𝑑𝑛),π‘πœ™(𝑑𝑛+1)); Simulate π‘πœ™(𝑑𝑛+1)βˆ’π‘πœ™(𝑑𝑛) independent random variables πœ‰π‘– with law 𝑓(𝑣); obtain 𝑋𝑛+1=π‘‹βˆ’π‘›+1+π‘‹π‘›βˆ‘π‘πœ™(𝑑𝑛+1)𝑖=π‘πœ™(𝑑𝑛)+1πΌπ‘‘π‘›β‰€πœ„π‘–<𝑑𝑛+1πœ‰π‘–.Subsequently, we can get the results in Theorems 4.2 and 4.3.

Acknowledgment

This work is supported by the NSF of China (no. 11071050).

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Copyright © 2012 Hui Yu and Minghui Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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