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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 675781, 17 pages
http://dx.doi.org/10.1155/2012/675781
Research Article

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

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 15 January 2012; Revised 20 March 2012; Accepted 22 March 2012

Academic Editor: Said Abbasbandy

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.

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 𝛾{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 𝛾{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 𝑢1𝑢2=max{𝑢1,𝑢2} and 𝑢1𝑢2=min{𝑢1,𝑢2}. Let || and , be the Euclidean norm and the inner product of vectors in 𝐑𝑑,𝑑𝐍. 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 𝐿20(Ω;𝚁𝑑) denote the family of 𝐑𝑑-valued 0-measurable random variables 𝜉 with 𝐄|𝜉|2<. [𝑧] denotes the largest integer which is less than or equal to 𝑧 in 𝐑. I𝒜 denotes the indicator function of a set 𝒜.

The following 𝑑-dimensional SDE driven by Poisson random measure is considered in our paper: 𝑑𝑥(𝑡)=𝑎(𝑥(𝑡))𝑑𝑡+𝑏(𝑥(𝑡))𝑑𝑊(𝑡)+𝜀𝑐(𝑥(𝑡),𝑣)̃𝑝𝜙(𝑑𝑣×𝑑𝑡),(2.1) for 𝑡>0 with initial condition 𝑥(0)=𝑥(0)=𝑥0𝐿20(Ω;𝐑𝑑), where 𝑥(𝑡) denotes lim𝑠𝑡𝑥(𝑠) and ̃𝑝𝜙(𝑑𝑣×𝑑𝑡)=𝑝𝜙(𝑑𝑣×𝑑𝑡)𝜙(𝑑𝑣)𝑑𝑡.

The drift coefficient 𝑎𝐑𝑑𝐑𝑑, the diffusion coefficient 𝑏𝐑𝑑𝐑𝑑×𝑚, and the jump coefficient 𝑐𝐑𝑑×𝜀𝐑𝑑 are assumed to be Borel measurable functions.

The randomness of (2.1) is generated by the following (see [9]). An 𝑚-dimensional Wiener process 𝑊={𝑊(𝑡)=(𝑊1(𝑡),,𝑊𝑚(𝑡))𝑇} with independent scalar components is defined on a filtered probability space (Ω𝑊,𝑊,(𝑊𝑡)𝑡0,𝐏𝑊). A Poisson random measure 𝑝𝜙(𝜔,𝑑𝑣×𝑑𝑡) is on Ω𝐽×𝜀×[0,), where 𝜀𝐑𝑟{0} with 𝑟𝐍, and its deterministic compensated measure 𝜙(𝑑𝑣)𝑑𝑡=𝜆𝑓(𝑣)𝑑𝑣𝑑𝑡, that is, 𝐄(𝑝𝜙(𝑑𝑣×𝑑𝑡))=𝜙(𝑑𝑣)𝑑𝑡. 𝑓(𝑣) is a probability density, and we require finite intensity 𝜆=𝜙(𝜀)<. The Poisson random measure is defined on a filtered probability space (Ω𝐽,𝐽,(𝐽𝑡)𝑡0,𝐏𝐽). The Wiener process and the Poisson random measure are mutually independent. The process 𝑥(𝑡) is thus defined on a product space (Ω,,(𝑡)𝑡0,𝐏), where Ω=Ω𝑊×Ω𝐽,=𝑊×𝐽,(𝑡)𝑡0=(𝑊𝑡)𝑡0×(𝐽𝑡)𝑡0,𝐏=𝐏𝑊×𝐏𝐽 and 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 𝑘1, there exists a positive constant 𝐶𝑘, dependent on 𝑘, such that ||||𝑎(𝑥)𝑎(𝑦)2||||𝑏(𝑥)𝑏(𝑦)2𝐶𝑘||||𝑥𝑦2,(2.2) for 𝑥,𝑦𝐑𝑑 with |𝑥||𝑦|𝑘,𝑘𝐍. And there exists a positive constant 𝐶 such that 𝜀||||𝑐(𝑥,𝑣)𝑐(𝑦,𝑣)2||||𝜙(𝑑𝑣)𝐶𝑥𝑦2,(2.3) for 𝑥,𝑦𝐑𝑑.

Assumption 2.2. There exists a positive constant 𝐿 such that ||||2𝑥,𝑎(𝑥)+𝑏(𝑥)2+𝜀||||𝑐(𝑥,𝑣)2𝜙(𝑑𝑣)𝐿1+|𝑥|2,(2.4) for 𝑥𝐑𝑑.

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

Assumption 2.3. Consider ||||𝑎(0)2+||||𝑏(0)2+𝜀||||𝑐(0,𝑣)2𝜙(𝑑𝑣)𝐿,𝐿>0.(2.5)

Actually, Assumptions 2.1 and 2.3 imply the linear growth conditions ||||𝑎(𝑥)2||||𝑏(𝑥)2𝐶𝑘1+|𝑥|2,(2.6) for 𝑥𝐑𝑑 with |𝑥|𝑘 and 𝐶𝑘>0, and 𝜀||||𝑐(𝑥,𝑣)2𝐶𝜙(𝑑𝑣)1+|𝑥|2,(2.7) for 𝑥𝐑𝑑 and 𝐶>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 𝜖(0,1) and 𝑇>0, there exists a sufficiently large integer 𝑘, dependent on 𝜖 and 𝑇, such that 𝐏𝜏𝑘𝑇𝜖,𝑘𝑘,(2.8) where 𝜏𝑘=inf{𝑡0|𝑥(𝑡)|𝑘} for 𝑘1.

Proof. Using Itô's formula (see [1]) to |𝑥(𝑡)|2, for 𝑡0, we have ||||𝑥(𝑡)2=||𝑥0||2+𝑡0||||2𝑥(𝑠),𝑎(𝑥(𝑠))+𝑏(𝑥(𝑠))2+𝑑s𝑡0𝜀||||𝑥(𝑠)+𝑐(𝑥(𝑠),𝑣)2||||𝑥(𝑠)2+2𝑥(𝑠),𝑐(𝑥(𝑠),𝑣)𝜙(𝑑𝑣)𝑑𝑠𝑡02𝑥(𝑠),𝑏(𝑥(𝑠))𝑑𝑊(𝑠)+𝑡0𝜀||||𝑥(𝑠)+𝑐(𝑥(𝑠),𝑣)2||||𝑥(𝑠)2̃𝑝𝜙(𝑑𝑣×𝑑𝑠),(2.9) which gives 𝐄||𝑥𝑡𝜏𝑘||2||𝑥=𝐄0||2+𝐄𝑡𝜏𝑘0||||2𝑥(𝑠),𝑎(𝑥(𝑠))+𝑏(𝑥(𝑠))2𝑑𝑠+𝐄𝑡𝜏𝑘0𝜀||||𝑐(𝑥(𝑠),𝑣)2||𝑥𝜙(𝑑𝑣)𝑑𝑠=𝐄0||2+𝐄𝑡𝜏𝑘0||||2𝑥(𝑠),𝑎(𝑥(𝑠))+𝑏(𝑥(𝑠))2+𝜀||||𝑐(𝑥(𝑠),𝑣)2𝜙(𝑑𝑣)𝑑𝑠,(2.10) for 𝑡[0,𝑇]. By Assumption 2.2, we thus have 𝐄||𝑥𝑡𝜏𝑘||2||𝑥𝐄0||2+𝐄𝑡𝜏𝑘0𝐿||||1+𝑥(𝑠)2||𝑥𝑑𝑠𝐄0||2+𝐿𝑇+𝐿𝑡0𝐄||𝑥𝑠𝜏𝑘||2𝑑𝑠,(2.11) for 𝑡[0,𝑇]. Consequently by using the Gronwall inequality (see [13]), we obtain 𝐄||𝑥𝑡𝜏𝑘||2𝐄||𝑥0||2𝑒+𝐿𝑇𝐿𝑇,(2.12) for 𝑡[0,𝑇]. We therefore get 𝐄||𝑥0||2𝑒+𝐿𝑇𝐿𝑇||𝑥𝐄𝑇𝜏𝑘||2||𝑥𝜏𝐄𝑘||2𝐼{𝜏𝑘𝑇}𝑘2𝐏𝜏𝑘𝑇,(2.13) which means 𝐏𝜏𝑘𝑒𝑇𝐿𝑇𝑘2𝐄||𝑥0||2.+𝐿𝑇(2.14) So for any 𝜖(0,1), we can choose 𝑘=𝑒𝐿𝑇𝐄||𝑥0||2+𝐿𝑇𝑒𝐿𝑇𝜖+1,(2.15) such that 𝐏𝜏𝑘𝑇𝜖,𝑘𝑘.(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 Δ𝑡(0,1), the Euler method applied to (2.1) computes approximation 𝑋𝑛𝑥(𝑡𝑛), where 𝑡𝑛=𝑛Δ𝑡,𝑛=0,1,, by setting 𝑋0=𝑥0 and forming 𝑋𝑛+1=𝑋𝑛𝑋+𝑎𝑛𝑋Δ𝑡+𝑏𝑛Δ𝑊𝑛+𝑡𝑛+1𝑡𝑛𝜀𝑐𝑋𝑛,𝑣̃𝑝𝜙(𝑑𝑣×𝑑𝑡),(3.1) where Δ𝑊𝑛=𝑊(𝑡𝑛+1)𝑊(𝑡𝑛).

The continuous-time Euler method is then defined by 𝑋(𝑡)=𝑋0+𝑡0𝑎(𝑍(𝑠))𝑑𝑠+𝑡0𝑏(𝑍(𝑠))𝑑𝑊(𝑠)+𝑡0𝜀𝑐(𝑍(𝑠),𝑣)̃𝑝𝜙(𝑑𝑣×𝑑𝑠),(3.2) where 𝑍(𝑡)=𝑋𝑛 for 𝑡[𝑡𝑛,𝑡𝑛+1),𝑛=0,1,.

Actually, we can see in [8], 𝑝𝜙={𝑝𝜙(𝑡)=𝑝𝜙(𝜀×[0,𝑡])} is a stochastic process that counts the number of jumps until some given time. The Poisson random measure 𝑝𝜙(𝑑𝑣×𝑑𝑡) generates a sequence of pairs {(𝜄𝑖,𝜉𝑖),𝑖{1,2,,𝑝𝜙(𝑇)}} for a given finite positive constant 𝑇 if 𝜆<. Here {𝜄𝑖Ω𝐑+,𝑖{1,2,,𝑝𝜙(𝑇)}} is a sequence of increasing nonnegative random variables representing the jump times of a standard Poisson process with intensity 𝜆, and {𝜉𝑖Ω𝜀,𝑖{1,2,,𝑝𝜙(𝑇)} is a sequence of independent identically distributed random variables, where 𝜉𝑖 is distributed according to 𝜙(𝑑𝑣)/𝜙(𝜀). Then (3.1) can equivalently be the following form: 𝑋𝑛+1=𝑋𝑛+𝑎𝑋𝑛𝜀𝑐𝑋𝑛𝑋,𝑣𝜙(𝑑𝑣)Δ𝑡+𝑏𝑛Δ𝑊𝑛+𝑝𝜙(𝑡𝑛+1)𝑖=𝑝𝜙(𝑡𝑛)+1𝑐𝑋𝑛,𝜉𝑖.(3.3)

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

Lemma 3.1. Under Assumptions 2.1 and 2.3, for any 𝑇>0, there exists a positive constant 𝐾1(𝑘), dependent on 𝑘 and independent of Δ𝑡, such that for all Δ𝑡(0,1) the continuous-time Euler method (3.2) satisfies 𝐄||||𝑋(𝑡)𝑍(𝑡)2𝐾1(𝑘)Δ𝑡,(3.4) for 0𝑡𝑇𝜏𝑘𝜌𝑘, where 𝜌𝑘=inf{𝑡0|𝑋(𝑡)|𝑘} for 𝑘1 and 𝜏𝑘 is defined in Lemma 2.4.

Proof. For 0𝑡𝑇𝜏𝑘𝜌𝑘, there is an integer 𝑛 such that 𝑡[𝑡𝑛,𝑡𝑛+1). So it follows from (3.2) that 𝑋(𝑡)𝑍(𝑡)=𝑋𝑛+𝑡𝑡𝑛𝑎(𝑍(𝑠))𝑑𝑠+𝑡𝑡𝑛𝑏(𝑍(𝑠))𝑑𝑊(𝑠)+𝑡𝑡𝑛𝜀𝑐(𝑍(𝑠),𝑣)̃𝑝𝜙(𝑑𝑣×𝑑𝑠)𝑋𝑛.(3.5) Thus, by taking expectations and using the Cauchy-Schwarz inequality and the martingale properties of 𝑑𝑊(𝑡) and ̃𝑝𝜙(𝑑𝑣×𝑑𝑡), we have 𝐄||||𝑋(𝑡)𝑍(𝑡)2||||3𝐄𝑡𝑡𝑛||||𝑎(𝑍(𝑠))𝑑𝑠2||||+3𝐄𝑡𝑡𝑛||||𝑏(𝑍(𝑠))𝑑𝑊(𝑠)2||||+3𝐄𝑡𝑡𝑛𝜀𝑐(𝑍(𝑠),𝑣)̃𝑝𝜙||||(𝑑𝑣×𝑑𝑠)23𝐄𝑡𝑡𝑛12𝑑𝑠𝑡𝑡𝑛||||𝑎(𝑍(𝑠))2𝑑𝑠+3𝐄𝑡𝑡𝑛||||𝑏(𝑍(𝑠))2𝑑𝑠+3𝐄𝑡𝑡𝑛𝜀||||𝑐(𝑍(𝑠),𝑣)2𝜙(𝑑𝑣)𝑑𝑠3Δ𝑡𝐄𝑡𝑡𝑛||||𝑎(𝑍(𝑠))2𝑑𝑠+3𝐄𝑡𝑡𝑛||||𝑏(𝑍(𝑠))2𝑑𝑠+3𝐄𝑡𝑡𝑛𝜀||||𝑐(𝑍(𝑠),𝑣)2𝜙(𝑑𝑣)𝑑𝑠,(3.6) where the inequality |𝑢1+𝑢2+𝑢3|23|𝑢1|2+3|𝑢2|2+3|𝑢3|2 for 𝑢1,𝑢2,𝑢3𝐑𝑑 is used. Therefore, by applying Assumptions 2.1 and 2.3, we get 𝐄𝑡𝑡𝑛||||𝑎(𝑍(𝑠))2𝐶𝑑𝑠𝑘𝐄𝑡𝑡𝑛||||1+𝑍(𝑠)2𝐶𝑑𝑠𝑘𝐶Δ𝑡+𝑘𝑘2𝐄Δ𝑡,𝑡𝑡𝑛||||𝑏(𝑍(𝑠))2𝐶𝑑𝑠𝑘𝐶Δ𝑡+𝑘𝑘2𝐄Δ𝑡,𝑡𝑡𝑛𝜀||||𝑐(𝑍(𝑠),𝑣)2𝜙(𝑑𝑣)𝑑𝑠𝐶Δ𝑡+𝐶𝑘2Δ𝑡,(3.7) which lead to 𝐄||||𝑋(𝑡)𝑍(𝑡)23𝐶Δ𝑡𝑘Δ𝑡+3𝑘2𝐶𝑘𝐶Δ𝑡+3𝑘+3𝑘2𝐶𝑘+3𝐶+3𝑘2𝐶,(3.8) for 𝑡[0,𝑇𝜏𝑘𝜌𝑘]. Therefore, we obtain the result (3.4) by choosing 𝐾1𝐶(𝑘)=6𝑘+6𝑘2𝐶𝑘+3𝐶+3𝑘2𝐶.(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 𝜖(0,1) and 𝑇>0, there exist a sufficiently large 𝑘 and a sufficiently small Δ𝑡1 such that 𝐏𝜌𝑘𝑇𝜖,Δ𝑡Δ𝑡1,(3.10) where 𝜌𝑘 is defined in Lemma 3.1.

Proof . Applying generalized Itô's formula (see [1]) to |𝑋(𝑡)|2, for 𝑡0, yields ||||𝑋(𝑡)2=||𝑋0||2+𝑡02+||||𝑋(𝑠),𝑎(𝑍(𝑠))𝑏(𝑍(𝑠))2+𝑑𝑠𝑡0𝜀||||𝑋(𝑠)+𝑐(𝑍(𝑠),𝑣)2||||𝑋(𝑠)22+𝑋(𝑠),𝑐(𝑍(𝑠),𝑣)𝜙(𝑑𝑣)𝑑𝑠𝑡02𝑋(𝑠),𝑏(𝑍(𝑠))𝑑𝑊(𝑠)+𝑡0𝜀||||𝑋(𝑠)+𝑐(𝑍(𝑠),𝑣)2||||𝑋(𝑠)2̃𝑝𝜙(𝑑𝑣×𝑑𝑠).(3.11) By taking expectations, we thus have 𝐄||𝑋𝑡𝜌𝑘||2||𝑋=𝐄0||2+𝐄𝑡𝜌𝑘02+||||𝑋(𝑠),𝑎(𝑍(𝑠))𝑏(𝑍(𝑠))2+𝜀||||𝑐(𝑍(𝑠),𝑣)2||𝑋𝜙(𝑑𝑣)𝑑𝑠=𝐄0||2+E𝑡𝜌𝑘02𝑋(𝑠),𝑎+|||𝑏𝑋(𝑠)|||𝑋(𝑠)2+𝜀|||𝑐𝑋|||(𝑠),𝑣2𝜙(𝑑𝑣)𝑑𝑠+𝐄𝑡𝜌𝑘02𝑋(𝑠),𝑎(𝑍(𝑠))𝑎𝑋(𝑠)𝑑𝑠+𝐄𝑡𝜌𝑘0||||𝑏(𝑍(𝑠))2|||𝑏|||𝑋(𝑠)2𝑑𝑠+𝐄𝑡𝜌𝑘0𝜀||||𝑐(𝑍(𝑠),𝑣)2|||𝑐|||𝑋(𝑠),𝑣2𝜙(𝑑𝑣)𝑑𝑠.(3.12) For 𝑡[0,𝑇]. Now, by using the inequalities 𝑢1,𝑢2|𝑢1||𝑢2| for 𝑢1,𝑢2𝐑𝑑, (2.2) in Assumption 2.1, Fubini's theorem, Cauchy-Schwarz’s inequality, and Lemma 3.1, we get 𝐄𝑡𝜌𝑘02𝑋(𝑠),𝑎(𝑍(𝑠))𝑎𝑋(𝑠)𝑑𝑠2𝐄𝑡𝜌𝑘0|||||||𝑋(𝑠)𝑎(𝑍(𝑠))𝑎|||𝑋(𝑠)𝑑𝑠2𝑘𝐶𝑘𝑡0𝐄||𝑍𝑠𝜌𝑘𝑋𝑠𝜌𝑘||𝑑𝑠2𝑘𝐶𝑘𝑡0𝐄||𝑍𝑠𝜌𝑘𝑋𝑠𝜌𝑘||21/2𝑑𝑠2𝑘𝑇𝐶𝑘𝐾1(𝑘)Δ𝑡.(3.13) And, similarly as above, we have 𝐄𝑡𝜌𝑘0||||𝑏(𝑍(𝑠))2|||𝑏|||𝑋(𝑠)2𝑑𝑠𝐄𝑡𝜌𝑘0||||+|||𝑏𝑏(𝑍(𝑠))|||×|||||||𝑏𝑋(𝑠)𝑏(𝑍(𝑠))|||𝑋(𝑠)𝑑𝑠2𝐶𝑘1+𝑘2𝐄𝑡𝜌𝑘0|||𝑏(𝑍(𝑠))𝑏|||𝑋(𝑠)𝑑𝑠2𝐶𝑘𝐶𝑘1+𝑘2𝑡0𝐄||𝑍𝑠𝜌𝑘𝑋𝑠𝜌𝑘||𝑑𝑠2𝑇𝐶𝑘𝐶𝑘𝐾1(𝑘)1+𝑘2Δ𝑡.(3.14) Moreover, in the same way, we obtain 𝐄𝑡𝜌𝑘0𝜀||||𝑐(𝑍(𝑠),𝑣)2|||𝑐|||𝑋(𝑠),𝑣2𝜙(𝑑𝑣)𝑑𝑠=𝐄𝑡𝜌𝑘0𝜀|||𝑐(𝑍(𝑠),𝑣)𝑐𝑋(𝑠),𝑣+𝑐|||𝑋(𝑠),𝑣2|||𝑐|||𝑋(𝑠),𝑣2𝜙(𝑑𝑣)𝑑𝑠𝐄𝑡𝜌𝑘0𝜀2|||𝑐(𝑍(𝑠),𝑣)𝑐|||𝑋(𝑠),𝑣2+|||𝑐|||𝑋(𝑠),𝑣2𝜙(𝑑𝑣)𝑑𝑠2𝐶𝐄𝑡𝜌𝑘0||𝑍(𝑠)||𝑋(𝑠)2𝑑𝑠+𝐶𝐄𝑡𝜌𝑘0||1+||𝑋(𝑠)2𝑑𝑠2𝐶𝑡0𝐄||𝑍𝑠𝜌𝑘𝑋𝑠𝜌𝑘||2𝑑𝑠+𝐶𝐄𝑡𝜌𝑘0||1+||𝑋(𝑠)2𝑑𝑠2𝐶𝑇𝐾1(𝑘)Δ𝑡+𝐶𝑇+𝐶𝐄𝑡𝜌𝑘0||||𝑋(𝑠)2𝑑𝑠,(3.15) where the inequality |𝑢1+𝑢2|22|𝑢1|2+2|𝑢2|2 for 𝑢1,𝑢2𝐑𝑑, (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||𝑋𝑡𝜌𝑘||2||𝑋𝐄0||2+𝐿𝐄𝑡𝜌𝑘0||1+||𝑋(𝑠)2𝑑𝑠+𝐶𝐄𝑡𝜌𝑘0||||𝑋(𝑠)2𝑑𝑠+2𝑘𝑇𝐶𝑘𝐾1(𝑘)Δ𝑡+2𝑇𝐶𝑘𝐶𝑘𝐾1(𝑘)1+𝑘2Δ𝑡+2𝐶𝑇𝐾1𝐶(𝑘)Δ𝑡+𝐶𝑇𝐿+𝑡0𝐄||𝑋𝑠𝜌𝑘||2||𝑋𝑑𝑠+𝐄0||2++𝐿𝑇+𝐶𝑇2𝑘𝑇𝐶𝑘𝐾1(𝑘)+2𝑇𝐶𝑘𝐶𝑘𝐾1(𝑘)1+𝑘2Δ𝑡+2𝐶𝑇𝐾1(𝑘)Δ𝑡,(3.16) for 0𝑡𝑇. Therefore, by the Gronwall inequality (see [13]), for 0𝑡𝑇, we get 𝐄||𝑋𝑡𝜌𝑘||2𝛼1𝛼4+𝛼4𝛼2(𝑘)Δ𝑡+𝛼4𝛼3(𝑘)Δ𝑡,(3.17) where 𝛼1||𝑋=𝐄0||2𝛼+𝐿𝑇+𝐶𝑇,2(𝑘)=2𝑘𝑇𝐶𝑘𝐾1(𝑘)+2𝑇𝐶𝑘𝐶𝑘𝐾1(𝑘)1+𝑘2,𝛼3(𝑘)=2𝐶𝑇𝐾1𝛼(𝑘),4.=exp𝐿𝑇+𝐶𝑇(3.18) We thus obtain that 𝑘2𝐏𝜌𝑘||𝑇𝐄𝑋𝜌𝑘||2𝐼{𝜌𝑘𝑇}||𝐄𝑋𝑇𝜌𝑘||2𝛼1𝛼4+𝛼4𝛼2(𝑘)Δ𝑡+𝛼4𝛼3(𝑘)Δ𝑡.(3.19) So for any 𝜖(0,1), we can choose sufficiently large integer 𝑘=𝑘 such that 𝛼1𝛼4𝑘2𝜖2,(3.20) and choose sufficiently small Δ𝑡1(0,1) such that 𝛼4𝛼2𝑘Δ𝑡1+𝛼4𝛼3𝑘Δ𝑡1𝑘2𝜖2.(3.21) Hence, we have 𝐏𝜌𝑘𝑇𝜖,Δ𝑡Δ𝑡1.(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(𝑘), dependent on 𝑘 and independent of Δ𝑡, such that for all Δ𝑡(0,1) the solution of (2.1) and the continuous-time Euler method (3.2) satisfy 𝐄sup0𝑡𝑇||𝑥𝑡𝜏𝑘𝜌𝑘𝑋𝑡𝜏𝑘𝜌𝑘||2𝐾2(𝑘)Δ𝑡,(4.1) where 𝜏𝑘 and 𝜌𝑘 are defined in Lemmas 2.4 and 3.1, respectively.

Proof. From (2.1) and (3.2), for any 0𝑡𝑇, we have 𝐄sup0𝑡𝑡||𝑥𝑡𝜏𝑘𝜌𝑘𝑋𝑡𝜏𝑘𝜌𝑘||23𝐄sup0𝑡𝑡||||𝑡𝜏𝑘𝜌𝑘0||||(𝑎(𝑥(𝑠))𝑎(𝑍(𝑠)))𝑑𝑠2+3𝐄sup0𝑡𝑡||||𝑡𝜏𝑘𝜌𝑘0||||(𝑏(𝑥(𝑠))𝑏(𝑍(𝑠)))𝑑𝑊(𝑠)2+3𝐄sup0𝑡𝑡||||𝑡𝜏𝑘𝜌𝑘0𝜀(𝑐(𝑥(𝑠),𝑣)𝑐(𝑍(𝑠),𝑣))̃𝑝𝜙(||||𝑑𝑣×𝑑𝑠)2,(4.2) where the inequality |𝑢1+𝑢2+𝑢3|23|𝑢1|2+3|𝑢2|2+3|𝑢3|2 for 𝑢1,𝑢2,𝑢3𝐑𝑑 is used. Therefore, by using the Cauchy-Schwarz inequality, (2.2) in Assumption 2.1, Lemma 3.1 and Fubini's theorem, we obtain 𝐄sup0𝑡𝑡||||𝑡𝜏𝑘𝜌𝑘0||||(𝑎(𝑥(𝑠))𝑎(𝑍(𝑠)))𝑑𝑠2𝐄sup0𝑡𝑡𝑡𝜏𝑘𝜌𝑘012𝑑𝑠𝑡𝜏𝑘𝜌𝑘0||||𝑎(𝑥(𝑠))𝑎(𝑍(𝑠))2𝑑𝑠𝑇𝐄𝑡𝜏𝑘𝜌𝑘0||||𝑎(𝑥(𝑠))𝑎(𝑍(𝑠))2𝑑𝑠2𝑇𝐶𝑘𝐄𝑡𝜏𝑘𝜌𝑘0||||𝑋(𝑠)𝑍(𝑠)2𝑑𝑠+2𝑇𝐶𝑘𝐄𝑡𝜏𝑘𝜌𝑘0||𝑥(𝑠)||𝑋(𝑠)2𝑑𝑠2𝑇𝐶𝑘𝑡0𝐄||𝑋𝑠𝜏𝑘𝜌𝑘𝑍𝑠𝜏𝑘𝜌𝑘||2𝑑𝑠+2𝑇𝐶𝑘𝑡0𝐄||𝑥𝑠𝜏𝑘𝜌𝑘𝑋𝑠𝜏𝑘𝜌𝑘||2𝑑𝑠2𝑇2𝐶𝑘𝐾1(𝑘)Δ𝑡+2𝑇𝐶𝑘𝑡0𝐄sup0𝑢𝑠||𝑥𝑢𝜏𝑘𝜌𝑘𝑋𝑢𝜏𝑘𝜌𝑘||2𝑑𝑠.(4.3) Moreover, by using the martingale properties of 𝑑𝑊(𝑡) and ̃𝑝𝜙(𝑑𝑣×𝑑𝑡), Assumption 2.1, Lemma 3.1, and Fubini's theorem, we have 𝐄sup0𝑡𝑡||||𝑡𝜏𝑘𝜌𝑘0||||(𝑏(𝑥(𝑠))𝑏(𝑍(𝑠)))𝑑𝑊(𝑠)24𝐄𝑡𝜏𝑘𝜌𝑘0||||𝑏(𝑥(𝑠))𝑏(𝑍(𝑠))2𝑑𝑠8𝐶𝑘𝐄𝑡𝜏𝑘𝜌𝑘0||||𝑋(𝑠)𝑍(𝑠)2𝑑𝑠+8𝐶𝑘𝐄𝑡𝜏𝑘𝜌𝑘0||𝑥(𝑠)||𝑋(𝑠)2𝑑𝑠8𝐶𝑘𝑡0𝐄||𝑋𝑠𝜏𝑘𝜌𝑘𝑍𝑠𝜏𝑘𝜌𝑘||2𝑑𝑠+8𝐶𝑘𝑡0𝐄||𝑥𝑠𝜏𝑘𝜌𝑘𝑋𝑠𝜏𝑘𝜌𝑘||2𝑑𝑠8𝑇𝐶𝑘𝐾1(𝑘)Δ𝑡+8𝐶𝑘𝑡0𝐄sup0𝑢𝑠||𝑥𝑢𝜏𝑘𝜌𝑘𝑋𝑢𝜏𝑘𝜌𝑘||2𝐄𝑑𝑠,sup0𝑡𝑡||||𝑡𝜏𝑘𝜌𝑘0𝜀(𝑐(𝑥(𝑠),𝑣)𝑐(𝑍(𝑠),𝑣))̃𝑝𝜙||||(𝑑𝑣×𝑑𝑠)2||||4𝐄𝑡𝜏𝑘𝜌𝑘0𝜀(𝑐(𝑥(𝑠),𝑣)𝑐(𝑍(𝑠),𝑣))̃𝑝𝜙||||(𝑑𝑣×𝑑𝑠)2=4𝐄𝑡𝜏𝑘𝜌𝑘0𝜀||||𝑐(𝑥(𝑠),𝑣)𝑐(𝑍(𝑠),𝑣)2𝜙(𝑑𝑣)𝑑𝑠8𝑇𝐶𝐾1(𝑘)Δ𝑡+8𝐶𝑡0𝐄sup0𝑢𝑠||𝑥𝑢𝜏𝑘𝜌𝑘𝑋𝑢𝜏𝑘𝜌𝑘||2𝑑𝑠.(4.4) Hence, by substituting (4.3) and (4.4) into (4.2), we get 𝐄sup0𝑡𝑡||𝑥𝑡𝜏𝑘𝜌𝑘𝑋𝑡𝜏𝑘𝜌𝑘||2Δ𝑡6𝑇2𝐶𝑘𝐾1(𝑘)+24𝑇𝐶𝑘𝐾1(𝑘)+24𝑇𝐶𝐾1(+𝑘)6𝑇𝐶𝑘+24𝐶𝑘×+24𝐶𝑡0𝐄sup0𝑢𝑠||𝑥𝑢𝜏𝑘𝜌𝑘𝑋𝑢𝜏𝑘𝜌𝑘||2𝑑𝑠.(4.5) So using the Gronwall inequality (see [13]), we have the result (4.1) by choosing 𝐾2(k)=6𝑇2𝐶𝑘𝐾1(𝑘)+24𝑇𝐶𝑘𝐾1(𝑘)+24𝑇𝐶𝐾1(𝑘)exp6𝑇2𝐶𝑘+24𝑇𝐶𝑘.+24𝑇𝐶(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 𝜖,𝜍(0,1), there is a Δ𝑡 such that for all Δ𝑡<Δ𝑡𝐏sup0𝑡𝑇||𝑥(𝑡)||𝑋(𝑡)2𝜍𝜖,(4.7) for any 𝑇>0.

Proof. For sufficiently small 𝜖,𝜍(0,1), we define Ω=𝜔sup0𝑡𝑇||𝑥(𝑡)||𝑋(𝑡)2.𝜍(4.8) According to Lemmas 2.4 and 3.2, there exists a pair of 𝑘 and Δ𝑡1 such that 𝐏𝜏𝑘𝜖𝑇3,𝐏𝜌𝑘𝜖𝑇3,Δ𝑡Δ𝑡1.(4.9) We thus have 𝐏Ω𝐏𝜏Ω𝑘𝜌𝑘𝜏>𝑇+𝐏𝑘𝜌𝑘𝑇𝐏𝜏Ω𝑘𝜌𝑘𝜏>𝑇+𝐏𝑘𝜌𝑇+𝐏𝑘𝑇𝐏𝜏Ω𝑘𝜌𝑘+>𝑇2𝜖3,(4.10) for Δ𝑡Δ𝑡1. Moreover, according to Lemma 4.1, we have 𝜍𝐏𝜏Ω𝑘𝜌𝑘𝐼>𝑇𝐄{𝜏𝑘𝜌𝑘>𝑇}sup0𝑡𝑇||𝑥(𝑡)||𝑋(𝑡)2𝐄sup0𝑡𝑇||𝑥𝑡𝜏𝑘𝜌𝑘𝑋𝑡𝜏𝑘𝜌𝑘||2𝐾2𝑘Δ𝑡,(4.11) which leads to 𝐏𝜏Ω𝑘𝜌𝑘𝜖>𝑇3,(4.12) for Δ𝑡Δ𝑡2. Therefore, from the inequalities above, we obtain 𝐏Ω𝜖,(4.13) for Δ𝑡Δ𝑡, where Δ𝑡=min{Δ𝑡1,Δ𝑡2}.

We remark that the continuous-time Euler solution 𝑋(𝑡) (3.2) cannot be computed, since it requires knowledge of the entire Brownian motion and Poisson random measure paths, not just only their Δ𝑡-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 𝜖,𝜍(0,1), there is a Δ𝑡 such that for all Δ𝑡<Δ𝑡𝐏||||𝑥(𝑡)𝑍(𝑡)2𝜍,0𝑡𝑇𝜖,(4.14) for any 𝑇>0.

Proof. For sufficiently small 𝜖,𝜍(0,1), we define ||||Ω=𝜔𝑥(𝑡)𝑍(𝑡)2𝜍,0𝑡𝑇.(4.15) A similar analysis as Theorem 4.2 gives 𝐏Ω𝜏𝐏Ω𝑘𝜌𝑘+>𝑇2𝜖3.(4.16) Recalling that 𝜏𝜍𝐏Ω𝑘𝜌𝑘||||>𝑇𝐄𝑥(𝑇)𝑍(𝑇)2𝐼{𝜏𝑘𝜌𝑘>𝑇}||𝑥𝐄𝑇𝜏𝑘𝜌𝑘𝑍𝑇𝜏𝑘𝜌𝑘||22𝐄sup0𝑡𝑇||𝑥𝑡𝜏𝑘𝜌𝑘𝑋𝑡𝜏𝑘𝜌𝑘||2||+2𝐄𝑋𝑇𝜏𝑘𝜌𝑘𝑍𝑇𝜏𝑘𝜌𝑘||22𝐾1𝑘Δ𝑡+2𝐾2𝑘Δ𝑡,(4.17) and using Lemmas 3.1 and 4.1, we get that 𝐏𝜏Ω𝑘𝜌𝑘𝜖>𝑇3,(4.18) for sufficiently small Δ𝑡. Consequently, the inequalities above show that 𝐏Ω𝜖,(4.19) for all sufficiently small Δ𝑡.
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: 𝑑𝑥(𝑡)=𝑎(𝑥(𝑡))𝑑𝑡+𝑏(𝑥(𝑡))𝑑𝑊(𝑡)+𝜀𝑐(𝑥(𝑡),𝑣)̃𝑝𝜙(𝑑𝑣×𝑑𝑡),𝑡>0,(5.1) with 𝑥(0)=𝑥(0)=0, where 𝑑=𝑚=𝑟=1. The coefficients of this equation have the form 1𝑎(𝑥)=2𝑥𝑥3,𝑏(𝑥)=𝑥2,𝑐(𝑥,𝑣)=𝑥𝑣.(5.2) The compensated measure of the Poisson random measure 𝑝𝜙(𝑑𝑣×𝑑𝑡) is given by 𝜙(𝑑𝑣)𝑑𝑡=𝜆𝑓(𝑣)𝑑𝑣𝑑𝑡, where 𝜆=5 and 1𝑓(𝑣)=2𝜋𝑣exp(ln𝑣)22,0𝑣<(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 ||||2𝑥,𝑎(𝑥)+𝑏(𝑥)2+𝜀||||𝑐(𝑥,𝑣)2𝜙(𝑑𝑣)=𝑥𝑥𝑥3+𝑥4+𝜀𝑥2𝑣2𝜆12𝜋𝑣exp(ln𝑣)22𝑑𝑣1+5𝑒21+𝑥2,(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 Δ𝑡, we can have the Euler method 𝑋𝑛+1=𝑋𝑛+12𝑋𝑛𝑋3𝑛Δ𝑡+𝑋2𝑛Δ𝑊𝑛+𝑋𝑛𝑡𝑛+1𝑡𝑛𝜀𝑣̃𝑝𝜙(𝑑𝑣×𝑑𝑡),(5.5) with 𝑋0=0.

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

Algorithm Simulate 𝑋𝑛+1=𝑋𝑛+(1/2)(𝑋𝑛𝑋3𝑛10𝑒𝑋𝑛)Δ𝑡+𝑋2𝑛Δ𝑊𝑛;Simulate variable 𝑝𝜙(𝑡𝑛+1)𝑝𝜙(𝑡𝑛), where 𝑝𝜙(𝑡𝑛) is from Poisson distribution with parameter 𝜆𝑡𝑛;Simulate 𝑝𝜙(𝑡𝑛+1)𝑝𝜙(𝑡𝑛) independent random variables 𝜄𝑖 uniformly distributed on the interval [𝑝𝜙(𝑡𝑛),𝑝𝜙(𝑡𝑛+1));Simulate 𝑝𝜙(𝑡𝑛+1)𝑝𝜙(𝑡𝑛) independent random variables 𝜉𝑖 with law 𝑓(𝑣);obtain 𝑋𝑛+1=𝑋𝑛+1+𝑋𝑛𝑝𝜙(𝑡𝑛+1)𝑖=𝑝𝜙(𝑡𝑛)+1𝐼𝑡𝑛𝜄𝑖<𝑡𝑛+1𝜉𝑖.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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