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 š›¾āˆˆ{0.5,1,1.5,ā€¦} 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 Taylorā€™s methods under weaker conditions than the currently known. In [5, 7], the drift-implicit schemes which achieve strong order š›¾āˆˆ{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 Eulerā€™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 š‘¢1āˆØš‘¢2=max{š‘¢1,š‘¢2} and š‘¢1āˆ§š‘¢2=min{š‘¢1,š‘¢2}. Let |ā‹…| and āŸØā‹…,ā‹…āŸ© be the Euclidean norm and the inner product of vectors in š‘š‘‘,š‘‘āˆˆš. If š“ is a vector or matrix, its transpose is denoted by š“š‘‡. If š“ is a matrix, its trace norm is denoted by āˆš|š“|=trace(š“š‘‡š“). Let šæ2ā„±0(Ī©;ššš‘‘) denote the family of š‘š‘‘-valued ā„±0-measurable random variables šœ‰ with š„|šœ‰|2<āˆž. [š‘§] denotes the largest integer which is less than or equal to š‘§ in š‘. Iš’œ denotes the indicator function of a set š’œ.

The following š‘‘-dimensional SDE driven by Poisson random measure is considered in our paper: ī€œš‘‘š‘„(š‘”)=š‘Ž(š‘„(š‘”āˆ’))š‘‘š‘”+š‘(š‘„(š‘”āˆ’))š‘‘š‘Š(š‘”)+šœ€š‘(š‘„(š‘”āˆ’),š‘£)Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘”),(2.1) for š‘”>0 with initial condition š‘„(0āˆ’)=š‘„(0)=š‘„0āˆˆšæ2ā„±0(Ī©;š‘š‘‘), where š‘„(š‘”āˆ’) denotes limš‘ ā†’š‘”āˆ’š‘„(š‘ ) and Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘”)āˆ¶=š‘šœ™(š‘‘š‘£Ć—š‘‘š‘”)āˆ’šœ™(š‘‘š‘£)š‘‘š‘”.

The drift coefficient š‘Žāˆ¶š‘š‘‘ā†’š‘š‘‘, the diffusion coefficient š‘āˆ¶š‘š‘‘ā†’š‘š‘‘Ć—š‘š, and the jump coefficient š‘āˆ¶š‘š‘‘Ć—šœ€ā†’š‘š‘‘ are assumed to be Borel measurable functions.

The randomness of (2.1) is generated by the following (see [9]). An š‘š-dimensional Wiener process š‘Š={š‘Š(š‘”)=(š‘Š1(š‘”),ā€¦,š‘Šš‘š(š‘”))š‘‡} with independent scalar components is defined on a filtered probability space (Ī©š‘Š,ā„±š‘Š,(ā„±š‘Šš‘”)š‘”ā‰„0,šš‘Š). A Poisson random measure š‘šœ™(šœ”,š‘‘š‘£Ć—š‘‘š‘”) is on Ī©š½Ć—šœ€Ć—[0,āˆž), where šœ€āŠ†š‘š‘Ÿā§µ{0} with š‘Ÿāˆˆš, and its deterministic compensated measure šœ™(š‘‘š‘£)š‘‘š‘”=šœ†š‘“(š‘£)š‘‘š‘£š‘‘š‘”, that is, š„(š‘šœ™(š‘‘š‘£Ć—š‘‘š‘”))=šœ™(š‘‘š‘£)š‘‘š‘”. š‘“(š‘£) is a probability density, and we require finite intensity šœ†=šœ™(šœ€)<āˆž. The Poisson random measure is defined on a filtered probability space (Ī©š½,ā„±š½,(ā„±š½š‘”)š‘”ā‰„0,šš½). The Wiener process and the Poisson random measure are mutually independent. The process š‘„(š‘”) is thus defined on a product space (Ī©,ā„±,(ā„±š‘”)š‘”ā‰„0,š), where Ī©=Ī©š‘ŠĆ—Ī©š½,ā„±=ā„±š‘ŠĆ—ā„±š½,(ā„±š‘”)š‘”ā‰„0=(ā„±š‘Šš‘”)š‘”ā‰„0Ɨ(ā„±š½š‘”)š‘”ā‰„0,š=šš‘ŠĆ—šš½ and ā„±0 contains all š-null sets.

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

Assumption 2.1. For each integer š‘˜ā‰„1, there exists a positive constant š¶š‘˜, dependent on š‘˜, such that ||||š‘Ž(š‘„)āˆ’š‘Ž(š‘¦)2āˆØ||||š‘(š‘„)āˆ’š‘(š‘¦)2ā‰¤š¶š‘˜||||š‘„āˆ’š‘¦2,(2.2) for š‘„,š‘¦āˆˆš‘š‘‘ with |š‘„|āˆØ|š‘¦|ā‰¤š‘˜,š‘˜āˆˆš. And there exists a positive constant š¶ such that ī€œšœ€||||š‘(š‘„,š‘£)āˆ’š‘(š‘¦,š‘£)2||||šœ™(š‘‘š‘£)ā‰¤š¶š‘„āˆ’š‘¦2,(2.3) for š‘„,š‘¦āˆˆš‘š‘‘.

Assumption 2.2. There exists a positive constant šæ such that ||||2āŸØš‘„,š‘Ž(š‘„)āŸ©+š‘(š‘„)2+ī€œšœ€||||š‘(š‘„,š‘£)2ī€·šœ™(š‘‘š‘£)ā‰¤šæ1+|š‘„|2ī€ø,(2.4) for š‘„āˆˆš‘š‘‘.

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

Assumption 2.3. Consider ||||š‘Ž(0)2+||||š‘(0)2+ī€œšœ€||||š‘(0,š‘£)2ī‚ī‚šœ™(š‘‘š‘£)ā‰¤šæ,šæ>0.(2.5)

Actually, Assumptions 2.1 and 2.3 imply the linear growth conditions ||||š‘Ž(š‘„)2āˆØ||||š‘(š‘„)2ā‰¤ī‚‹š¶š‘˜ī€·1+|š‘„|2ī€ø,(2.6) for š‘„āˆˆš‘š‘‘ with |š‘„|ā‰¤š‘˜ and ī‚‹š¶š‘˜>0, and ī€œšœ€||||š‘(š‘„,š‘£)2ī‚š¶ī€·šœ™(š‘‘š‘£)ā‰¤1+|š‘„|2ī€ø,(2.7) for š‘„āˆˆš‘š‘‘ and ī‚š¶>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 šœ–āˆˆ(0,1) and š‘‡>0, there exists a sufficiently large integer š‘˜āˆ—, dependent on šœ– and š‘‡, such that šī€·šœš‘˜ī€øā‰¤š‘‡ā‰¤šœ–,āˆ€š‘˜ā‰„š‘˜āˆ—,(2.8) where šœš‘˜=inf{š‘”ā‰„0āˆ¶|š‘„(š‘”)|ā‰„š‘˜} for š‘˜ā‰„1.

Proof. Using ItĆ“'s formula (see [1]) to |š‘„(š‘”)|2, for š‘”ā‰„0, we have ||||š‘„(š‘”)2=||š‘„0||2+ī€œš‘”0ī‚€||||āŸØ2š‘„(š‘ āˆ’),š‘Ž(š‘„(š‘ āˆ’))āŸ©+š‘(š‘„(š‘ āˆ’))2ī‚+ī€œš‘‘sš‘”0ī€œšœ€ī‚€||||š‘„(š‘ āˆ’)+š‘(š‘„(š‘ āˆ’),š‘£)2āˆ’||||š‘„(š‘ āˆ’)2ī‚+ī€œāˆ’āŸØ2š‘„(š‘ āˆ’),š‘(š‘„(š‘ āˆ’),š‘£)āŸ©šœ™(š‘‘š‘£)š‘‘š‘ š‘”0ī€œāŸØ2š‘„(š‘ āˆ’),š‘(š‘„(š‘ āˆ’))āŸ©š‘‘š‘Š(š‘ )+š‘”0ī€œšœ€ī‚€||||š‘„(š‘ āˆ’)+š‘(š‘„(š‘ āˆ’),š‘£)2āˆ’||||š‘„(š‘ āˆ’)2ī‚Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘ ),(2.9) which gives š„||š‘„ī€·š‘”āˆ§šœš‘˜ī€ø||2||š‘„=š„0||2ī€œ+š„š‘”āˆ§šœš‘˜0ī‚€||||āŸØ2š‘„(š‘ āˆ’),š‘Ž(š‘„(š‘ āˆ’))āŸ©+š‘(š‘„(š‘ āˆ’))2ī‚ī€œš‘‘š‘ +š„š‘”āˆ§šœš‘˜0ī€œšœ€||||š‘(š‘„(š‘ āˆ’),š‘£)2||š‘„šœ™(š‘‘š‘£)š‘‘š‘ =š„0||2ī€œ+š„š‘”āˆ§šœš‘˜0ī‚µ||||āŸØ2š‘„(š‘ āˆ’),š‘Ž(š‘„(š‘ āˆ’))āŸ©+š‘(š‘„(š‘ āˆ’))2+ī€œšœ€||||š‘(š‘„(š‘ āˆ’),š‘£)2ī‚¶šœ™(š‘‘š‘£)š‘‘š‘ ,(2.10) for š‘”āˆˆ[0,š‘‡]. By Assumption 2.2, we thus have š„||š‘„ī€·š‘”āˆ§šœš‘˜ī€ø||2||š‘„ā‰¤š„0||2ī€œ+š„š‘”āˆ§šœš‘˜0šæī‚€||||1+š‘„(š‘ āˆ’)2ī‚||š‘„š‘‘š‘ ā‰¤š„0||2ī€œ+šæš‘‡+šæš‘”0š„||š‘„ī€·š‘ āˆ§šœš‘˜āˆ’ī€ø||2š‘‘š‘ ,(2.11) for š‘”āˆˆ[0,š‘‡]. Consequently by using the Gronwall inequality (see [13]), we obtain š„||š‘„ī€·š‘”āˆ§šœš‘˜ī€ø||2ā‰¤ī‚€š„||š‘„0||2ī‚š‘’+šæš‘‡šæš‘‡,(2.12) for š‘”āˆˆ[0,š‘‡]. We therefore get ī‚€š„||š‘„0||2ī‚š‘’+šæš‘‡šæš‘‡||š‘„ī€·ā‰„š„š‘‡āˆ§šœš‘˜ī€ø||2ī‚€||š‘„ī€·šœā‰„š„š‘˜ī€ø||2š¼{šœš‘˜ā‰¤š‘‡}ī‚ā‰„š‘˜2šī€·šœš‘˜ī€øā‰¤š‘‡,(2.13) which means šī€·šœš‘˜ī€øā‰¤š‘’ā‰¤š‘‡šæš‘‡š‘˜2ī‚€š„||š‘„0||2ī‚.+šæš‘‡(2.14) So for any šœ–āˆˆ(0,1), we can choose š‘˜āˆ—=āŽ”āŽ¢āŽ¢āŽ£īƒŽš‘’šæš‘‡š„||š‘„0||2+šæš‘‡š‘’šæš‘‡šœ–āŽ¤āŽ„āŽ„āŽ¦+1,(2.15) such that šī€·šœš‘˜ī€øā‰¤š‘‡ā‰¤šœ–,āˆ€š‘˜ā‰„š‘˜āˆ—.(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,š‘‡], where š‘‡ is a positive number.

Given a step size Ī”š‘”āˆˆ(0,1), the Euler method applied to (2.1) computes approximation š‘‹š‘›ā‰ˆš‘„(š‘”š‘›), where š‘”š‘›=š‘›Ī”š‘”,š‘›=0,1,ā€¦, by setting š‘‹0=š‘„0 and forming š‘‹š‘›+1=š‘‹š‘›ī€·š‘‹+š‘Žš‘›ī€øī€·š‘‹Ī”š‘”+š‘š‘›ī€øĪ”š‘Šš‘›+ī€œš‘”š‘›+1š‘”š‘›ī€œšœ€š‘ī€·š‘‹š‘›ī€ø,š‘£Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘”),(3.1) where Ī”š‘Šš‘›=š‘Š(š‘”š‘›+1)āˆ’š‘Š(š‘”š‘›).

The continuous-time Euler method is then defined by š‘‹(š‘”)āˆ¶=š‘‹0+ī€œš‘”0ī€œš‘Ž(š‘(š‘ ))š‘‘š‘ +š‘”0ī€œš‘(š‘(š‘ ))š‘‘š‘Š(š‘ )+š‘”0ī€œšœ€š‘(š‘(š‘ ),š‘£)Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘ ),(3.2) where š‘(š‘”)=š‘‹š‘› for š‘”āˆˆ[š‘”š‘›,š‘”š‘›+1),š‘›=0,1,ā€¦.

Actually, we can see in [8], š‘šœ™={š‘šœ™(š‘”)āˆ¶=š‘šœ™(šœ€Ć—[0,š‘”])} is a stochastic process that counts the number of jumps until some given time. The Poisson random measure š‘šœ™(š‘‘š‘£Ć—š‘‘š‘”) generates a sequence of pairs {(šœ„š‘–,šœ‰š‘–),š‘–āˆˆ{1,2,ā€¦,š‘šœ™(š‘‡)}} for a given finite positive constant š‘‡ if šœ†<āˆž. Here {šœ„š‘–āˆ¶Ī©ā†’š‘+,š‘–āˆˆ{1,2,ā€¦,š‘šœ™(š‘‡)}} is a sequence of increasing nonnegative random variables representing the jump times of a standard Poisson process with intensity šœ†, and {šœ‰š‘–āˆ¶Ī©ā†’šœ€,š‘–āˆˆ{1,2,ā€¦,š‘šœ™(š‘‡)} is a sequence of independent identically distributed random variables, where šœ‰š‘– is distributed according to šœ™(š‘‘š‘£)/šœ™(šœ€). Then (3.1) can equivalently be the following form: š‘‹š‘›+1=š‘‹š‘›+ī‚µš‘Žī€·š‘‹š‘›ī€øāˆ’ī€œšœ€š‘ī€·š‘‹š‘›ī€øī‚¶ī€·š‘‹,š‘£šœ™(š‘‘š‘£)Ī”š‘”+š‘š‘›ī€øĪ”š‘Šš‘›+š‘šœ™(š‘”š‘›+1)ī“š‘–=š‘šœ™(š‘”š‘›)+1š‘ī€·š‘‹š‘›,šœ‰š‘–ī€ø.(3.3)

The following lemma shows the close relation between the continuous-time Euler method (3.2) and its step function š‘(š‘”).

Lemma 3.1. Under Assumptions 2.1 and 2.3, for any š‘‡>0, there exists a positive constant š¾1(š‘˜), dependent on š‘˜ and independent of Ī”š‘”, such that for all Ī”š‘”āˆˆ(0,1) the continuous-time Euler method (3.2) satisfies š„||||š‘‹(š‘”)āˆ’š‘(š‘”)2ā‰¤š¾1(š‘˜)Ī”š‘”,(3.4) for 0ā‰¤š‘”ā‰¤š‘‡āˆ§šœš‘˜āˆ§šœŒš‘˜, where šœŒš‘˜=inf{š‘”ā‰„0āˆ¶|š‘‹(š‘”)|ā‰„š‘˜} for š‘˜ā‰„1 and šœš‘˜ is defined in Lemma 2.4.

Proof. For 0ā‰¤š‘”ā‰¤š‘‡āˆ§šœš‘˜āˆ§šœŒš‘˜, there is an integer š‘› such that š‘”āˆˆ[š‘”š‘›,š‘”š‘›+1). So it follows from (3.2) that š‘‹(š‘”)āˆ’š‘(š‘”)=š‘‹š‘›+ī€œš‘”š‘”š‘›ī€œš‘Ž(š‘(š‘ ))š‘‘š‘ +š‘”š‘”š‘›ī€œš‘(š‘(š‘ ))š‘‘š‘Š(š‘ )+š‘”š‘”š‘›ī€œšœ€š‘(š‘(š‘ ),š‘£)Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘ )āˆ’š‘‹š‘›.(3.5) Thus, by taking expectations and using the Cauchy-Schwarz inequality and the martingale properties of š‘‘š‘Š(š‘”) and Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘”), we have š„||||š‘‹(š‘”)āˆ’š‘(š‘”)2||||ī€œā‰¤3š„š‘”š‘”š‘›||||š‘Ž(š‘(š‘ ))š‘‘š‘ 2||||ī€œ+3š„š‘”š‘”š‘›||||š‘(š‘(š‘ ))š‘‘š‘Š(š‘ )2||||ī€œ+3š„š‘”š‘”š‘›ī€œšœ€š‘(š‘(š‘ ),š‘£)Ģƒš‘šœ™||||(š‘‘š‘£Ć—š‘‘š‘ )2ī‚µī€œā‰¤3š„š‘”š‘”š‘›12ī€œš‘‘š‘ š‘”š‘”š‘›||||š‘Ž(š‘(š‘ ))2ī‚¶ī€œš‘‘š‘ +3š„š‘”š‘”š‘›||||š‘(š‘(š‘ ))2ī€œš‘‘š‘ +3š„š‘”š‘”š‘›ī€œšœ€||||š‘(š‘(š‘ ),š‘£)2ī€œšœ™(š‘‘š‘£)š‘‘š‘ ā‰¤3Ī”š‘”š„š‘”š‘”š‘›||||š‘Ž(š‘(š‘ ))2ī€œš‘‘š‘ +3š„š‘”š‘”š‘›||||š‘(š‘(š‘ ))2ī€œš‘‘š‘ +3š„š‘”š‘”š‘›ī€œšœ€||||š‘(š‘(š‘ ),š‘£)2šœ™(š‘‘š‘£)š‘‘š‘ ,(3.6) where the inequality |š‘¢1+š‘¢2+š‘¢3|2ā‰¤3|š‘¢1|2+3|š‘¢2|2+3|š‘¢3|2 for š‘¢1,š‘¢2,š‘¢3āˆˆš‘š‘‘ is used. Therefore, by applying Assumptions 2.1 and 2.3, we get š„ī€œš‘”š‘”š‘›||||š‘Ž(š‘(š‘ ))2ī‚‹š¶š‘‘š‘ ā‰¤š‘˜š„ī€œš‘”š‘”š‘›ī‚€||||1+š‘(š‘ )2ī‚ī‚‹š¶š‘‘š‘ ā‰¤š‘˜ī‚š¶Ī”š‘”+š‘˜š‘˜2š„ī€œĪ”š‘”,š‘”š‘”š‘›||||š‘(š‘(š‘ ))2ī‚‹š¶š‘‘š‘ ā‰¤š‘˜ī‚š¶Ī”š‘”+š‘˜š‘˜2š„ī€œĪ”š‘”,š‘”š‘”š‘›ī€œšœ€||||š‘(š‘(š‘ ),š‘£)2ī‚ī‚šœ™(š‘‘š‘£)š‘‘š‘ ā‰¤š¶Ī”š‘”+š¶š‘˜2Ī”š‘”,(3.7) which lead to š„||||š‘‹(š‘”)āˆ’š‘(š‘”)2ī‚€3ī‚‹š¶ā‰¤Ī”š‘”š‘˜Ī”š‘”+3š‘˜2ī‚‹š¶š‘˜ī‚‹š¶Ī”š‘”+3š‘˜+3š‘˜2ī‚‹š¶š‘˜ī‚+3š¶+3š‘˜2ī‚š¶ī‚,(3.8) for š‘”āˆˆ[0,š‘‡āˆ§šœš‘˜āˆ§šœŒš‘˜]. Therefore, we obtain the result (3.4) by choosing š¾1ī‚‹š¶(š‘˜)=6š‘˜+6š‘˜2ī‚‹š¶š‘˜ī‚+3š¶+3š‘˜2ī‚š¶.(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 šœ–āˆˆ(0,1) and š‘‡>0, there exist a sufficiently large š‘˜āˆ— and a sufficiently small Ī”š‘”āˆ—1 such that šī€·šœŒš‘˜āˆ—ī€øā‰¤š‘‡ā‰¤šœ–,āˆ€Ī”š‘”ā‰¤Ī”š‘”āˆ—1,(3.10) where šœŒš‘˜āˆ— is defined in Lemma 3.1.

Proof . Applying generalized ItĆ“'s formula (see [1]) to |š‘‹(š‘”)|2, for š‘”ā‰„0, yields ||||š‘‹(š‘”)2=||š‘‹0||2+ī€œš‘”02ī‚€ī‚¬ī‚­+||||š‘‹(š‘ ),š‘Ž(š‘(š‘ ))š‘(š‘(š‘ ))2ī‚+ī€œš‘‘š‘ š‘”0ī€œšœ€ī‚€||||š‘‹(š‘ )+š‘(š‘(š‘ ),š‘£)2āˆ’||||š‘‹(š‘ )2āˆ’ī‚¬2+ī€œš‘‹(š‘ ),š‘(š‘(š‘ ),š‘£)ī‚­ī‚šœ™(š‘‘š‘£)š‘‘š‘ š‘”0ī‚¬2ī‚­ī€œš‘‹(š‘ ),š‘(š‘(š‘ ))š‘‘š‘Š(š‘ )+š‘”0ī€œšœ€ī‚€||||š‘‹(š‘ )+š‘(š‘(š‘ ),š‘£)2āˆ’||||š‘‹(š‘ )2ī‚Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘ ).(3.11) By taking expectations, we thus have š„||š‘‹ī€·š‘”āˆ§šœŒš‘˜ī€ø||2||š‘‹=š„0||2ī€œ+š„š‘”āˆ§šœŒš‘˜0ī‚µī‚¬2ī‚­+||||š‘‹(š‘ ),š‘Ž(š‘(š‘ ))š‘(š‘(š‘ ))2+ī€œšœ€||||š‘(š‘(š‘ ),š‘£)2ī‚¶||š‘‹šœ™(š‘‘š‘£)š‘‘š‘ =š„0||2ī€œ+Eš‘”āˆ§šœŒš‘˜0ī‚µī‚¬2ī‚€š‘‹(š‘ ),š‘Ž+|||š‘ī‚€š‘‹(š‘ )ī‚ī‚­ī‚|||š‘‹(š‘ )2+ī€œšœ€|||š‘ī‚€š‘‹ī‚|||(š‘ ),š‘£2šœ™ī‚¶ī€œ(š‘‘š‘£)š‘‘š‘ +š„š‘”āˆ§šœŒš‘˜0ī‚¬2š‘‹ī‚€(š‘ ),š‘Ž(š‘(š‘ ))āˆ’š‘Žš‘‹ī€œ(š‘ )ī‚ī‚­š‘‘š‘ +š„š‘”āˆ§šœŒš‘˜0ī‚µ||||š‘(š‘(š‘ ))2āˆ’|||š‘ī‚€ī‚|||š‘‹(š‘ )2ī‚¶ī€œš‘‘š‘ +š„š‘”āˆ§šœŒš‘˜0ī€œšœ€ī‚µ||||š‘(š‘(š‘ ),š‘£)2āˆ’|||š‘ī‚€ī‚|||š‘‹(š‘ ),š‘£2ī‚¶šœ™(š‘‘š‘£)š‘‘š‘ .(3.12) For š‘”āˆˆ[0,š‘‡]. Now, by using the inequalities āŸØš‘¢1,š‘¢2āŸ©ā‰¤|š‘¢1||š‘¢2| for š‘¢1,š‘¢2āˆˆš‘š‘‘, (2.2) in Assumption 2.1, Fubini's theorem, Cauchy-Schwarzā€™s inequality, and Lemma 3.1, we get š„ī€œš‘”āˆ§šœŒš‘˜0ī‚¬2ī‚€š‘‹(š‘ ),š‘Ž(š‘(š‘ ))āˆ’š‘Žī€œš‘‹(š‘ )ī‚ī‚­š‘‘š‘ ā‰¤2š„š‘”āˆ§šœŒš‘˜0|||||||ī‚€š‘‹(š‘ )š‘Ž(š‘(š‘ ))āˆ’š‘Žī‚|||āˆšš‘‹(š‘ )š‘‘š‘ ā‰¤2š‘˜š¶š‘˜ī€œš‘”0š„||š‘ī€·š‘ āˆ§šœŒš‘˜ī€øāˆ’š‘‹ī€·š‘ āˆ§šœŒš‘˜ī€ø||āˆšš‘‘š‘ ā‰¤2š‘˜š¶š‘˜ī€œš‘”0ī‚€š„||š‘ī€·š‘ āˆ§šœŒš‘˜ī€øāˆ’š‘‹ī€·š‘ āˆ§šœŒš‘˜ī€ø||2ī‚1/2āˆšš‘‘š‘ ā‰¤2š‘˜š‘‡š¶š‘˜š¾1(š‘˜)Ī”š‘”.(3.13) And, similarly as above, we have š„ī€œš‘”āˆ§šœŒš‘˜0ī‚µ||||š‘(š‘(š‘ ))2āˆ’|||š‘ī‚€ī‚|||š‘‹(š‘ )2ī‚¶ī€œš‘‘š‘ ā‰¤š„š‘”āˆ§šœŒš‘˜0ī‚€||||+|||š‘ī‚€š‘(š‘(š‘ ))ī‚|||ī‚Ć—ī‚€||||āˆ’|||š‘ī‚€š‘‹(š‘ )š‘(š‘(š‘ ))ī‚|||ī‚ī”š‘‹(š‘ )š‘‘š‘ ā‰¤2ī‚‹š¶š‘˜ī€·1+š‘˜2ī€øš„ī€œš‘”āˆ§šœŒš‘˜0|||ī‚€š‘(š‘(š‘ ))āˆ’š‘ī‚|||ī”š‘‹(š‘ )š‘‘š‘ ā‰¤2š¶š‘˜ī‚‹š¶š‘˜ī€·1+š‘˜2ī€øī€œš‘”0š„||š‘ī€·š‘ āˆ§šœŒš‘˜ī€øāˆ’š‘‹ī€·š‘ āˆ§šœŒš‘˜ī€ø||ī”š‘‘š‘ ā‰¤2š‘‡š¶š‘˜ī‚‹š¶š‘˜š¾1ī€·(š‘˜)1+š‘˜2ī€øĪ”š‘”.(3.14) Moreover, in the same way, we obtain š„ī€œš‘”āˆ§šœŒš‘˜0ī€œšœ€ī‚µ||||š‘(š‘(š‘ ),š‘£)2āˆ’|||š‘ī‚€ī‚|||š‘‹(š‘ ),š‘£2ī‚¶ī€œšœ™(š‘‘š‘£)š‘‘š‘ =š„š‘”āˆ§šœŒš‘˜0ī€œšœ€ī‚µ|||ī‚€š‘(š‘(š‘ ),š‘£)āˆ’š‘ī‚ī‚€š‘‹(š‘ ),š‘£+š‘ī‚|||š‘‹(š‘ ),š‘£2āˆ’|||š‘ī‚€ī‚|||š‘‹(š‘ ),š‘£2ī‚¶ī€œšœ™(š‘‘š‘£)š‘‘š‘ ā‰¤š„š‘”āˆ§šœŒš‘˜0ī€œšœ€ī‚µ2|||ī‚€š‘(š‘(š‘ ),š‘£)āˆ’š‘ī‚|||š‘‹(š‘ ),š‘£2+|||š‘ī‚€ī‚|||š‘‹(š‘ ),š‘£2ī‚¶ī€œšœ™(š‘‘š‘£)š‘‘š‘ ā‰¤2š¶š„š‘”āˆ§šœŒš‘˜0||š‘(š‘ )āˆ’||š‘‹(š‘ )2ī‚ī€œš‘‘š‘ +š¶š„š‘”āˆ§šœŒš‘˜0ī‚€||1+||š‘‹(š‘ )2ī‚ī€œš‘‘š‘ ā‰¤2š¶š‘”0š„||š‘ī€·š‘ āˆ§šœŒš‘˜ī€øāˆ’š‘‹ī€·š‘ āˆ§šœŒš‘˜ī€ø||2ī‚ī€œš‘‘š‘ +š¶š„š‘”āˆ§šœŒš‘˜0ī‚€||1+||š‘‹(š‘ )2ī‚š‘‘š‘ ā‰¤2š¶š‘‡š¾1ī‚ī‚ī€œ(š‘˜)Ī”š‘”+š¶š‘‡+š¶š„š‘”āˆ§šœŒš‘˜0||||š‘‹(š‘ )2š‘‘š‘ ,(3.15) where the inequality |š‘¢1+š‘¢2|2ā‰¤2|š‘¢1|2+2|š‘¢2|2 for š‘¢1,š‘¢2āˆˆš‘š‘‘, (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 š„||š‘‹ī€·š‘”āˆ§šœŒš‘˜ī€ø||2||š‘‹ā‰¤š„0||2ī€œ+šæš„š‘”āˆ§šœŒš‘˜0ī‚€||1+||š‘‹(š‘ )2ī‚ī‚ī€œš‘‘š‘ +š¶š„š‘”āˆ§šœŒš‘˜0||||š‘‹(š‘ )2āˆšš‘‘š‘ +2š‘˜š‘‡š¶š‘˜š¾1ī”(š‘˜)Ī”š‘”+2š‘‡š¶š‘˜ī‚‹š¶š‘˜š¾1ī€·(š‘˜)1+š‘˜2ī€øĪ”š‘”+2š¶š‘‡š¾1ī‚ā‰¤ī‚€ī‚š¶ī‚ī€œ(š‘˜)Ī”š‘”+š¶š‘‡šæ+š‘”0š„||š‘‹ī€·š‘ āˆ§šœŒš‘˜ī€ø||2||š‘‹š‘‘š‘ +š„0||2ī‚+ī‚µāˆš+šæš‘‡+š¶š‘‡2š‘˜š‘‡š¶š‘˜š¾1ī”(š‘˜)+2š‘‡š¶š‘˜ī‚‹š¶š‘˜š¾1ī€·(š‘˜)1+š‘˜2ī€øī‚¶āˆšĪ”š‘”+2š¶š‘‡š¾1(š‘˜)Ī”š‘”,(3.16) for 0ā‰¤š‘”ā‰¤š‘‡. Therefore, by the Gronwall inequality (see [13]), for 0ā‰¤š‘”ā‰¤š‘‡, we get š„||š‘‹ī€·š‘”āˆ§šœŒš‘˜ī€ø||2ā‰¤š›¼1š›¼4+š›¼4š›¼2āˆš(š‘˜)Ī”š‘”+š›¼4š›¼3(š‘˜)Ī”š‘”,(3.17) where š›¼1||š‘‹=š„0||2ī‚š›¼+šæš‘‡+š¶š‘‡,2āˆš(š‘˜)=2š‘˜š‘‡š¶š‘˜š¾1ī”(š‘˜)+2š‘‡š¶š‘˜ī‚‹š¶š‘˜š¾1ī€·(š‘˜)1+š‘˜2ī€ø,š›¼3(š‘˜)=2š¶š‘‡š¾1š›¼(š‘˜),4ī‚€ī‚ī‚.=expšæš‘‡+š¶š‘‡(3.18) We thus obtain that š‘˜2šī€·šœŒš‘˜ī€øī‚€||ā‰¤š‘‡ā‰¤š„š‘‹ī€·šœŒš‘˜ī€ø||2š¼{šœŒš‘˜ā‰¤š‘‡}ī‚||ā‰¤š„š‘‹ī€·š‘‡āˆ§šœŒš‘˜ī€ø||2ā‰¤š›¼1š›¼4+š›¼4š›¼2āˆš(š‘˜)Ī”š‘”+š›¼4š›¼3(š‘˜)Ī”š‘”.(3.19) So for any šœ–āˆˆ(0,1), we can choose sufficiently large integer š‘˜=š‘˜āˆ— such that š›¼1š›¼4š‘˜āˆ—2ā‰¤šœ–2,(3.20) and choose sufficiently small Ī”š‘”āˆ—1āˆˆ(0,1) such that š›¼4š›¼2ī€·š‘˜āˆ—ī€øāˆšĪ”š‘”āˆ—1+š›¼4š›¼3ī€·š‘˜āˆ—ī€øĪ”š‘”āˆ—1š‘˜āˆ—2ā‰¤šœ–2.(3.21) Hence, we have šī€·šœŒš‘˜āˆ—ī€øā‰¤š‘‡ā‰¤šœ–,āˆ€Ī”š‘”ā‰¤Ī”š‘”āˆ—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,š‘‡].

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

Lemma 4.1. Under Assumptions 2.1 and 2.3, for any š‘‡>0, there exists a positive constant š¾2(š‘˜), dependent on š‘˜ and independent of Ī”š‘”, such that for all Ī”š‘”āˆˆ(0,1) the solution of (2.1) and the continuous-time Euler method (3.2) satisfy š„ī‚µsup0ā‰¤š‘”ā‰¤š‘‡||š‘„ī€·š‘”āˆ§šœš‘˜āˆ§šœŒš‘˜ī€øāˆ’š‘‹ī€·š‘”āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2ī‚¶ā‰¤š¾2(š‘˜)Ī”š‘”,(4.1) where šœš‘˜ and šœŒš‘˜ are defined in Lemmas 2.4 and 3.1, respectively.

Proof. From (2.1) and (3.2), for any 0ā‰¤š‘”ā€²ā‰¤š‘‡, we have š„īƒ©sup0ā‰¤š‘”ā‰¤š‘”ā€²||š‘„ī€·š‘”āˆ§šœš‘˜āˆ§šœŒš‘˜ī€øāˆ’š‘‹ī€·š‘”āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2īƒŖīƒ©ā‰¤3š„sup0ā‰¤š‘”ā‰¤š‘”ā€²||||ī€œš‘”āˆ§šœš‘˜āˆ§šœŒš‘˜0||||(š‘Ž(š‘„(š‘ āˆ’))āˆ’š‘Ž(š‘(š‘ )))š‘‘š‘ 2īƒŖīƒ©+3š„sup0ā‰¤š‘”ā‰¤š‘”ā€²||||ī€œš‘”āˆ§šœš‘˜āˆ§šœŒš‘˜0||||(š‘(š‘„(š‘ āˆ’))āˆ’š‘(š‘(š‘ )))š‘‘š‘Š(š‘ )2īƒŖīƒ©+3š„sup0ā‰¤š‘”ā‰¤š‘”ā€²||||ī€œš‘”āˆ§šœš‘˜āˆ§šœŒš‘˜0ī€œšœ€(š‘(š‘„(š‘ āˆ’),š‘£)āˆ’š‘(š‘(š‘ ),š‘£))Ģƒš‘šœ™(||||š‘‘š‘£Ć—š‘‘š‘ )2īƒŖ,(4.2) where the inequality |š‘¢1+š‘¢2+š‘¢3|2ā‰¤3|š‘¢1|2+3|š‘¢2|2+3|š‘¢3|2 for š‘¢1,š‘¢2,š‘¢3āˆˆš‘š‘‘ is used. Therefore, by using the Cauchy-Schwarz inequality, (2.2) in Assumption 2.1, Lemma 3.1 and Fubini's theorem, we obtain š„īƒ©sup0ā‰¤š‘”ā‰¤š‘”ā€²||||ī€œš‘”āˆ§šœš‘˜āˆ§šœŒš‘˜0||||(š‘Ž(š‘„(š‘ āˆ’))āˆ’š‘Ž(š‘(š‘ )))š‘‘š‘ 2īƒŖīƒ©ā‰¤š„sup0ā‰¤š‘”ā‰¤š‘”ā€²ī€œš‘”āˆ§šœš‘˜āˆ§šœŒš‘˜012ī€œš‘‘š‘ š‘”āˆ§šœš‘˜āˆ§šœŒš‘˜0||||š‘Ž(š‘„(š‘ āˆ’))āˆ’š‘Ž(š‘(š‘ ))2īƒŖīƒ©ī€œš‘‘š‘ ā‰¤š‘‡š„š‘”ā€²āˆ§šœš‘˜āˆ§šœŒš‘˜0||||š‘Ž(š‘„(š‘ āˆ’))āˆ’š‘Ž(š‘(š‘ ))2īƒŖš‘‘š‘ ā‰¤2š‘‡š¶š‘˜š„īƒ©ī€œš‘”ā€²āˆ§šœš‘˜āˆ§šœŒš‘˜0||||š‘‹(š‘ )āˆ’š‘(š‘ )2īƒŖš‘‘š‘ +2š‘‡š¶š‘˜š„īƒ©ī€œš‘”ā€²āˆ§šœš‘˜āˆ§šœŒš‘˜0||š‘„(š‘ āˆ’)āˆ’||š‘‹(š‘ )2īƒŖš‘‘š‘ ā‰¤2š‘‡š¶š‘˜ī€œš‘”ā€²0š„||š‘‹ī€·š‘ āˆ§šœš‘˜āˆ§šœŒš‘˜ī€øī€·āˆ’š‘š‘ āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2š‘‘š‘ +2š‘‡š¶š‘˜ī€œš‘”ā€²0š„||š‘„ī€·š‘ āˆ§šœš‘˜āˆ§šœŒš‘˜āˆ’ī€øāˆ’š‘‹ī€·š‘ āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2š‘‘š‘ ā‰¤2š‘‡2š¶š‘˜š¾1(š‘˜)Ī”š‘”+2š‘‡š¶š‘˜ī€œš‘”ā€²0š„ī‚µsup0ā‰¤š‘¢ā‰¤š‘ ||š‘„ī€·š‘¢āˆ§šœš‘˜āˆ§šœŒš‘˜āˆ’ī€øāˆ’š‘‹ī€·š‘¢āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2ī‚¶š‘‘š‘ .(4.3) Moreover, by using the martingale properties of š‘‘š‘Š(š‘”) and Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘”), Assumption 2.1, Lemma 3.1, and Fubini's theorem, we have š„īƒ©sup0ā‰¤š‘”ā‰¤š‘”ā€²||||ī€œš‘”āˆ§šœš‘˜āˆ§šœŒš‘˜0||||(š‘(š‘„(š‘ āˆ’))āˆ’š‘(š‘(š‘ )))š‘‘š‘Š(š‘ )2īƒŖī€œā‰¤4š„š‘”ā€²āˆ§šœš‘˜āˆ§šœŒš‘˜0||||š‘(š‘„(š‘ āˆ’))āˆ’š‘(š‘(š‘ ))2š‘‘š‘ ā‰¤8š¶š‘˜š„ī€œš‘”ā€²āˆ§šœš‘˜āˆ§šœŒš‘˜0||||š‘‹(š‘ )āˆ’š‘(š‘ )2š‘‘š‘ +8š¶š‘˜š„ī€œš‘”ā€²āˆ§šœš‘˜āˆ§šœŒš‘˜0||š‘„(š‘ āˆ’)āˆ’||š‘‹(š‘ )2š‘‘š‘ ā‰¤8š¶š‘˜ī€œš‘”ā€²0š„||š‘‹ī€·š‘ āˆ§šœš‘˜āˆ§šœŒš‘˜ī€øī€·āˆ’š‘š‘ āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2š‘‘š‘ +8š¶š‘˜ī€œš‘”ā€²0š„||š‘„ī€·š‘ āˆ§šœš‘˜āˆ§šœŒš‘˜āˆ’ī€øāˆ’š‘‹ī€·š‘ āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2š‘‘š‘ ā‰¤8š‘‡š¶š‘˜š¾1(š‘˜)Ī”š‘”+8š¶š‘˜ī€œš‘”ā€²0š„ī‚µsup0ā‰¤š‘¢ā‰¤š‘ ||š‘„ī€·š‘¢āˆ§šœš‘˜āˆ§šœŒš‘˜āˆ’ī€øāˆ’š‘‹ī€·š‘¢āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2ī‚¶š„īƒ©š‘‘š‘ ,sup0ā‰¤š‘”ā‰¤š‘”ā€²||||ī€œš‘”āˆ§šœš‘˜āˆ§šœŒš‘˜0ī€œšœ€(š‘(š‘„(š‘ āˆ’),š‘£)āˆ’š‘(š‘(š‘ āˆ’),š‘£))Ģƒš‘šœ™||||(š‘‘š‘£Ć—š‘‘š‘ )2īƒŖ||||ī€œā‰¤4š„š‘”ā€²āˆ§šœš‘˜āˆ§šœŒš‘˜0ī€œšœ€(š‘(š‘„(š‘ āˆ’),š‘£)āˆ’š‘(š‘(š‘ āˆ’),š‘£))Ģƒš‘šœ™||||(š‘‘š‘£Ć—š‘‘š‘ )2ī€œ=4š„š‘”ā€²āˆ§šœš‘˜āˆ§šœŒš‘˜0ī€œšœ€||||š‘(š‘„(š‘ āˆ’),š‘£)āˆ’š‘(š‘(š‘ āˆ’),š‘£)2šœ™(š‘‘š‘£)š‘‘š‘ ā‰¤8š‘‡š¶š¾1(ī€œš‘˜)Ī”š‘”+8š¶š‘”ā€²0š„ī‚µsup0ā‰¤š‘¢ā‰¤š‘ ||š‘„ī€·š‘¢āˆ§šœš‘˜āˆ§šœŒš‘˜āˆ’ī€øāˆ’š‘‹ī€·š‘¢āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2ī‚¶š‘‘š‘ .(4.4) Hence, by substituting (4.3) and (4.4) into (4.2), we get š„īƒ©sup0ā‰¤š‘”ā‰¤š‘”ā€²||š‘„ī€·š‘”āˆ§šœš‘˜āˆ§šœŒš‘˜ī€øāˆ’š‘‹ī€·š‘”āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2īƒŖī€·ā‰¤Ī”š‘”6š‘‡2š¶š‘˜š¾1(š‘˜)+24š‘‡š¶š‘˜š¾1(š‘˜)+24š‘‡š¶š¾1(ī€ø+ī€·š‘˜)6š‘‡š¶š‘˜+24š¶š‘˜ī€øƗī€œ+24š¶š‘”ā€²0š„ī‚µsup0ā‰¤š‘¢ā‰¤š‘ ||š‘„ī€·š‘¢āˆ§šœš‘˜āˆ§šœŒš‘˜āˆ’ī€øāˆ’š‘‹ī€·š‘¢āˆ§šœš‘˜āˆ§šœŒš‘˜ī€ø||2ī‚¶š‘‘š‘ .(4.5) So using the Gronwall inequality (see [13]), we have the result (4.1) by choosing š¾2ī€·(k)=6š‘‡2š¶š‘˜š¾1(š‘˜)+24š‘‡š¶š‘˜š¾1(š‘˜)+24š‘‡š¶š¾1ī€øī€·(š‘˜)exp6š‘‡2š¶š‘˜+24š‘‡š¶š‘˜ī€ø.+24š‘‡š¶(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 šœ–,šœāˆˆ(0,1), there is a Ī”š‘”āˆ— such that for all Ī”š‘”<Ī”š‘”āˆ—šī‚µsup0ā‰¤š‘”ā‰¤š‘‡||š‘„(š‘”)āˆ’||š‘‹(š‘”)2ī‚¶ā‰„šœā‰¤šœ–,(4.7) for any š‘‡>0.

Proof. For sufficiently small šœ–,šœāˆˆ(0,1), we define ī‚»Ī©=šœ”āˆ¶sup0ā‰¤š‘”ā‰¤š‘‡||š‘„(š‘”)āˆ’||š‘‹(š‘”)2ī‚¼.ā‰„šœ(4.8) According to Lemmas 2.4 and 3.2, there exists a pair of š‘˜ā‹† and Ī”š‘”ā‹†1 such that šī€·šœš‘˜ā‹†ī€øā‰¤šœ–ā‰¤š‘‡3,šī€·šœŒš‘˜ā‹†ī€øā‰¤šœ–ā‰¤š‘‡3,āˆ€Ī”š‘”ā‰¤Ī”š‘”ā‹†1.(4.9) We thus have šī‚€Ī©ī‚ī‚€ā‰¤šī€½šœĪ©āˆ©š‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€¾ī‚ī€·šœ>š‘‡+šš‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€øī‚€ā‰¤š‘‡ā‰¤šī€½šœĪ©āˆ©š‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€¾ī‚ī€·šœ>š‘‡+šš‘˜ā‹†ī€øī€·šœŒā‰¤š‘‡+šš‘˜ā‹†ī€øī‚€ā‰¤š‘‡ā‰¤šī€½šœĪ©āˆ©š‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€¾ī‚+>š‘‡2šœ–3,(4.10) for Ī”š‘”ā‰¤Ī”š‘”ā‹†1. Moreover, according to Lemma 4.1, we have ī‚€šœšī€½šœĪ©āˆ©š‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€¾ī‚ī‚µš¼>š‘‡ā‰¤š„{šœš‘˜ā‹†āˆ§šœŒš‘˜ā‹†>š‘‡}sup0ā‰¤š‘”ā‰¤š‘‡||š‘„(š‘”)āˆ’||š‘‹(š‘”)2ī‚¶ī‚µā‰¤š„sup0ā‰¤š‘”ā‰¤š‘‡||š‘„ī€·š‘”āˆ§šœš‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€øāˆ’š‘‹ī€·š‘”āˆ§šœš‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€ø||2ī‚¶ā‰¤š¾2ī€·š‘˜ā‹†ī€øĪ”š‘”,(4.11) which leads to šī‚€ī€½šœĪ©āˆ©š‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€¾ī‚ā‰¤šœ–>š‘‡3,(4.12) for Ī”š‘”ā‰¤Ī”š‘”ā‹†2. Therefore, from the inequalities above, we obtain šī‚€Ī©ī‚ā‰¤šœ–,(4.13) for Ī”š‘”ā‰¤Ī”š‘”ā‹†, where Ī”š‘”ā‹†=min{Ī”š‘”ā‹†1,Ī”š‘”ā‹†2}.

We remark that the continuous-time Euler solution š‘‹(š‘”) (3.2) cannot be computed, since it requires knowledge of the entire Brownian motion and Poisson random measure paths, not just only their Ī”š‘”-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 šœ–,šœāˆˆ(0,1), there is a Ī”š‘”āˆ— such that for all Ī”š‘”<Ī”š‘”āˆ—šī‚€||||š‘„(š‘”)āˆ’š‘(š‘”)2ī‚ā‰„šœ,0ā‰¤š‘”ā‰¤š‘‡ā‰¤šœ–,(4.14) for any š‘‡>0.

Proof. For sufficiently small šœ–,šœāˆˆ(0,1), we define ī‚ī‚†||||Ī©=šœ”āˆ¶š‘„(š‘”)āˆ’š‘(š‘”)2ī‚‡ā‰„šœ,0ā‰¤š‘”ā‰¤š‘‡.(4.15) A similar analysis as Theorem 4.2 gives šī‚€ī‚Ī©ī‚ī‚€ī‚ī€½šœā‰¤šĪ©āˆ©š‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€¾ī‚+>š‘‡2šœ–3.(4.16) Recalling that ī‚€ī‚ī€½šœšœšĪ©āˆ©š‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€¾ī‚ī‚€||||>š‘‡ā‰¤š„š‘„(š‘‡)āˆ’š‘(š‘‡)2š¼{šœš‘˜ā‹†āˆ§šœŒš‘˜ā‹†>š‘‡}ī‚||š‘„ī€·ā‰¤š„š‘‡āˆ§šœš‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€øī€·āˆ’š‘š‘‡āˆ§šœš‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€ø||2ī‚µā‰¤2š„sup0ā‰¤š‘”ā‰¤š‘‡||š‘„ī€·š‘”āˆ§šœš‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€øāˆ’š‘‹ī€·š‘”āˆ§šœš‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€ø||2ī‚¶||+2š„š‘‹ī€·š‘‡āˆ§šœš‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€øī€·āˆ’š‘š‘‡āˆ§šœš‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€ø||2ā‰¤2š¾1ī€·š‘˜ā‹†ī€øĪ”š‘”+2š¾2ī€·š‘˜ā‹†ī€øĪ”š‘”,(4.17) and using Lemmas 3.1 and 4.1, we get that šī‚€ī‚ī€½šœĪ©āˆ©š‘˜ā‹†āˆ§šœŒš‘˜ā‹†ī€¾ī‚ā‰¤šœ–>š‘‡3,(4.18) for sufficiently small Ī”š‘”. Consequently, the inequalities above show that šī‚€ī‚Ī©ī‚ā‰¤šœ–,(4.19) for all sufficiently small Ī”š‘”.
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: ī€œš‘‘š‘„(š‘”)=š‘Ž(š‘„(š‘”āˆ’))š‘‘š‘”+š‘(š‘„(š‘”āˆ’))š‘‘š‘Š(š‘”)+šœ€š‘(š‘„(š‘”āˆ’),š‘£)Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘”),š‘”>0,(5.1) with š‘„(0)=š‘„(0āˆ’)=0, where š‘‘=š‘š=š‘Ÿ=1. The coefficients of this equation have the form 1š‘Ž(š‘„)=2ī€·š‘„āˆ’š‘„3ī€ø,š‘(š‘„)=š‘„2,š‘(š‘„,š‘£)=š‘„š‘£.(5.2) The compensated measure of the Poisson random measure š‘šœ™(š‘‘š‘£Ć—š‘‘š‘”) is given by šœ™(š‘‘š‘£)š‘‘š‘”=šœ†š‘“(š‘£)š‘‘š‘£š‘‘š‘”, where šœ†=5 and 1š‘“(š‘£)=āˆšī‚µāˆ’2šœ‹š‘£exp(lnš‘£)22ī‚¶,0ā‰¤š‘£<āˆž(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 ||||2āŸØš‘„,š‘Ž(š‘„)āŸ©+š‘(š‘„)2+ī€œšœ€||||š‘(š‘„,š‘£)2ī€·šœ™(š‘‘š‘£)=š‘„š‘„āˆ’š‘„3ī€ø+š‘„4+ī€œšœ€š‘„2š‘£2šœ†1āˆšī‚µāˆ’2šœ‹š‘£exp(lnš‘£)22ī‚¶ā‰¤ī€·š‘‘š‘£1+5š‘’2ī€øī€·1+š‘„2ī€ø,(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,āˆž).

Given the stepsize Ī”š‘”, we can have the Euler method š‘‹š‘›+1=š‘‹š‘›+12ī€·š‘‹š‘›āˆ’š‘‹3š‘›ī€øĪ”š‘”+š‘‹2š‘›Ī”š‘Šš‘›+š‘‹š‘›ī€œš‘”š‘›+1š‘”š‘›ī€œšœ€š‘£Ģƒš‘šœ™(š‘‘š‘£Ć—š‘‘š‘”),(5.5) with š‘‹0=0.

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

Algorithm ā€‰Simulate š‘‹āˆ’š‘›+1āˆ¶=š‘‹š‘›+(1/2)(š‘‹š‘›āˆ’š‘‹3š‘›āˆšāˆ’10š‘’š‘‹š‘›)Ī”š‘”+š‘‹2š‘›Ī”š‘Šš‘›;ā€‰Simulate variable š‘šœ™(š‘”š‘›+1)āˆ’š‘šœ™(š‘”š‘›), where š‘šœ™(š‘”š‘›) is from Poisson distribution with parameter šœ†š‘”š‘›;ā€‰Simulate š‘šœ™(š‘”š‘›+1)āˆ’š‘šœ™(š‘”š‘›) independent random variables šœ„š‘– uniformly distributed on the interval [š‘šœ™(š‘”š‘›),š‘šœ™(š‘”š‘›+1));ā€‰Simulate š‘šœ™(š‘”š‘›+1)āˆ’š‘šœ™(š‘”š‘›) independent random variables šœ‰š‘– with law š‘“(š‘£);ā€‰obtain š‘‹š‘›+1=š‘‹āˆ’š‘›+1+š‘‹š‘›āˆ‘š‘šœ™(š‘”š‘›+1)š‘–=š‘šœ™(š‘”š‘›)+1š¼š‘”š‘›ā‰¤šœ„š‘–<š‘”š‘›+1šœ‰š‘–.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).