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

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 𝑢₁∨𝑢₂=max{𝑢₁,𝑢₂} and 𝑢₁∧𝑢₂=min{𝑢₁,𝑢₂}. Let |⋅| and ⟨⋅,⋅⟩ be the Euclidean norm and the inner product of vectors in 𝐑^𝑑,𝑑∈𝐍. If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^𝑇. If 𝐴 is a matrix, its trace norm is denoted by |𝐴|=√trace(𝐴^𝑇𝐴). Let 𝐿^2ℱ₀(Ω;𝚁^𝑑) denote the family of 𝐑^𝑑-valued ℱ₀-measurable random variables 𝜉 with 𝐄|𝜉|²<∞. [𝑧] denotes the largest integer which is less than or equal to 𝑧 in 𝐑. I_𝒜 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)𝑝_𝜙̃(𝑑𝑣×𝑑𝑡),(2.1) for 𝑡>0 with initial condition 𝑥(0−)=𝑥(0)=𝑥₀∈𝐿^2ℱ₀(Ω;𝐑^𝑑), where 𝑥(𝑡−) denotes lim_{𝑠→𝑡−}𝑥(𝑠) and 𝑝_𝜙̃(𝑑𝑣×𝑑𝑡)∶=𝑝_𝜙(𝑑𝑣×𝑑𝑡)−𝜙(𝑑𝑣)𝑑𝑡.

The drift coefficient 𝑎∶𝐑^𝑑→𝐑^𝑑, the diffusion coefficient 𝑏∶𝐑^𝑑→𝐑^𝑑×𝑚, and the jump coefficient 𝑐∶𝐑^𝑑×𝜀→𝐑^𝑑 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),…,𝑊^𝑚(𝑡))^𝑇} with independent scalar components is defined on a filtered probability space (Ω^𝑊,ℱ^𝑊,(ℱ^𝑊𝑡)_𝑡≥0,𝐏^𝑊). A Poisson random measure 𝑝_𝜙(𝜔,𝑑𝑣×𝑑𝑡) is on Ω^𝐽×𝜀×[0,∞), where 𝜀⊆𝐑^𝑟⧵{0} with 𝑟∈𝐍, and its deterministic compensated measure 𝜙(𝑑𝑣)𝑑𝑡=𝜆𝑓(𝑣)𝑑𝑣𝑑𝑡, that is, 𝐄(𝑝_𝜙(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 (Ω^𝐽,ℱ^𝐽,(ℱ^𝐽𝑡)_𝑡≥0,𝐏^𝐽). The Wiener process and the Poisson random measure are mutually independent. The process 𝑥(𝑡) is thus defined on a product space (Ω,ℱ,(ℱ_𝑡)_𝑡≥0,𝐏), where Ω=Ω^𝑊×Ω^𝐽,ℱ=ℱ^𝑊×ℱ^𝐽,(ℱ_𝑡)_𝑡≥0=(ℱ^𝑊𝑡)_𝑡≥0×(ℱ^𝐽𝑡)_𝑡≥0,𝐏=𝐏^𝑊×𝐏^𝐽 and ℱ₀ contains all 𝐏-null sets.

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

Assumption 2.1. For each integer 𝑘≥1, there exists a positive constant 𝐶_𝑘, dependent on 𝑘, such that ||𝑎(𝑥)−𝑎(𝑦)||²∨||𝑏(𝑥)−𝑏(𝑦)||²≤𝐶_𝑘||𝑥−𝑦||²,(2.2) for 𝑥,𝑦∈𝐑^𝑑 with |𝑥|∨|𝑦|≤𝑘,𝑘∈𝐍. And there exists a positive constant 𝐶 such that _𝜀||𝑐(𝑥,𝑣)−𝑐(𝑦,𝑣)||²𝜙(𝑑𝑣)≤𝐶||𝑥−𝑦||²,(2.3) for 𝑥,𝑦∈𝐑^𝑑.

Assumption 2.2. There exists a positive constant 𝐿 such that 2⟨𝑥,𝑎(𝑥)⟩+||𝑏(𝑥)||²+_𝜀||𝑐(𝑥,𝑣)||²𝜙(𝑑𝑣)≤𝐿1+|𝑥|²,(2.4) for 𝑥∈𝐑^𝑑.

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)||²≤𝐶_𝑘1+|𝑥|²,(2.6) for 𝑥∈𝐑^𝑑 with |𝑥|≤𝑘 and 𝐶_𝑘>0, and _𝜀||𝑐(𝑥,𝑣)||²𝜙(𝑑𝑣)≤𝐶1+|𝑥|²,(2.7) for 𝑥∈𝐑^𝑑 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 𝐏𝜏_𝑘≤𝑇≤𝜖,∀𝑘≥𝑘^∗,(2.8) where 𝜏_𝑘=inf{𝑡≥0∶|𝑥(𝑡)|≥𝑘} for 𝑘≥1.

Proof. Using Itô's formula (see [1]) to |𝑥(𝑡)|², for 𝑡≥0, we have ||𝑥(𝑡)||²=||𝑥₀||²+^𝑡0+⟨2𝑥(𝑠−),𝑎(𝑥(𝑠−))⟩+||𝑏(𝑥(𝑠−))||²𝑑s_𝑡0_𝜀||𝑥(𝑠−)+𝑐(𝑥(𝑠−),𝑣)||²+−||𝑥(𝑠−)||²−⟨2𝑥(𝑠−),𝑐(𝑥(𝑠−),𝑣)⟩𝜙(𝑑𝑣)𝑑𝑠_𝑡0⟨2𝑥(𝑠−),𝑏(𝑥(𝑠−))⟩𝑑𝑊(𝑠)+^𝑡0_𝜀||𝑥(𝑠−)+𝑐(𝑥(𝑠−),𝑣)||²−||𝑥(𝑠−)||²𝑝^𝜙̃(𝑑𝑣×𝑑𝑠),(2.9) which gives 𝐄||𝑥𝑡∧𝜏_𝑘||²=𝐄+||𝑥𝐄⁰||²_{𝑡∧𝜏𝑘0}+𝐄^{𝑡∧𝜏𝑘}₀⟨2𝑥(𝑠−),𝑎(𝑥(𝑠−))⟩+||𝑏(𝑥(𝑠−))||²𝑑𝑠_𝜀=𝐄||𝑥₀||𝑐(𝑥(𝑠−),𝑣)||²𝜙(𝑑𝑣)𝑑𝑠||²+𝐄^{𝑡∧𝜏𝑘}₀⟨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||𝑥𝑡∧𝜏_𝑘||²≤≤𝐄𝐄||||𝑥𝑥⁰₀||²+𝐄^{𝑡∧𝜏𝑘}₀𝐿1+||𝑥(𝑠−)||²𝑑𝑠||²+𝐿𝑇+𝐿^𝑡0𝐄||𝑥𝑠∧𝜏_𝑘−||²𝑑𝑠,(2.11) for 𝑡∈[0,𝑇]. Consequently by using the Gronwall inequality (see [13]), we obtain 𝐄||𝑥𝑡∧𝜏_𝑘||²≤𝐄||𝑥₀||²+𝐿𝑇𝑒^𝐿𝑇,(2.12) for 𝑡∈[0,𝑇]. We therefore get 𝐄||𝑥⁰||²+𝐿𝑇𝑒^𝐿𝑇≥𝐄||𝑥𝑇∧𝜏_𝑘||²≥𝐄||𝑥𝜏_𝑘||²𝐼_{{𝜏𝑘≤𝑇}}≥𝑘²𝐏𝜏_𝑘≤𝑇,(2.13) which means 𝐏𝜏_𝑘≤𝑇≤𝑒^𝐿𝑇𝑘²𝐄||𝑥⁰||²+𝐿𝑇.(2.14) So for any 𝜖∈(0,1), we can choose 𝑘^∗=⎡⎢⎢⎣𝑒^𝐿𝑇𝐄||𝑥₀||²+𝐿𝑇𝑒^𝐿𝑇𝜖⎤⎥⎥⎦+1,(2.15) such that 𝐏𝜏_𝑘≤𝑇≤𝜖,∀𝑘≥𝑘^∗.(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 𝑋_𝑛≈𝑥(𝑡_𝑛), where 𝑡_𝑛=𝑛Δ𝑡,𝑛=0,1,…, by setting 𝑋₀=𝑥₀ and forming 𝑋_𝑛+1=𝑋_𝑛+𝑎𝑋_𝑛Δ𝑡+𝑏𝑋_𝑛Δ𝑊_𝑛+^𝑡𝑛+1_𝑡𝑛_𝜀𝑐𝑋_𝑛,𝑣𝑝_𝜙̃(𝑑𝑣×𝑑𝑡),(3.1) where Δ𝑊_𝑛=𝑊(𝑡_𝑛+1)−𝑊(𝑡_𝑛).

The continuous-time Euler method is then defined by 𝑋(𝑡)∶=𝑋₀+^𝑡0𝑎(𝑍(𝑠))𝑑𝑠+^𝑡0𝑏(𝑍(𝑠))𝑑𝑊(𝑠)+^𝑡0_𝜀𝑐(𝑍(𝑠),𝑣)𝑝_𝜙̃(𝑑𝑣×𝑑𝑠),(3.2) where 𝑍(𝑡)=𝑋_𝑛 for 𝑡∈[𝑡_𝑛,𝑡_𝑛+1),𝑛=0,1,….

Actually, we can see in [8], 𝑝_𝜙={𝑝_𝜙(𝑡)∶=𝑝_𝜙(𝜀×[0,𝑡])} is a stochastic process that counts the number of jumps until some given time. The Poisson random measure 𝑝_𝜙(𝑑𝑣×𝑑𝑡) generates a sequence of pairs {(𝜄_𝑖,𝜉_𝑖),𝑖∈{1,2,…,𝑝_𝜙(𝑇)}} for a given finite positive constant 𝑇 if 𝜆<∞. Here {𝜄_𝑖∶Ω→𝐑₊,𝑖∈{1,2,…,𝑝_𝜙(𝑇)}} is a sequence of increasing nonnegative random variables representing the jump times of a standard Poisson process with intensity 𝜆, and {𝜉_𝑖∶Ω→𝜀,𝑖∈{1,2,…,𝑝_𝜙(𝑇)} is a sequence of independent identically distributed random variables, where 𝜉_𝑖 is distributed according to 𝜙(𝑑𝑣)/𝜙(𝜀). Then (3.1) can equivalently be the following form: 𝑋_𝑛+1=𝑋_𝑛+𝑎𝑋_𝑛−_𝜀𝑐𝑋_𝑛,𝑣𝜙(𝑑𝑣)Δ𝑡+𝑏𝑋_𝑛Δ𝑊_𝑛+^{𝑝𝜙(𝑡𝑛+1)}_𝑖=𝑝_{𝜙(𝑡𝑛)+1}𝑐𝑋_𝑛,𝜉_𝑖.(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)||²≤𝐾₁(𝑘)Δ𝑡,(3.4) for 0≤𝑡≤𝑇∧𝜏_𝑘∧𝜌_𝑘, where 𝜌_𝑘=inf{𝑡≥0∶|𝑋(𝑡)|≥𝑘} for 𝑘≥1 and 𝜏_𝑘 is defined in Lemma 2.4.

Proof. For 0≤𝑡≤𝑇∧𝜏_𝑘∧𝜌_𝑘, there is an integer 𝑛 such that 𝑡∈[𝑡_𝑛,𝑡_𝑛+1). So it follows from (3.2) that 𝑋(𝑡)−𝑍(𝑡)=𝑋_𝑛+^𝑡𝑡_𝑛𝑎(𝑍(𝑠))𝑑𝑠+^𝑡𝑡_𝑛𝑏(𝑍(𝑠))𝑑𝑊(𝑠)+^𝑡𝑡_𝑛_𝜀𝑐(𝑍(𝑠),𝑣)𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)−𝑋_𝑛.(3.5) Thus, by taking expectations and using the Cauchy-Schwarz inequality and the martingale properties of 𝑑𝑊(𝑡) and 𝑝_𝜙̃(𝑑𝑣×𝑑𝑡), we have 𝐄||𝑋(𝑡)−𝑍(𝑡)||²≤3𝐄||||^𝑡𝑡_𝑛𝑎(𝑍(𝑠))𝑑𝑠||||²+3𝐄||||^𝑡𝑡_𝑛𝑏(𝑍(𝑠))𝑑𝑊(𝑠)||||²+3𝐄||||^𝑡𝑡_𝑛_𝜀𝑐(𝑍(𝑠),𝑣)𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)||||²≤3+𝐄3𝐄^𝑡𝑡_{𝑛𝑡𝑡𝑛}1²𝑑𝑠^𝑡𝑡_𝑛||𝑎(𝑍(𝑠))||²𝑑𝑠+3𝐄^𝑡𝑡_𝑛||𝑏(𝑍(𝑠))||²𝑑𝑠_𝜀≤3Δ𝑡𝐄_𝑡𝑡𝑛||𝑐(𝑍(𝑠),𝑣)||²𝜙(𝑑𝑣)𝑑𝑠||𝑎(𝑍(𝑠))||²𝑑𝑠+3𝐄^𝑡𝑡_𝑛||𝑏(𝑍(𝑠))||²𝑑𝑠+3𝐄^𝑡𝑡_𝑛_𝜀||𝑐(𝑍(𝑠),𝑣)||²𝜙(𝑑𝑣)𝑑𝑠,(3.6) where the inequality |𝑢₁+𝑢₂+𝑢₃|²≤3|𝑢₁|²+3|𝑢₂|²+3|𝑢₃|² for 𝑢₁,𝑢₂,𝑢₃∈𝐑^𝑑 is used. Therefore, by applying Assumptions 2.1 and 2.3, we get 𝐄^𝑡𝑡_𝑛||𝑎(𝑍(𝑠))||𝐄²𝑑𝑠≤𝐶_𝑘𝐄^𝑡𝑡_𝑛1+||𝑍(𝑠)||²𝑑𝑠≤𝐶^𝑘Δ𝑡+𝐶_𝑘𝑘²Δ𝑡,_𝑡𝑡𝑛𝐄||𝑏(𝑍(𝑠))||²𝑑𝑠≤𝐶_𝑘Δ𝑡+𝐶_𝑘𝑘²Δ𝑡,_𝑡𝑡𝑛_𝜀||𝑐(𝑍(𝑠),𝑣)||²𝜙(𝑑𝑣)𝑑𝑠≤𝐶Δ𝑡+𝐶𝑘²Δ𝑡,(3.7) which lead to 𝐄||𝑋(𝑡)−𝑍(𝑡)||²≤Δ𝑡3𝐶_𝑘Δ𝑡+3𝑘²𝐶_𝑘Δ𝑡+3𝐶_𝑘+3𝑘²𝐶_𝑘+3𝐶+3𝑘²𝐶,(3.8) for 𝑡∈[0,𝑇∧𝜏_𝑘∧𝜌_𝑘]. Therefore, we obtain the result (3.4) by choosing 𝐾₁(𝑘)=6𝐶_𝑘+6𝑘²𝐶_𝑘+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 Δ𝑡^∗1 such that 𝐏𝜌_𝑘^∗≤𝑇≤𝜖,∀Δ𝑡≤Δ𝑡^∗1,(3.10) where 𝜌_𝑘^∗ is defined in Lemma 3.1.

Proof . Applying generalized Itô's formula (see [1]) to |𝑋(𝑡)|², for 𝑡≥0, yields ||𝑋(𝑡)||²=||𝑋₀||²+^𝑡0+2𝑋(𝑠),𝑎(𝑍(𝑠))+||𝑏(𝑍(𝑠))||²𝑑𝑠_𝑡0_𝜀||+𝑋(𝑠)+𝑐(𝑍(𝑠),𝑣)||²−||𝑋(𝑠)||²−2𝑋(𝑠),𝑐(𝑍(𝑠),𝑣)𝜙(𝑑𝑣)𝑑𝑠_𝑡02𝑋(𝑠),𝑏(𝑍(𝑠))𝑑𝑊(𝑠)+^𝑡0_𝜀||𝑋(𝑠)+𝑐(𝑍(𝑠),𝑣)||²−||𝑋(𝑠)||²𝑝^𝜙̃(𝑑𝑣×𝑑𝑠).(3.11) By taking expectations, we thus have 𝐄||𝑋𝑡∧𝜌_𝑘||²=𝐄||𝑋₀||²+𝐄^{𝑡∧𝜌𝑘}₀2=𝐄||𝑋₀𝑋(𝑠),𝑎(𝑍(𝑠))+||𝑏(𝑍(𝑠))||²+_𝜀||𝑐(𝑍(𝑠),𝑣)||²𝜙(𝑑𝑣)𝑑𝑠||²+E^{𝑡∧𝜌𝑘}₀2𝑋(𝑠),𝑎𝑋(𝑠)+|||𝑏𝑋(𝑠)|||²+_𝜀|||𝑐+𝐄_{𝑡∧𝜌𝑘0}𝑋(𝑠),𝑣|||²𝜙(𝑑𝑣)𝑑𝑠+𝐄_{𝑡∧𝜌𝑘0}2𝑋(𝑠),𝑎(𝑍(𝑠))−𝑎𝑋(𝑠)𝑑𝑠||𝑏(𝑍(𝑠))||²−|||𝑏𝑋(𝑠)|||²+𝐄_{𝑡∧𝜌𝑘0}𝑑𝑠_𝜀||𝑐(𝑍(𝑠),𝑣)||²−|||𝑐𝑋(𝑠),𝑣|||²𝜙(𝑑𝑣)𝑑𝑠.(3.12) For 𝑡∈[0,𝑇]. Now, by using the inequalities ⟨𝑢₁,𝑢₂⟩≤|𝑢₁||𝑢₂| for 𝑢₁,𝑢₂∈𝐑^𝑑, (2.2) in Assumption 2.1, Fubini's theorem, Cauchy-Schwarz’s inequality, and Lemma 3.1, we get 𝐄^{𝑡∧𝜌𝑘}₀2𝑋(𝑠),𝑎(𝑍(𝑠))−𝑎𝑋(𝑠)𝑑𝑠≤≤22𝐄𝑘√^{𝑡∧𝜌𝑘}₀||𝑋(𝑠)|||||𝑎(𝑍(𝑠))−𝑎𝑋(𝑠)|||𝑑𝑠𝐶_𝑘^𝑡0≤2𝑘√𝐄||𝑍𝑠∧𝜌_𝑘−𝑋𝑠∧𝜌_𝑘||𝑑𝑠𝐶_𝑘^𝑡0𝐄||𝑍𝑠∧𝜌^𝑘−≤2𝑘𝑇√𝑋𝑠∧𝜌_𝑘||²^1/2𝑑𝑠𝐶_𝑘𝐾₁(𝑘)Δ𝑡.(3.13) And, similarly as above, we have 𝐄^{𝑡∧𝜌𝑘}₀||𝑏(𝑍(𝑠))||²−|||𝑏𝑋(𝑠)|||²𝑑𝑠≤𝐄^{𝑡∧𝜌𝑘}₀||𝑏(𝑍(𝑠))||+|||𝑏×||𝑏(𝑍(𝑠))||−|||𝑏𝑋(𝑠)|||≤2𝑋(𝑠)|||𝑑𝑠≤2𝐶_𝑘1+𝑘²𝐄^{𝑡∧𝜌𝑘}₀|||𝑏(𝑍(𝑠))−𝑏𝑋(𝑠)|||𝑑𝑠≤2𝑇𝐶_𝑘𝐶_𝑘1+𝑘²^𝑡0𝐄||𝑍𝑠∧𝜌_𝑘−𝑋𝑠∧𝜌_𝑘||𝑑𝑠𝐶_𝑘𝐶_𝑘𝐾₁(𝑘)1+𝑘²Δ𝑡.(3.14) Moreover, in the same way, we obtain 𝐄^{𝑡∧𝜌𝑘}₀_𝜀||𝑐(𝑍(𝑠),𝑣)||²−|||𝑐𝑋(𝑠),𝑣|||²=𝐄_{𝑡∧𝜌𝑘0}𝜙(𝑑𝑣)𝑑𝑠_𝜀|||𝑐(𝑍(𝑠),𝑣)−𝑐𝑋(𝑠),𝑣+𝑐𝑋(𝑠),𝑣|||²−|||𝑐𝑋(𝑠),𝑣|||²≤𝐄_{𝑡∧𝜌𝑘0}𝜙(𝑑𝑣)𝑑𝑠_𝜀2|||𝑐(𝑍(𝑠),𝑣)−𝑐𝑋(𝑠),𝑣|||²+|||𝑐𝑋(𝑠),𝑣|||²≤2𝐶𝐄_{𝑡∧𝜌𝑘0}𝜙(𝑑𝑣)𝑑𝑠||𝑍(𝑠)−𝑋(𝑠)||²𝑑𝑠+𝐶𝐄^{𝑡∧𝜌𝑘}₀≤2𝐶_𝑡01+||𝑋(𝑠)||²𝑑𝑠≤2𝐶𝑇𝐾𝐄₁||𝑍𝑠∧𝜌_𝑘−𝑋𝑠∧𝜌_𝑘||²𝑑𝑠+𝐶𝐄^{𝑡∧𝜌𝑘}₀1+||𝑋(𝑠)||²𝑑𝑠(𝑘)Δ𝑡+𝐶𝑇+𝐶𝐄^{𝑡∧𝜌𝑘}₀||𝑋(𝑠)||²𝑑𝑠,(3.15) where the inequality |𝑢₁+𝑢₂|²≤2|𝑢₁|²+2|𝑢₂|² for 𝑢₁,𝑢₂∈𝐑^𝑑, (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||𝑋𝑡∧𝜌_𝑘||²≤𝐄+||𝑋2𝑘₀||𝑇²√+𝐿𝐄^{𝑡∧𝜌𝑘}₀1+||𝑋(𝑠)||²𝑑𝑠+𝐶𝐄^{𝑡∧𝜌𝑘}₀||𝑋(𝑠)||²𝑑𝑠≤𝐿+𝐶𝐶_𝑘𝑡0𝐾₁(𝑘)Δ𝑡+2𝑇𝐶_𝑘𝐶_𝑘𝐾₁(𝑘)1+𝑘²Δ𝑡+2𝐶𝑇𝐾₁(𝑘)Δ𝑡+𝐶𝑇+2𝑘𝑇√𝐄||𝑋𝑠∧𝜌_𝑘||²𝑑𝑠+𝐄||𝑋₀||²+𝐿𝑇+𝐶𝑇𝐶_𝑘𝐾₁(𝑘)+2𝑇𝐶_𝑘𝐶_𝑘𝐾₁(𝑘)1+𝑘²√Δ𝑡+2𝐶𝑇𝐾₁(𝑘)Δ𝑡,(3.16) for 0≤𝑡≤𝑇. Therefore, by the Gronwall inequality (see [13]), for 0≤𝑡≤𝑇, we get 𝐄||𝑋𝑡∧𝜌_𝑘||²≤𝛼₁𝛼₄+𝛼₄𝛼₂(𝑘)√Δ𝑡+𝛼₄𝛼₃(𝑘)Δ𝑡,(3.17) where 𝛼𝛼₁=𝐄||𝑋₀||²+𝐿𝑇+𝐶𝑇,₂(𝑘)=2𝑘𝑇√𝐶𝛼_𝑘𝐾₁₃(𝑘)+2𝑇𝐶_𝑘𝐶_𝑘𝐾₁(𝑘)1+𝑘²,𝛼(𝑘)=2𝐶𝑇𝐾₁(𝑘),₄=exp𝐿𝑇+𝐶𝑇.(3.18) We thus obtain that 𝑘²𝐏𝜌_𝑘≤𝑇≤𝐄||𝑋𝜌_𝑘||²𝐼_{{𝜌𝑘≤𝑇}}≤𝐄||𝑋𝑇∧𝜌_𝑘||²≤𝛼₁𝛼₄+𝛼₄𝛼₂(𝑘)√Δ𝑡+𝛼₄𝛼₃(𝑘)Δ𝑡.(3.19) So for any 𝜖∈(0,1), we can choose sufficiently large integer 𝑘=𝑘^∗ such that 𝛼₁𝛼₄𝑘^∗2≤𝜖2,(3.20) and choose sufficiently small Δ𝑡^∗1∈(0,1) such that 𝛼₄𝛼₂𝑘^∗√Δ𝑡^∗1+𝛼₄𝛼₃𝑘^∗Δ𝑡^∗1𝑘^∗2≤𝜖2.(3.21) Hence, we have 𝐏𝜌_𝑘^∗≤𝑇≤𝜖,∀Δ𝑡≤Δ𝑡^∗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(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 𝐄sup_{0≤𝑡≤𝑇}||𝑥𝑡∧𝜏_𝑘∧𝜌_𝑘−𝑋𝑡∧𝜏_𝑘∧𝜌_𝑘||²≤𝐾²(𝑘)Δ𝑡,(4.1) where 𝜏_𝑘 and 𝜌_𝑘 are defined in Lemmas 2.4 and 3.1, respectively.

Proof. From (2.1) and (3.2), for any 0≤𝑡′≤𝑇, we have 𝐄sup_{0≤𝑡≤𝑡′}||𝑥𝑡∧𝜏_𝑘∧𝜌_𝑘−𝑋𝑡∧𝜏_𝑘∧𝜌_𝑘||²≤3𝐄sup_{0≤𝑡≤𝑡′}||||^𝑡∧𝜏_𝑘^∧𝜌_𝑘0(𝑎(𝑥(𝑠−))−𝑎(𝑍(𝑠)))𝑑𝑠||||²+3𝐄sup_{0≤𝑡≤𝑡′}||||^𝑡∧𝜏_𝑘^∧𝜌_𝑘0(𝑏(𝑥(𝑠−))−𝑏(𝑍(𝑠)))𝑑𝑊(𝑠)||||²+3𝐄sup_{0≤𝑡≤𝑡′}||||^𝑡∧𝜏_𝑘^∧𝜌_𝑘0_𝜀(𝑐(𝑥(𝑠−),𝑣)−𝑐(𝑍(𝑠),𝑣))𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)||||²,(4.2) where the inequality |𝑢₁+𝑢₂+𝑢₃|²≤3|𝑢₁|²+3|𝑢₂|²+3|𝑢₃|² for 𝑢₁,𝑢₂,𝑢₃∈𝐑^𝑑 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sup_{0≤𝑡≤𝑡′}||||^𝑡∧𝜏_𝑘^∧𝜌_𝑘0(𝑎(𝑥(𝑠−))−𝑎(𝑍(𝑠)))𝑑𝑠||||²≤𝐄sup_{0≤𝑡≤𝑡′}^{𝑡∧𝜏𝑘∧𝜌𝑘}₀≤𝑇𝐄_{𝑡′∧𝜏𝑘∧𝜌𝑘0}1²𝑑𝑠^{𝑡∧𝜏𝑘∧𝜌𝑘}₀||𝑎(𝑥(𝑠−))−𝑎(𝑍(𝑠))||²𝑑𝑠≤2𝑇𝐶_𝑘||𝑎(𝑥(𝑠−))−𝑎(𝑍(𝑠))||²𝑑𝑠≤2𝑇𝐶_𝑘𝐄^{𝑡′∧𝜏𝑘∧𝜌𝑘}₀||𝑋(𝑠)−𝑍(𝑠)||²𝑑𝑠+2𝑇𝐶_𝑘𝐄^{𝑡′∧𝜏𝑘∧𝜌𝑘}₀||𝑥(𝑠−)−𝑋(𝑠)||²𝑑𝑠^𝑡′₀+2𝑇𝐶_𝑘𝐄||𝑋𝑠∧𝜏_𝑘∧𝜌_𝑘−𝑍𝑠∧𝜏_𝑘∧𝜌_𝑘||²𝑑𝑠^𝑡′₀≤2𝑇₂𝐄||𝑥𝑠∧𝜏_𝑘∧𝜌_𝑘−−𝑋𝑠∧𝜏_𝑘∧𝜌_𝑘||²𝑑𝑠𝐶_𝑘𝐾₁(𝑘)Δ𝑡+2𝑇𝐶_𝑘^𝑡′₀𝐄sup_{0≤𝑢≤𝑠}||𝑥𝑢∧𝜏_𝑘∧𝜌_𝑘−−𝑋𝑢∧𝜏_𝑘∧𝜌_𝑘||²𝑑𝑠.(4.3) Moreover, by using the martingale properties of 𝑑𝑊(𝑡) and 𝑝_𝜙̃(𝑑𝑣×𝑑𝑡), Assumption 2.1, Lemma 3.1, and Fubini's theorem, we have 𝐄sup_{0≤𝑡≤𝑡′}||||^𝑡∧𝜏_𝑘^∧𝜌_𝑘0(𝑏(𝑥(𝑠−))−𝑏(𝑍(𝑠)))𝑑𝑊(𝑠)||||²≤4𝐄_{𝑡′∧𝜏𝑘∧𝜌𝑘0}≤8𝐶_𝑘||𝑏(𝑥(𝑠−))−𝑏(𝑍(𝑠))||²𝑑𝑠≤8𝐶_𝑘𝐄^{𝑡′∧𝜏𝑘∧𝜌𝑘}₀||𝑋(𝑠)−𝑍(𝑠)||²𝑑𝑠+8𝐶_𝑘𝐄^{𝑡′∧𝜏𝑘∧𝜌𝑘}₀||𝑥(𝑠−)−𝑋(𝑠)||²𝑑𝑠^𝑡′₀+8𝐶_𝑘𝐄||𝑋𝑠∧𝜏_𝑘∧𝜌_𝑘−𝑍𝑠∧𝜏_𝑘∧𝜌_𝑘||²𝑑𝑠^𝑡′₀≤8𝑇𝐶_𝑘𝐄||𝑥𝑠∧𝜏_𝑘∧𝜌_𝑘−−𝑋𝑠∧𝜏_𝑘∧𝜌_𝑘||²𝑑𝑠𝐾₁(𝑘)Δ𝑡+8𝐶_𝑘^𝑡′₀𝐄sup_{0≤𝑢≤𝑠}||𝑥𝑢∧𝜏_𝑘∧𝜌_𝑘−−𝑋𝑢∧𝜏_𝑘∧𝜌_𝑘||²𝐄sup_{0≤𝑡≤𝑡′}𝑑𝑠,||||^𝑡∧𝜏_𝑘^∧𝜌_𝑘0_𝜀(𝑐(𝑥(𝑠−),𝑣)−𝑐(𝑍(𝑠−),𝑣))𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)||||²≤4𝐄||||_{𝑡′∧𝜏𝑘∧𝜌𝑘0}_𝜀(𝑐(𝑥(𝑠−),𝑣)−𝑐(𝑍(𝑠−),𝑣))𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)||||²=4𝐄^{𝑡′∧𝜏𝑘∧𝜌𝑘}₀_𝜀≤8𝑇𝐶𝐾₁||𝑐(𝑥(𝑠−),𝑣)−𝑐(𝑍(𝑠−),𝑣)||²𝜙(𝑑𝑣)𝑑𝑠(𝑘)Δ𝑡+8𝐶^𝑡′₀𝐄sup_{0≤𝑢≤𝑠}||𝑥𝑢∧𝜏_𝑘∧𝜌_𝑘−−𝑋𝑢∧𝜏_𝑘∧𝜌_𝑘||²𝑑𝑠.(4.4) Hence, by substituting (4.3) and (4.4) into (4.2), we get 𝐄sup_{0≤𝑡≤𝑡′}||𝑥𝑡∧𝜏_𝑘∧𝜌_𝑘−𝑋𝑡∧𝜏_𝑘∧𝜌_𝑘||²≤Δ𝑡6𝑇₂×𝐶_𝑘𝐾₁(𝑘)+24𝑇𝐶_𝑘𝐾₁(𝑘)+24𝑇𝐶𝐾₁(𝑘)+6𝑇𝐶_𝑘+24𝐶_𝑘+24𝐶_𝑡′0𝐄sup_{0≤𝑢≤𝑠}||𝑥𝑢∧𝜏_𝑘∧𝜌_𝑘−−𝑋𝑢∧𝜏_𝑘∧𝜌_𝑘||²𝑑𝑠.(4.5) So using the Gronwall inequality (see [13]), we have the result (4.1) by choosing 𝐾₂(k)=6𝑇²𝐶_𝑘𝐾₁(𝑘)+24𝑇𝐶_𝑘𝐾₁(𝑘)+24𝑇𝐶𝐾₁(𝑘)exp6𝑇²𝐶_𝑘+24𝑇𝐶_𝑘+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<Δ𝑡^∗𝐏sup_{0≤𝑡≤𝑇}||𝑥(𝑡)−𝑋(𝑡)||²≥𝜍≤𝜖,(4.7) for any 𝑇>0.

Proof. For sufficiently small 𝜖,𝜍∈(0,1), we define Ω=𝜔∶sup_{0≤𝑡≤𝑇}||𝑥(𝑡)−𝑋(𝑡)||²≥𝜍.(4.8) According to Lemmas 2.4 and 3.2, there exists a pair of 𝑘^⋆ and Δ𝑡^⋆1 such that 𝐏𝜏_𝑘^⋆≤𝑇≤𝜖𝐏𝜌_𝑘⋆3,≤𝑇≤𝜖3,∀Δ𝑡≤Δ𝑡^⋆1.(4.9) We thus have 𝐏Ω≤≤𝐏𝐏Ω∩𝜏_𝑘^⋆∧𝜌_𝑘^⋆>𝑇+𝐏𝜏_𝑘^⋆∧𝜌_𝑘^⋆≤𝑇≤𝐏Ω∩𝜏_𝑘^⋆∧𝜌_𝑘^⋆>𝑇+𝐏𝜏_𝑘^⋆≤𝑇+𝐏𝜌_𝑘^⋆≤𝑇Ω∩𝜏_𝑘^⋆∧𝜌_𝑘^⋆>𝑇+2𝜖3,(4.10) for Δ𝑡≤Δ𝑡^⋆1. Moreover, according to Lemma 4.1, we have 𝜍𝐏Ω∩𝜏_𝑘^⋆∧𝜌_𝑘^⋆>𝑇≤𝐄𝐼_{{𝜏𝑘⋆∧𝜌𝑘⋆>𝑇}}sup_{0≤𝑡≤𝑇}||𝑥(𝑡)−𝑋(𝑡)||²≤𝐄sup_{0≤𝑡≤𝑇}||𝑥𝑡∧𝜏_𝑘⋆∧𝜌_𝑘^⋆−𝑋𝑡∧𝜏_𝑘^⋆∧𝜌_𝑘^⋆||²≤𝐾₂𝑘^⋆Δ𝑡,(4.11) which leads to 𝐏Ω∩𝜏_𝑘^⋆∧𝜌_𝑘^⋆>𝑇≤𝜖3,(4.12) for Δ𝑡≤Δ𝑡^⋆2. Therefore, from the inequalities above, we obtain 𝐏Ω≤𝜖,(4.13) for Δ𝑡≤Δ𝑡^⋆, where Δ𝑡^⋆=min{Δ𝑡^⋆1,Δ𝑡^⋆2}.

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<Δ𝑡^∗𝐏||𝑥(𝑡)−𝑍(𝑡)||²≥𝜍,0≤𝑡≤𝑇≤𝜖,(4.14) for any 𝑇>0.

Proof. For sufficiently small 𝜖,𝜍∈(0,1), we define Ω=𝜔∶||𝑥(𝑡)−𝑍(𝑡)||²≥𝜍,0≤𝑡≤𝑇.(4.15) A similar analysis as Theorem 4.2 gives 𝐏Ω≤𝐏Ω∩𝜏_𝑘^⋆∧𝜌_𝑘^⋆>𝑇+2𝜖3.(4.16) Recalling that 𝜍𝐏Ω∩𝜏_𝑘^⋆∧𝜌_𝑘^⋆>𝑇≤≤𝐄𝐄||𝑥||𝑥𝑇(𝑇∧)𝜏−_𝑘⋆𝑍(𝑇)||²𝐼_{{𝜏𝑘⋆∧𝜌𝑘⋆>𝑇}}∧𝜌_𝑘^⋆−𝑍𝑇∧𝜏_𝑘^⋆∧𝜌_𝑘^⋆||²≤2𝐄sup_{0≤𝑡≤𝑇}||𝑥𝑡∧𝜏_𝑘⋆∧𝜌_𝑘^⋆−𝑋𝑡∧𝜏_𝑘^⋆∧𝜌_𝑘^⋆||²+2𝐄||𝑋𝑇∧𝜏_𝑘^⋆∧𝜌_𝑘^⋆−𝑍𝑇∧𝜏_𝑘^⋆∧𝜌_𝑘^⋆||²≤2𝐾₁𝑘^⋆Δ𝑡+2𝐾₂𝑘^⋆Δ𝑡,(4.17) and using Lemmas 3.1 and 4.1, we get that 𝐏Ω∩𝜏_𝑘^⋆∧𝜌_𝑘^⋆>𝑇≤𝜖3,(4.18) for sufficiently small Δ𝑡. Consequently, the inequalities above show that 𝐏Ω≤𝜖,(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)𝑝_𝜙̃(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 𝑎(𝑥)=12𝑥−𝑥³,𝑏(𝑥)=𝑥²,𝑐(𝑥,𝑣)=𝑥𝑣.(5.2) The compensated measure of the Poisson random measure 𝑝_𝜙(𝑑𝑣×𝑑𝑡) is given by 𝜙(𝑑𝑣)𝑑𝑡=𝜆𝑓(𝑣)𝑑𝑣𝑑𝑡, where 𝜆=5 and 𝑓(𝑣)=1√2𝜋𝑣exp−(ln2𝑣)²,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⟨𝑥,𝑎(𝑥)⟩+||𝑏(𝑥)||²+_𝜀||𝑐(𝑥,𝑣)||²𝜙(𝑑𝑣)=𝑥𝑥−𝑥³+𝑥⁴+_𝜀𝑥²𝑣²𝜆1≤1+5𝑒₂√2𝜋𝑣exp−(ln2𝑣)²𝑑𝑣1+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 𝑋_𝑛+1=𝑋_𝑛+12𝑋^𝑛−𝑋^3𝑛Δ𝑡+𝑋^2𝑛Δ𝑊_𝑛+𝑋_𝑛^𝑡𝑛+1_𝑡𝑛_𝜀𝑣𝑝_𝜙̃(𝑑𝑣×𝑑𝑡),(5.5) with 𝑋₀=0.

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

Algorithm  Simulate 𝑋^−𝑛+1∶=𝑋_𝑛+(1/2)(𝑋_𝑛−𝑋^3𝑛−10√𝑒𝑋_𝑛)Δ𝑡+𝑋^2𝑛Δ𝑊_𝑛; Simulate variable 𝑝_𝜙(𝑡_𝑛+1)−𝑝_𝜙(𝑡_𝑛), where 𝑝_𝜙(𝑡_𝑛) is from Poisson distribution with parameter 𝜆𝑡_𝑛; Simulate 𝑝_𝜙(𝑡_𝑛+1)−𝑝_𝜙(𝑡_𝑛) independent random variables 𝜄_𝑖 uniformly distributed on the interval [𝑝_𝜙(𝑡_𝑛),𝑝_𝜙(𝑡_𝑛+1)); Simulate 𝑝_𝜙(𝑡_𝑛+1)−𝑝_𝜙(𝑡_𝑛) independent random variables 𝜉_𝑖 with law 𝑓(𝑣); obtain 𝑋_𝑛+1=𝑋^−𝑛+1+𝑋_𝑛∑^{𝑝𝜙(𝑡𝑛+1)𝑖=𝑝}_{𝜙(𝑡𝑛)+1}𝐼_{𝑡𝑛≤𝜄𝑖<𝑡𝑛+1}𝜉_𝑖.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).