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

The numerical methods in the current known literature require the stochastic di ﬀ erential equations (cid:3) SDEs (cid:4) 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 coe ﬃ cients 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 coe ﬃ cients. Numerical example is given to demonstrate our results.


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-Liberati and 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

The SDEs Driven by Poisson Random Measure with Non-Lipschitz Coefficients
Throughout this paper, unless specified, we use the following notations. Let u 1 ∨ u 2 max{u 1 , u 2 } and u 1 ∧ u 2 min{u 1 , u 2 }. 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 T A . Let L 2 F 0 Ω; R d denote the family of R d -valued F 0 -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: 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 J t 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 W t t≥0 × F J t t≥0 , P P W × P J and F 0 contains all P-null sets. Now, the condition of non-Lipschitz coefficients is given by the following assumptions.
Actually, Assumptions 2.1 and 2.3 imply the linear growth conditions for x ∈ R d with |x| ≤ k and C k > 0, and for x ∈ R d and C > 0.

Journal of Applied Mathematics
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 ∈ 0, 1 and T > 0, there exists a sufficiently large integer k * , dependent on and T , such that Proof. Using Itô's formula see 1 to |x t | 2 , for t ≥ 0, we have for t ∈ 0, T . By Assumption 2.2, we thus have

2.11
Journal of Applied Mathematics 5 for t ∈ 0, T . Consequently by using the Gronwall inequality see 13 , we obtain for t ∈ 0, T . We therefore get 14 So for any ∈ 0, 1 , we can choose Hence, we have the result 2.8 .

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 X 0 x 0 and forming The continuous-time Euler method is then defined by where Z t X n for t ∈ t n , t n 1 , n 0, 1, . . ..

Journal of Applied Mathematics
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 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: The following lemma shows the close relation between the continuous-time Euler method 3.2 and its step function Z t .
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

3.5
Thus, by taking expectations and using the Cauchy-Schwarz inequality and the martingale properties of dW t and p φ dv × dt , we have where the inequality |u 1 u 2 u 3 | 2 ≤ 3|u 1 | 2 3|u 2 | 2 3|u 3 | 2 for u 1 , u 2 , u 3 ∈ R d is used. Therefore, by applying Assumptions 2.1 and 2.3, we get for t ∈ 0, T ∧ τ k ∧ ρ k . Therefore, we obtain the result 3.4 by choosing 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. Assumptions 2.1, 2.2, and 2.3, for any pair of ∈ 0, 1 and T > 0, there exist a sufficiently large k * and a sufficiently small Δt * 1 such that

Lemma 3.2. Under
where ρ k * is defined in Lemma 3.1.

Journal of Applied Mathematics
Proof . Applying generalized Itô's formula see 1 to |X t | 2 , for t ≥ 0, yields

3.11
By taking expectations, we thus have

3.13
Journal of Applied Mathematics 9 And, similarly as above, we have 3.14 Moreover, in the same way, we obtain

3.15
where the inequality Journal of Applied Mathematics

3.16
for 0 ≤ t ≤ T . Therefore, by the Gronwall inequality see 13 , for 0 ≤ t ≤ T , we get

3.18
We thus obtain that

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.
where τ k and ρ k are defined in Lemmas 2.4 and 3.1, respectively.
Moreover, by using the martingale properties of dW t and p φ dv × dt , Assumption 2.1, Lemma 3.1, and Fubini's theorem, we have

4.5
So using the Gronwall inequality see 13 , we have the result 4.1 by choosing Now, let's state our theorem which demonstrates the convergence in probability of the continuous-time Euler method 3.2 .
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 . Under Assumptions 2.1, 2.2, and 2.3, for sufficiently small , ς ∈ 0, 1 , there is a Δt * such that for all Δt < Δt * P |x t − Z t | 2 ≥ ς, 0 ≤ t ≤ T ≤ , 4.14 for any T > 0.