Research Article | Open Access
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 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.
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 .
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 , a convergence result for strong approximations of any given order was presented. Moreover, N. Bruti-Liberatiand E. Platen (see ) obtain the jump-adapted order 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 ) 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 are given. Recently, Mordecki et al.  improved adaptive time stepping algorithms based on a jump augmented Monte Carlo Euler-Maruyama method, which achieve a prescribed precision. M. Wei  demonstrates the convergence of numerical solutions for variable delay differential equations driven by Poisson random measure. In , 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 , 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  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 and . Let and be the Euclidean norm and the inner product of vectors in . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by . Let denote the family of -valued -measurable random variables with .  denotes the largest integer which is less than or equal to in . denotes the indicator function of a set .
The following -dimensional SDE driven by Poisson random measure is considered in our paper: for with initial condition , where denotes and .
The drift coefficient , the diffusion coefficient , and the jump coefficient are assumed to be Borel measurable functions.
The randomness of (2.1) is generated by the following (see ). An -dimensional Wiener process with independent scalar components is defined on a filtered probability space . A Poisson random measure is on , where with , and its deterministic compensated measure , that is, . is a probability density, and we require finite intensity . The Poisson random measure is defined on a filtered probability space . The Wiener process and the Poisson random measure are mutually independent. The process is thus defined on a product space , where and contains all -null sets.
Now, the condition of non-Lipschitz coefficients is given by the following assumptions.
Assumption 2.1. For each integer , there exists a positive constant , dependent on , such that for with . And there exists a positive constant such that for .
Assumption 2.2. There exists a positive constant such that for .
Assumption 2.3. Consider
The following result shows that the solution of (2.1) keeps in a compact set with a large probability.
Proof. Using Itô's formula (see ) to , for , we have which gives for . By Assumption 2.2, we thus have for . Consequently by using the Gronwall inequality (see ), we obtain for . We therefore get which means So for any , we can choose such that 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 , where is a positive number.
Given a step size , the Euler method applied to (2.1) computes approximation , where , by setting and forming where .
The continuous-time Euler method is then defined by where for .
Actually, we can see in , is a stochastic process that counts the number of jumps until some given time. The Poisson random measure generates a sequence of pairs for a given finite positive constant if . Here is a sequence of increasing nonnegative random variables representing the jump times of a standard Poisson process with intensity , and is a sequence of independent identically distributed random variables, where is distributed according to . 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 .
Lemma 3.1. Under Assumptions 2.1 and 2.3, for any , there exists a positive constant , dependent on and independent of , such that for all the continuous-time Euler method (3.2) satisfies for , where for and is defined in Lemma 2.4.
Proof. For , there is an integer such that . So it follows from (3.2) that Thus, by taking expectations and using the Cauchy-Schwarz inequality and the martingale properties of and , we have where the inequality for is used. Therefore, by applying Assumptions 2.1 and 2.3, we get which lead to for . Therefore, we obtain the result (3.4) by choosing
Proof . Applying generalized Itô's formula (see ) to , for , yields By taking expectations, we thus have For . Now, by using the inequalities for , (2.2) in Assumption 2.1, Fubini's theorem, Cauchy-Schwarz’s inequality, and Lemma 3.1, we get And, similarly as above, we have Moreover, in the same way, we obtain where the inequality for , (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 for . Therefore, by the Gronwall inequality (see ), for , we get where We thus obtain that So for any , we can choose sufficiently large integer such that and choose sufficiently small such that Hence, we have
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 .
At the beginning, we give a lemma based on Lemma 3.1.
Lemma 4.1. Under Assumptions 2.1 and 2.3, for any , there exists a positive constant , dependent on and independent of , such that for all the solution of (2.1) and the continuous-time Euler method (3.2) satisfy where and are defined in Lemmas 2.4 and 3.1, respectively.
Proof. From (2.1) and (3.2), for any , we have where the inequality for is used. Therefore, by using the Cauchy-Schwarz inequality, (2.2) in Assumption 2.1, Lemma 3.1 and Fubini's theorem, we obtain Moreover, by using the martingale properties of and , Assumption 2.1, Lemma 3.1, and Fubini's theorem, we have Hence, by substituting (4.3) and (4.4) into (4.2), we get So using the Gronwall inequality (see ), 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).
Proof. For sufficiently small , we define According to Lemmas 2.4 and 3.2, there exists a pair of and such that We thus have for . Moreover, according to Lemma 4.1, we have which leads to for . Therefore, from the inequalities above, we obtain for , where .
We remark that the continuous-time Euler solution (3.2) cannot be computed, since it requires knowledge of the entire Brownian motion and Poisson random measure paths, not just only their -increments. Therefore, the last theorem shows the convergence in probability of the discrete Euler solution (3.1).
Proof. For sufficiently small , we define
A similar analysis as Theorem 4.2 gives
and using Lemmas 3.1 and 4.1, we get that
for sufficiently small . Consequently, the inequalities above show that
for all sufficiently small .
So we complete the result (4.14).
5. Numerical Example
Now, we consider the following equation: with , where . The coefficients of this equation have the form The compensated measure of the Poisson random measure is given by , where and 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 that is to say, Assumptions 2.1, 2.2, and 2.3 in Section 2 are satisfied. Therefore, Albeverio et al.  guarantee that (5.1) has a unique global solution on .
Given the stepsize , we can have the Euler method with .
And in Matlab experiment, each discretized trajectory is actually given in detail by the following.
Algorithm Simulate ; Simulate variable , where is from Poisson distribution with parameter ; Simulate independent random variables uniformly distributed on the interval ; Simulate independent random variables with law ; obtain .Subsequently, we can get the results in Theorems 4.2 and 4.3.
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.