Journal of Applied Mathematics

Volume 2012 (2012), Article ID 675781, 17 pages

doi:10.1155/2012/675781

Research Article

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

Hui Yu and Minghui Song

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 was presented. Moreover, N. Bruti-Liberatiand E. Platen (see [7]) 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 [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 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 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 [9]). 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 .

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

Assumption 2.3. Consider

Actually, Assumptions 2.1 and 2.3 imply the linear growth conditions

for with and , and

for and .

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 and , there exists a sufficiently large integer , dependent on and , such that

where for .

Proof. Using Itô's formula (see [1]) to , for , we have

which gives

for . By Assumption 2.2, we thus have

for . Consequently by using the Gronwall inequality (see [13]), 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 [8], 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

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 and , there exist a sufficiently large and a sufficiently small such that

where is defined in Lemma 3.1.

Proof . Applying generalized Itô's formula (see [1]) 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 [13]), 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 [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).

Theorem 4.2. Under Assumptions 2.1, 2.2, and 2.3, for sufficiently small , there is a such that for all

for any .

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).

Theorem 4.3. Under Assumptions 2.1, 2.2, and 2.3, for sufficiently small , there is a such that for all

for any .

Proof. For sufficiently small , we define

A similar analysis as Theorem 4.2 gives

Recalling that

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

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:

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. [12] 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.

Acknowledgment

This work is supported by the NSF of China (no. 11071050).

References

P. J. Schöonbucher, Credit Derivatives Pricing Models: Models, Pricing and Implementation, Wiley, Chichester, UK, 2003.

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004.

R. C. Merton, “Option pricing when underlying stock returns are discintinuous,” The Journal of Financial Economics, vol. 2, pp. 125–144, 1976.

X. Q. Liu and C. W. Li, “Weak approximation and extrapolations of stochastic differential equations with jumps,” SIAM Journal on Numerical Analysis, vol. 37, no. 6, pp. 1747–1767, 2000.

N. Bruti-Liberati and E. Platen, “Approximation of jump diffusions in finance and economics,” Computational Economics, vol. 29, pp. 283–312, 2007.

E. Platen, “An approximation method for a class of Itô processes with jump component,” Lietuvos Matematikos Rinkinys, vol. 22, no. 2, pp. 124–136, 1982.

N. Bruti-Liberati and E. Platen, “On the strong approximation of jump-diffusion processes,” Tech. Rep., University of Technology, Sydney, Australia, 2005, Quantitative Finance Research Papers 157.

N. Bruti-Liberati and E. Platen, “Strong approximations of stochastic differential equations with jumps,” Journal of Computational and Applied Mathematics, vol. 205, no. 2, pp. 982–1001, 2007.

E. Mordecki, A. Szepessy, R. Tempone, and G. E. Zouraris, “Adaptive weak approximation of diffusions with jumps,” SIAM Journal on Numerical Analysis, vol. 46, no. 4, pp. 1732–1768, 2008.

M. Wei, “Convergence of numerical solutions for variable delay differential equations driven by Poisson random jump measure,” Applied Mathematics and Computation, vol. 212, no. 2, pp. 409–417, 2009.

E. Buckwar and M. G. Riedler, “Runge-Kutta methods for jump-diffusion differential equations,” Journal of Computational and Applied Mathematics, vol. 236, no. 6, pp. 1155–1182, 2011.

S. Albeverio, Z. Brzeźniak, and J. L. Wu, “Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 309–322, 2010.

X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006.