- Open Access
Convergence and stability of the compensated split-step θ-method for stochastic differential equations with jumps
© Tan et al.; licensee Springer. 2014
Received: 27 December 2013
Accepted: 8 July 2014
Published: 4 August 2014
In this paper, we develop a new compensated split-step θ (CSSθ) method for stochastic differential equations with jumps (SDEwJs). First, it is proved that the proposed method is convergent with strong order 1/2 in the mean-square sense. Then the condition of the mean-square (MS) stability of the CSSθ method is obtained. Finally, some scalar test equations are simulated to verify the results obtained from theory, and a comparison between the compensated stochastic theta (CST) method by Wang and Gan (Appl. Numer. Math. 60:877-887, 2010) and CSSθ is analyzed. Meanwhile, the results show the higher efficiency of the CSSθ method.
for , with , where denotes , , , , is a scalar standard Wiener process, and is a scalar Poisson process with intensity λ.
Recently, stochastic differential equations with jumps (SDEwJs) are becoming increasingly used to model real-world phenomena in different fields, such as economics, finance, biology, and physics. However, few analytical solutions have been proposed so far; thus, it is necessary to develop numerical methods for SDEwJs and study the properties of these methods. For example, Higham and Kloeden  studied the convergence and stability of the implicit method for jump-diffusion systems, and they further analyzed the strong convergence rates of the backward Euler method for a nonlinear jump-diffusion system . Chalmers and Higham  studied the convergence and stability for the implicit simulations of SDEs with random jump magnitudes. Higham and Kloeden  constructed the split-step backward Euler (SSBE) method and the compensated split-step backward Euler (CSSBE) method for nonlinear SDEwJs. Bruti-Liberati and Platen [5, 6] developed strong and weak approximations of SDEwJs.
Lately, Wang and Gan  started to focus on the CST method for stochastic differential equations with jumps. Hu and Gan  studied the convergence and stability of the balanced methods for SDEwJs. The split-step θ (SSθ) method was firstly developed by Ding et al.  to solve the stochastic differential equations. Thus, we will construct the compensated split-step θ method (CSSθ) for SDEwJs.
In this paper, we investigate the convergence and mean-square stability of the CSSθ method for SDEwJs. The outline of the paper is as follows. In Section 2, we introduce some notations and hypotheses and give the CSSθ method for SDEwJs. In Section 3, we prove that the numerical solutions produced by the CSSθ method converge to the true solutions with strong order 1/2. In Section 4, the mean-square stability of the CSSθ method for linear test equation is studied. At last, some numerical experiments are used to verify the results obtained from the theory.
2 The compensated split-step θ-method
For the existence and uniqueness of the solution for (1.1), we usually assume that f, g, and h satisfy the following assumptions:
where , is the numerical approximation of with . Moreover, the increments are independent Gaussian random variables with mean 0 and variance Δt; are independent Poisson distributed random variables with mean and variance .
If we give , the SSθ method becomes the SSBE method in . If , the SSθ method is an explicit method.
If we give , the CSSθ method becomes the CSSBE method in .
To answer the question of the existence of numerical solution, we will give the following lemma.
Lemma 2.1 Assume that satisfies (2.1), and let , , then equation (2.7) can be solved uniquely for , with probability 1.
Then the result follows from the classical Banach contraction mapping theorem . □
3 Strong convergence on a finite time interval
In this section, we prove the strong convergence of the CSSθ method for problem (1.1) on a finite time interval , where T is a constant.
where N is the largest number such that , and is the indicator function for the set A, i.e.,
It is easy to verify that , that is, and coincide with the discrete solutions at the gridpoints. Hence we refer to as a continuous-time extension of the discrete approximation . So our plan is to prove a strong convergence result for .
Now we begin the proof of the strong convergence of the CSSθ method, our first lemma shows the relationship between and .
where and are produced by (2.7) and (2.8).
where and . The proof is completed. □
The next lemma shows that the discrete numerical solutions and (), produced by the CSSθ method, have bounded second moments.
where and are two positive constants independent of Δt.
are both independent of Δt.
The next lemma shows that the continuous-time approximation in (3.4) remains close to the step functions and in the mean square sense.
where , , , and are defined by (3.1), (3.2), (3.4), respectively.
where . Thus we can prove (3.18).
Now we give the proof of (3.19).
where . Then we have proved (3.19). □
Now we use the above lemmas to prove a strong convergence result.
for any fixed , and Δt sufficiently small.
where is a positive constant independent of Δt.
4 Mean-square stability
for all .
the CSSθ method (2.7)-(2.8) applied to equation (4.1) is MS-stable.
By (4.2), we know that (4.15) holds for all , i.e., the CSSθ method is MS-stable for all . Note that if , the CSSθ method reduces to CSSBE, and (4.15) coincides with Theorem 7 which was studied in .
From (4.13), we know that the CSSθ method is MS-stable. This proves the theorem. □
5 Numerical experiments
see, for example, .
To illustrate the convergence order and the linear mean-square stability of the CSSθ method, we choose the following examples from the reference .
Example 5.1 , , , .
Example 5.2 , , , .
In this section, the data used in all figures are obtained by the mean square of data by 1,000 trajectories, that is, , ; in all figures denotes the mesh-point.
To illustrate the step size h on the mean-square stability of the CSSθ method, we applied the CSSθ method to Examples 5.1 and 5.2.
At last, Figure 6 (lower) shows that the numerical solution of the CSSθ method is still stable when . This implies that maybe the mean-square stability bound we obtained by Theorem 4.1 is not optimal.
This research was supported with funds provided by the National Natural Science Foundation of China (Nos. 11226321, 11272229 and 11102132). We thank two anonymous reviewers for their very valuable comments and helpful suggestions which improved this paper significantly.
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