Stochastic differential equations with jumps are of a wide application area especially in mathematical finance. In general, it is hard to obtain their analytical solutions and the construction of some numerical solutions with good performance is therefore an important task in practice. In this study, a compensated split-step

Stochastic differential equations (SDEs) for jump-diffusions arise in a variety of practical areas and have successfully been used to describe unexpected and abrupt changes in the present structure (for an overview, see [

Chalmers and Higham [

However, as pointed out by Chalmers and Higham [

Our work is motivated by [

Let

We firstly introduce the following assumptions for the establishment of the convergence and stability of the proposed numerical solution of the SDEVDRJMs in (

(i) The time delay

(ii) Global Lipschitz condition: there exists a positive constant

(iii) Quadratic growth condition: for any

(iv) For

Under the assumptions (i) and (iii), there exist positive constants

From (

Using the compensated Poisson process

For a given constant time step size

In this section, we will prove that the above numerical solutions converge in mean-square to the true solution of the SDEVDRJM with the approximate rate of

Replacing the numerical approximations with the exact solution values on the right-hand side of (

Under the assumptions (i)–(iv), there exists a constant

From (

To estimate

(1) If

(2) If

(3) If

(4) If

(5) If

Similarly, it can be proved that there exists a constant

From (

Moreover, it is known from Theorem 3.4 in [

Summing up the above conclusions, we can obtain that, for any

Otherwise, for any

Under assumptions (i)–(iv), there exists a constant

From the definitions of

From (

Combining the above conclusions, we have

(1) If

(2) If

(3) If

Combining the above three cases, it can be obtained that

In this section, we will discuss the stability of the analytical solutions of the SDEVDRJM and the numerical method introduced in Section

Consider the following scalar test equation:

In what follows, we give some sufficient conditions on the stability property of the analytical solutions of (

Assume that the constants

For any

Applying the compensated split-step

Assume that condition (

(i) If

(ii) If

(iii) If

(iv) If

Here,

From (

If

Furthermore,

if

if

which implies that (

If

if

if

if

If

Therefore, we have

The above results show that the compensated split-step

In this paper, A compensated split-step

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the anonymous referee and the editors for their careful reading of this paper and for their valuable comments. This research is funded by the National Science Foundation of China (no. 11201365).