The numerical approximation of exponential Euler method is constructed for semilinear stochastic differential equations (SDEs). The convergence and mean-square
(MS) stability of exponential Euler method are investigated. It is proved that the exponential Euler method is convergent with the strong order

Stochastic differential equations are utilized as mathematical models for physical application that possesses inherent noise and uncertainty. Such models have played an important role in a range of applications, including biology, chemistry, epidemiology, microelectronics, and finance. Many mathematicians have devoted their effort to develop it and have obtained a substantial body of achievements. In order to understand the dynamics of stochastic system, it is important to construct an efficient numerical simulation of SDEs. There are many results for numerical solutions of stochastic differential equations. The general introduction to numerical methods for SDEs can be found in [

The related concepts of

The phenomenon of stiffness appears in the process of applying a certain numerical method to solve ODEs and SEDs. Stiffness was also the reason for the introduction of exponential integrators, which have been proposed independently by many authors. The main contribution of exponential integrators is that it can solve exactly the linear part of the problem [

Lawson [

In this paper, the exponential Euler method as one of the simplest forms of exponential RK method is extended to semi-linear SDEs. The exponential Euler method is based on a discrete version of the variation of constants formula.

In Section

In Section

We then consider the exponentially MS stability of the exponential Euler method for scalar semi-linear SDEs in Section

Throughout this paper, unless otherwise specified, let

We consider the

In our analysis, it will be more natural to work with the equivalent expression

Now, we introduce the exponential Euler method for (

In this subsection, we are taking aim at the convergence of exponential Euler method applying to (

Assume that

The following lemma illustrates that the continuous exponential Euler approximate solution (

Under Assumption

From (

Letting

From the definition of

In the following, we show the convergence result of exponential Euler method for semi-linear SDE (

Under Assumption

From (

By Assumption

In this section, we focus on the MS stability of the exponential Euler as it is applied to scalar semi-linear SDEs. It is significantly helpful to describe the MS stability region of the exponential Euler method. In the following, scalar linear SDE as the test equation is used to calculate the MS stability region.

Consider the test equation

It is well known [

The MS stability region of (

If the adaptation of a numerical method to (

The MS stability regions of EM method and exponential Euler method are denoted by

The exponential Euler method described by (

Higham [

The MS stability region of exponential Euler method for (

The mean-square stability region of exponential Euler method for (

From (

From the inequality

In [

This subsection presents new result on the MS exponentially stability for scalar semi-linear stochastic differential equation

We have proved that the exponential Euler approximation solution preserves the MS exponential stability of the exact solution for any stepsize

Assume that there exist a positive constant

The condition (

The solution of (

Under Assumption

This lemma can be proved in the similar way as Theorem 4.1 proved in [

Now the original result about the MS exponential stability of exponential Euler method is given in the following theorem.

If (

The adaptation of exponential Euler method to (

Recall the inequality

From(

If

In this section, several numerical experiments are given to verify the conclusions of convergence and MS stability of exponential Euler method for semi-linear stochastic differential equations.

In order to make the notion of convergence precise, we must decide how to measure their difference. Using

A method is said to have strong order of convergence equal to

The parameter

Numerical approximation for strong convergence order of exponential Euler method.

If the inequality (

Consider the linear scaler stochastic differential equation

To examine MS stability of exponential Euler method, we solve (

The numerical solutions with

In the upper curves of Figure

Next, we use the parameter set

The numerical solutions with

Consider the scalar semi-linear stochastic differential equation:

The solutions of exponential Euler method and EM method with

Exponential Euler method.

EM method.

Exponential Euler method.

EM method.

From Figures

The classical explicit numerical methods for SDEs such as EM method and Milstein method and the semi-implicit method such as stochastic Theta method

It is proved that under the conditions where scalar semi-linear SDEs is MS exponentially stable, the exponential Euler method can preserve the MS stability for all stepsize

In this paper, the scalar Wiener process is considered and the corresponding results can be generalized to the multidimensional Winer process.

The MS exponential stability is investigated for scalar semi-linear SDEs in this paper. For

This paper is supported by the Fundamental Research Funds for the Central Universities (HIT.NSRIF.2013081).