The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations

and Applied Analysis 3 2.1. Exponential Euler Method We consider the n-dimensional semi-linear SDEs dX t FX t f t, X t dt g t, X t dW t , X 0 X0, 2.1 where initial dataX0 ∈ LpF0 Ω;R , F ∈ Rn×n is the generator of a strongly continuous analytic semigroup S S t t≥0 on a Banach space 17 , f : 0, T × R → R, g : 0, T × R → R and W t is a scalar Wiener process. In our analysis, it will be more natural to work with the equivalent expression X t eX0 ∫ t 0 e t−s f s,X s ds ∫ t 0 e t−s g s,X s dW s . 2.2 Now, we introduce the exponential Euler method for 2.1 . Given a stepsize h > 0, the exponential Euler approximate solution is defined by yk 1 eyk ef ( tk, yk ) h eg ( tk, yk ) ΔWk, 2.3 where yk is an approximation to X tk with tk kh, y0 X0 and ΔWk W tk 1 −W tk is the Wiener increment. It is convenient to use the continuous exponential Euler approximate solution and hence y t is defined by y t : ey0 ∫ t 0 e t−s f s, Y s ds ∫ t 0 e t−s g s, Y s dW s , 2.4 where s s/h h and x denote the largest integer, which is smaller than x and Y t is the step function which defined by Y t : ∞ ∑ k 0 I tk ,tk 1 t yk, 2.5 where I A is the indicator function of set A. Obviously, y tk Y tk yk for any integer k ≥ 0; that is the continuous exponential Euler solution y t and the step function Y t coincide with the discrete solution at the grid point. 2.2. Strong Convergence In this subsection, we are taking aim at the convergence of exponential Euler method applying to 2.1 . To show this, some conditions are imposed to the functions f and g in 2.1 . 4 Abstract and Applied Analysis Assumption 2.1. Assume that f and g satisfy the globally Lipschitz condition and the linear growth condition, that is, there exist two constants L1, L2 such that ∣ ∣f t, X − f t, Y ∣2 ∨ ∣g t, X − g t, Y ∣2 ≤ L1|X − Y |, 2.6 ∣ ∣f t, X ∣ ∣2 ∨ ∣g t, Y ∣2 ≤ L2 ( 1 |X| ) , 2.7 for allX,Y ∈ LpF0 Ω;R . Furthermore, f and g are supposed to satisfy the following property: ∣ ∣f t, X − f s,X ∣2 ∨ ∣g t, X − g s,X ∣2 ≤ L3 ( 1 |X| ) |t − s|, 2.8 where L3 is a constant and t, s ∈ 0, T with t > s. The following lemma illustrates that the continuous exponential Euler approximate solution 2.4 is bounded in MS sense and the relationship between continuous approximate solution 2.4 and the step function Y t . Lemma 2.2. Under Assumption 2.1, there exist two constants C1, C2 independent of h such that E ( sup 0≤t≤T ∣y t ∣2 ) ≤ C1, 2.9 E ∣y t − Y t ∣2 ) ≤ C2h, 2.10 for any t ∈ 0, T . Proof. From 2.4 and the elementary inequality a b c 2 ≤ 3 a2 b2 c2 , we have ∣y t ∣2 ≤ 3 ⎡ ⎣ ∣∣∣eFty0 ∣∣ 2 ∣∣∣∣ ∫ t 0 e t−s f s, Y s ds ∣∣∣∣ 2 ∣∣∣∣ ∫ t 0 e t−s g s, Y s dW s ∣∣∣∣ 2 ⎤ ⎦. 2.11 Taking the expectation on both sides and using the Hölder inequality and Doom’s martingale inequality yields E ( sup 0≤s≤t ∣y s ∣2 ) ≤ 3 [∣∣eFt ∣∣ 2 E ∣y0 ∣∣2 TE ∫ t 0 ∣∣eF t−s ∣∣ ∣f s, Y s ∣2ds 4E (∫ t 0 ∣∣eF t−s ∣∣ ∣g s, Y s ∣2ds )] . 2.12 Abstract and Applied Analysis 5 Letting M max{|eFT |, 1}, by the linear growth condition 2.7 , thenand Applied Analysis 5 Letting M max{|eFT |, 1}, by the linear growth condition 2.7 , then E ( sup 0≤s≤t ∣ ∣y s ∣ ∣2 ) ≤ 3M [ E ∣ ∣y0 ∣ ∣2 T ∫ t 0 E ∣ ∣f s, Y s ∣ ∣ds 4E ∫ t 0 E ∣ ∣g s, Y s ∣ ∣ds ] ≤ 3M [ E ∣y0 ∣∣2 L2T ∫ t 0 ( 1 E|Y s | ) ds 4L2 ∫ t 0 ( 1 E|Y s |ds )] ≤ 3M [ E ∣ ∣y0 ∣ ∣2 L2T T 4 L2 T 4 ∫ t 0 ( E|Y s |ds )] ≤ 3M ( E ∣y0 ∣2 L2T T 4 ) 3ML2 T 4 ∫ t 0 E ( sup 0≤r≤s ∣y r ∣2 ) ds. 2.13 Now using the Gronwall inequality yields that E ( sup 0≤s≤t ∣y s ∣∣2 ) ≤ C1, 2.14 where C1 3M E|y0| L2T T 4 e3ML2 T 4 T . From the definition of Y t and 2.4 , for t ∈ tk, tk 1 , we can obtain y t − Y t e t−tk yk ∫ t tk e t−s f ( tk, yk ) ds ∫ t tk e t−s g ( tk, yk ) dW s − yk. 2.15 Using Hölder inequality gives |y t − Y t | ≤ 3 ⎡ ⎣ ∣∣eF t−tk − In ∣∣ ∣yk ∣2 h ∫ t tk ∣∣eF t−s ∣∣ ∣f ( tk, yk ∣2ds ∣∣∣∣ ∫ t tk e t−s g ( tk, yk ) dW s ∣∣∣∣ 2 ⎤ ⎦, 2.16 where In is the n dimension identity matrix. Taking the expectation of both sides, we have E ∣y t − Y t ∣∣2 ≤ 3 [∣∣eF t−tk − In ∣∣ 2 E ∣yk ∣∣2 hE ∫ t tk ∣∣eF t−s ∣∣ ∣f ( tk, yk ∣2ds E ∫ t tk ∣∣eF t−s ∣∣ ∣g ( tk, yk ∣2ds ] . 2.17 6 Abstract and Applied Analysis In view of 2.7 , 2.9 , and |eF t−tk − In| ∼ O h2 , we can obtain E ∣ ∣y t − Y t ∣2 ≤ 3 [∣ ∣ ∣e t−tk − In ∣ ∣ ∣ 2 C1 hML2 1 C1 h ML2 1 C1 h ] ≤ 3M 1 C1 L2h O ( h2 ) . 2.18


Introduction
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 1-3 .
The related concepts of pth moment stability 0 < p ≤ 2 are attractive in its own right for analytical solution and numerical solution see 4, 5 .Especially, MS stability p 2 is

Exponential Euler Method and Strong Convergence
Throughout this paper, unless otherwise specified, let | • | be the Euclidean norm in R n .If A is a vector or matrix, its transpose is denoted by A T .If A is a matrix, its trace norm is denoted by |A| trace A T A .Let Ω, F, P be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions, that is right continuous and increasing, while F 0 contains all p-null sets.Let W t be a scalar Wiener process defined on the probability space.Let p > 0 and L p F 0 Ω; R n denote the family of R n -valued F 0 -measurable random variables ξ with E|ξ| p < ∞.
Abstract and Applied Analysis 3

Exponential Euler Method
We consider the n-dimensional semi-linear SDEs dX t FX t f t, X t dt g t, X t dW t , X 0 X 0 ,

2.1
where initial data X 0 ∈ L p F 0 Ω; R n , F ∈ R n×n is the generator of a strongly continuous analytic semigroup S S t t≥0 on a Banach space 17 , f : 0, T × R n → R n , g : 0, T × R n → R n and W t is a scalar Wiener process.
In our analysis, it will be more natural to work with the equivalent expression

Strong Convergence
In this subsection, we are taking aim at the convergence of exponential Euler method applying to 2.1 .To show this, some conditions are imposed to the functions f and g in 2.1 .Assumption 2.1.Assume that f and g satisfy the globally Lipschitz condition and the linear growth condition, that is, there exist two constants L 1 , L 2 such that for all X, Y ∈ L p F 0 Ω; R n .Furthermore, f and g are supposed to satisfy the following property: where L 3 is a constant and t, s ∈ 0, T with t > s. The

2.13
Now using the Gronwall inequality yields that where

2.18
Therefore, where In the following, we show the convergence result of exponential Euler method for semi-linear SDE 2.1 .

Theorem 2.3. Under Assumption 2.1, the numerical solution produced by the exponential Euler method converges to the exact solution of 2.1 in MS sense with the strong order
Proof.From 2.2 and 2.4 we know

2.22
By the H ölder inequality and Doom's martingale inequality, we have

2.23
Abstract and Applied Analysis 7 Consider the first argument in 2.23

2.24
By Assumption 2.1 and Lemma 2.2, it is obvious that

2.25
This implies

2.26
By the similar procedure, we can observe that

2.28
Since |e Fh − I n | ∼ O h 2 , we can show the following result by Gronwall inequality: , we can obtain the convergence result for any 0 ≤ t ≤ T .

Mean-Square Stability
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.The MS stability region of 3.1 is denoted by S SDE S SDE λ, μ and represents the set of parameter values for which the equilibrium solution of 3.1 is MS stable.The exponential Euler method for test equation 3.1 leads to the following type:

Test Equation
X n 1 e λh X n e λh μX n ΔW n e λh X n e λh μX n hZ e λh 1 μ hZ X n ,

3.3
where X n is the approximation of y t n with X 0 y 0 .Now we define MS stability region for the numerical method applied to 3.1 .The notations and definitions are similar to those in 8 .Definition 3.1.If the adaptation of a numerical method to 3.1 leads to a numerical process of the following type: Furthermore, if R h, λ, μ < 1, the numerical method is MS stable and the region of parameter values that satisfy R h, λ, μ < 1 is called the MS stability region of the numerical method, where Z ∼ N 0, 1 .
The MS stability regions of EM method and exponential Euler method are denoted by S EM , S EE , respectively.Definition 3.2.The exponential Euler method described by 2.3 is mean-square A-stable if for all h, S SDE ⊆ S EE .
3.5 Higham 19 proposed that the MS stability region for EM method is S EM { p, q | 0 < q < −p p 2 }, where p λh and q |μ| 2 h.According to 3.2 , the SDE 3.1 is MS stable only and if only the pair of parameters λ and μ belong to the region of S SDE { p, q | 0 < q < −2p}.
The MS stability region of exponential Euler method for 3.1 is given in the following theorem.By the comparing with the Euler method and stochastic Theta method, it is observed that the exponential Euler method as an explicit numerical method has desired property.Theorem 3.3.The mean-square stability region of exponential Euler method for 3.1 is where p λh and q |μ| 2 h.
Proof.From 3.3 , by using EZ 0 and EZ 2 1, the MS stability function of exponential Euler method is
Remark 3.4.From the inequality −p p 2 < −2p < e −2p −1, we can find that MS stability region of exponential Euler method contains that of EM method and contains MS stability region of the exact solution for 3.1 , that is,

3.10
According to the Definition 3.2, the exponential Euler method is mean-square A-stable.
Remark 3.5.In 9 the MS stability region of stochastic Theta method is denoted by S STM θ, h and had the conclusions that S STM θ, h ⊂ S SDE if 0 ≤ θ < 1/2 and S SDE ⊂ S STM θ, h if 1/2 < θ < 1.We know that stochastic Theta method is MS stable if and only if

3.12
Abstract and Applied Analysis 11

Scalar Semilinear SDEs
This subsection presents new result on the MS exponentially stability for scalar semi-linear stochastic differential equation dX t aX t f t, X t dt g t, X t dW t , 3.13 with initial value X 0 X 0 , where a < 0 is the linear argument of the drift coefficient.We have proved that the exponential Euler approximation solution preserves the MS exponential stability of the exact solution for any stepsize h under the following conditions.Assumption 3.6.Assume that there exist a positive constant K such that

3.14
The condition 3.14 implies f t, 0 0, g t, 0 0 and ensures that the analytical solution will never reach the origin with probability one.Definition 3.7 see 20 .The solution of 3.13 is said to be exponentially stable in MS sense if there is a pair of positive constants γ and C such that 3.15 Lemma 3.8.Under Assumption 3.6, if the analytic solution of 3.13 is exponentially stable in MS sense, that is, 3.17 This lemma can be proved in the similar way as Theorem 4.1 proved in 20 .We can obtain that if ρ < 0, the analytic solution is exponentially stable in MS sense.Now the original result about the MS exponential stability of exponential Euler method is given in the following theorem.Theorem 3.9.If 3.14 and 3.16 hold, then for any stepsize h > 0, the exponential Euler method for 3.13 is exponentially stable in MS sense, that is, where Proof.The adaptation of exponential Euler method to 3.13 leads to a numerical process of the following type:

3.22
From the linear condition 3.14 , we obtain that

3.23
Taking expectations on both sides yields From a < 0 and ρ < 0, it is easy to observe that f h 8/3 a 3 h K − 2a 2 < 0 when h > 0 and f 0 ρ < 0. Hence 3.26 holds and this implies 3.25 always holds when h > 0.
From 3.24 , it follows that
Remark 3.10.If 2a 2 √ K K < 0, the analytic solution of 3.13 is exponentially stable in MS sense.Under the same conditions, the numerical solution of exponential Euler method can preserve the exponential stable in MS sense for any stepsize h > 0, that is, the stability of exponential Euler method for 3.13 has no limitation to the stepsize h.

Numerical Experiments
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.

Strong Convergence of Exponential Euler Method
In order to make the notion of convergence precise, we must decide how to measure their difference.Using E|y n − X τ n | leads to the concept of strong convergence 19 .Definition 4.1.A method is said to have strong order of convergence equal to 1/2 if there exists a constant C, such that for any sufficiently small stepsize h and fixed τ nh ∈ 0, T .
The parameter λ 2, μ 1 and X 0 1 is used to look at the strong convergence of exponential Euler method for 3.1 .We compute 5000 different discrete Brownian paths over 0, 1 with δ 2 −9 .For each path, exponential Euler method is applied with differential stepsizes: h 2 p−1 δ for 1 ≤ p ≤ 5. We denote by y k,p the value of kth generated trajectory of numerical solution at the endpoint with h 2 p−1 δ for 1 ≤ p ≤ 5 and by X k,p the corresponding value of exact solution.It is easy to obtain the analytical solution of 3.1 .The average errors e h 1 5000 at the endpoint over 5000 sample paths are approximation for h 2 p−1 δ, 1 ≤ p ≤ 5. We plot the approximation to e h against h in blue on a log-log scale in Figure 1.For reference, a dashed red line of slope one-half is added.In Figure 1  that 4.1 is valid.Therefore, our results are consistent with a strong order of exponential Euler method equal to 1/2 from numerical experience.

Mean-Square Stability Region
Consider the linear scaler stochastic differential equation dX t λX t dt μX t dW t , X 0 X 0 .

4.4
To examine MS stability of exponential Euler method, we solve 4.4 with X 0 1 over 0, 20 for two parameters sets.The first set is λ −3 and μ √ 3.These values satisfy 3.2 , hence the problem is MS stable.Firstly, We apply EM method over 50000 discrete Brownian paths for the three differential stepsizes: h 1, h 1/2, h 1/4.Secondly, we apply exponential Euler method with the same stepsizes.Figure 2 depicts the plot of the sample average of y 2 j against t j jh.Note that the vertical axis is logarithmically scaled.In the upper curves of Figure 2, h 1 and h 1/2 curves increase with t while the h 1/4 curve decays toward zero.However, in the lower curves of Figure 2, all the curves decay toward zero whether h 1, h 1/2, or h 1/4.This implies that the MS stability region of exponential Euler method contains that of EM method.
Next, we use the parameter set λ −3 and μ 3. It is observed that 4.4 is not MS stable.The upper curves of Figure 3 are approximated by the EM method and the lower curves in Figure 3 are approximated by the exponential Euler method.The curves in the upper picture increase with t, while all the curves in the lower picture decrease toward zero.This implies that the MS stability region of exponential Euler method contains the MS region of test equation 4.4 .

Mean-Square Exponential Stability
Consider the scalar semi-linear stochastic differential equation: dx t −3x t sin x t dt x t dW t .

4.5
It is easy to verify that 4.5 has the properties of 3.14 and 3.16 .According to Theorem 3.9, the problem is exponentially stable in MS sense.To test MS exponential stability, we solve 4.5 with X 0 1 over 0, 100 .We apply the exponential Euler method and EM method, respectively, over 5000 different discrete Brownian paths with different large stepsize h 4, h 8, and the average over 5000 sample paths is approximated to |y n | 2 .The solutions of exponential Euler method and EM method with h 4 can be found in Figures 4 and 5.We can find that the curve in Figure 4 decreases significantly while the curve in Figure 5 increases sharply with t.It is observed that the exponential Euler method can preserve the MS exponential stability, but EM method does not have this property when h 4. Figures 6 and 7 demonstrate numerical solutions of the two methods when stepsize h 2 3 and the similar result can be obtained.Hence exponential Euler approximation solution shares the MS exponential stability of the exact solution.
From Figures 5 and 7, it is manifest that EM method does not preserve the stability with h 2 2 and h 2 3 .

Conclusions
The classical explicit numerical methods for SDEs such as EM method and Milstein method and the semi-implicit method such as stochastic Theta method 0 ≤ θ < 1/2 usually have limitations to the stepsize h, and the stability results are given for sufficient small stepsize h.In this paper, the exponential Euler method is extended to semi-linear SDEs and we proved that the stability results have fewer restrictions of stepsize and preserve the stability of SDEs.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 h > 0. For scaler linear test equation, the MS stable region of exponential Euler method is calculated, and it is observed that the MS stable region of exponential Euler method contains that of EM method and stochastic Theta method 0 ≤ θ < 1 and also contains that of the scalar linear test equation.According to Definition 3.2, exponential Euler method is MS A-stable.
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 n-dimensional SDEs n ≥ 2 , dX t f X t dt g X t dW t , X 0 X 0 , 5.1 whether exponential Euler method can be applied to obtain the numerical solution is our future work.

Figure 1 :
Figure 1: Numerical approximation for strong convergence order of exponential Euler method.
Fhy k e Fh f t k , y k h e Fh g t k , y k ΔW is the continuous exponential Euler solution y t and the step function Y t coincide with the discrete solution at the grid point.
s f s, X s ds t 0 e F t−s g s, X s dW s .2.2Now, we introduce the exponential Euler method for 2.1 .Given a stepsize h > 0, the exponential Euler approximate solution is defined byy k 1 e k , 2.3where y k is an approximation to X t k with t k kh, y 0 X 0 andΔW k W t k 1 − W tk is the Wiener increment.It is convenient to use the continuous exponential Euler approximate solution and hence y t is defined by y t : e Ft y 0 t 0 e F t−s f s, Y s ds t 0 e F t−s g s, Y s dW s , 2.4 where s s/h h and x denote the largest integer, which is smaller than x and Y t is the step function which defined by Y t : following lemma illustrates that the continuous exponential Euler approximate solution 2.4 is bounded in MS sense and the relationship between continuous approximate solution 2.4 and the step function Y t .Under Assumption 2.1, there exist two constants C 1 , C 2 independent of h such that 1 e ah y n e ah f t n , y n h e ah g t n , y n ΔW n ,

1
Kh 2 Kh 2 Furthermore, we see that the slope of the curve appears to 1/2.A least-squares power law fit produces the slope 0.5218, residual 0.0435 of the blue curve in Figure1.This suggests