# T-Stability of the Heun Method and Balanced Method for Solving Stochastic Differential Delay Equations

Authors: Xiaolin Zhu; Hu Peng
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

This paper studies the T-stability of the Heun method and balanced method for solving stochastic differential delay equations (SDDEs). Two T-stable conditions of the Heun method are obtained for two kinds of linear SDDEs. Moreover, two conditions under which the balanced method is T-stable are obtained for two kinds of linear SDDEs. Some numerical examples verify the theoretical results proposed.


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## Body

## 1. Introduction

Stochastic differential delay equations (SDDEs) are the promotion of stochastic differential equations (SDEs) and differential delay equations (DDEs). These kinds of equations consider not only the stochastic factors in the process of the development of a system, but also the impact of the delay. As an important mathematical model, SDDEs have been applied widely in many areas, such as stochastic control, economics, and biology. Since it is difficult to find the analytic solutions to SDDEs, to get the numerical solutions to SDDEs generated by some numerical methods is commonly used. For a numerical method, it is important to analyze its stability.The general form of SDDEs with Gaussian white noise is(1)dX(t)=f(t,X(t),X(t-τ))dt+g(t,X(t),X(t-τ))dW(t),t∈[0,T],X(t)=φ(t),t∈[-τ,0],
where τ>0,φ(t)∈C([-τ,0],Rm), f:R+×Rm×Rm→Rm, g:R+×Rm×Rm→Rm×d, and W(t) is a standard d-dimensional Wiener process. Equation (1) has a unique solution if f and g are sufficiently smooth and satisfy the following conditions:
(2)|f(t,x1,y1)-f(t,x2,y2)|∨|g(t,x1,y1)-g(t,x2,y2)|≤L(|x1-x2|+|y1-y2|),(3)|f(t,x,y)|2∨|g(t,x,y)|2≤K(1+|x|2+|y|2),
where t≥0,x,y,x1,y1,x2,y2∈Rm, and L and K are constants. The condition in (2) is calledLipschitz condition, and the condition in (3) is called thelinear growth condition.The main numerical methods for SDDEs are Euler-Maruyama method [1, 2] and Milstein method [3] at present. The mean square stability of these methods for SDDEs has been well studied. Cao et al. [1] studied the mean square stability of Euler-Maruyama method for linear SDDEs. Liu et al. [2] studied the mean square stability of the semi-implicit Euler method for linear SDDEs. Wang and Zhang [3] discussed the mean square stability of Milstein method for linear SDDEs. Wang and Chen [4, 5] studied the mean square stability of semi-implicit Euler method for nonlinear neutral SDDEs and that of Heun methods for nonlinear SDDEs. Tan et al. [6] discussed the mean square stability of balanced methods for SDDEs. T-stability is introduced by Saito et al. in [7–9], and it is another kind of stability with respect to the approximate sequence of sample path. Cao [10] studied the T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise. Rathinasamy and Balachandran [11] studied the T-stability of the split-step θ-methods for linear SDDEs. Yang and Liu [12] discussed the T-stability of the θ-method for a stochastic pantograph differential equation.Applying the Heun method [5] to (1) gives
(4)Xn+1=Xn+12[f(tn,Xn,Xn-m)hhhhhhhh+f(tn+1,Xn+hf(tn,Xn,Xn-m),Xn-m+1)]h+g(tn,Xn,Xn-m)ΔWn,
where h>0 is a step size with τ=mh for a positive integer m and tn=nh. Xn is an approximation of X(tn), and Xn=φ(tn) if tn≤0. ΔWn=W(tn+1)-W(tn)~N(0,h).The mean square stability of the Heun method in (4) was studied in [5], but there is no result about T-stability of the method at present. This paper gives two T-stable conditions of the Heun method (4) for two kinds of linear SDDEs.The balanced method for solving (1) is
(5)Xn+1=Xn+f(tn,Xn,Xn-m)h+g(tn,Xn,Xn-m)ΔWn+C(Xn,Xn-m)(Xn-Xn+1),
where C(Xn,Xn-m)=C0(Xn,Xn-m)h+C1(Xn,Xn-m)|ΔWn| and C0,C1 are d×d real matrix functions.LetM(x,y)=I+β0C0(x,y)+β1C1(x,y), where I is a unit matrix, β0∈[0,α], α≥h, β1≥0, and (x,y)∈Rd×Rd. Assume that M(x,y) is invertible with |(M(x,y))-1|≤K<∞.The mean square stability of the balanced method for SDEs and SDEs with jumps was studied in [13, 14], respectively. In 2011, Tan et al. [6] applied the balanced method to SDDEs and discussed the mean square convergence and stability of this method. However, there is no research result about T-stability of the balanced method (5) at present. In this paper, the conditions under which the balanced method (5) is T-stable are obtained for two kinds of linear SDDEs.Section2 introduces the stochastically asymptotically stable conditions in the large for two kinds of linear SDDEs. In Section 3, T-stability of the Heun method equipped with a specified driving process is discussed and the corresponding step size range is given. Section 4 studies T-stability of the balanced method, and Section 5 uses some numerical examples to verify the results given in this paper.


## 2. Asymptotic Stability of Analytical Solution

Consider the following two scalar linear test equations:(6)dX(t)=[aX(t)+bX(t-τ)]dt+cX(t)dW(t),t∈[0,T]X(t)=φ(t),t∈[-τ,0],(7)dX(t)=[aX(t)+bX(t-τ)]dt+[cX(t)+dX(t-τ)]dW(t),t∈[0,T]X(t)=φ(t),t∈[-τ,0].
Let (Ω,F,{Ft}t≥0,P) be a complete probability space with a filtration {Ft}t≥0, which is right continuous, and each Ft contains all P-null sets in  F. In (6) and (7), a,b,c,d∈R;τ>0; W(t) is a one-dimensional standard Wiener process; initial function φ(t)∈C([-τ,0],Rm) is F0-measurable and E∥φ∥2<∞. Equations (6) and (7) have unique strong solution if (6) and (7) meet Lipschitz condition in (2) and the linear growth condition in (3).Definition 1 (see [10]).
The solution of (1) isstochastically asymptotically stable in the large if
(8)P(lim⁡t→∞|X(t,φ)|=0)=1
for all initial functions φ.From Corollary3.2 in [15], we get Lemmas 2 and 3 as follows.Lemma 2.
The solution of (6) is stochastically asymptotically stable in the large if parameters a,b, and c in (6) satisfy
(9)a<-|b|-12c2.Lemma 3.
The solution of (7) is stochastically asymptotically stable in the large if parameters a,b,c, and d in (7) satisfy
(10)a<-|b|-12(|c|+|d|)2.


## 3. T-Stability of the Heun Method

Definition 4 (see [10]).
Suppose that the condition in (9) or in (10) is fulfilled. A numerical scheme equipped with a specified driving process is said to be T-stable if
(11)lim⁡n→∞|Xn|⟶0a.s.
for the driving process, where Xn is the numerical solution generated by the numerical scheme applied to the test equation (6) or (7).For analyzing T-stability, we focus our attention on the trajectory of numerical solution. A specified driving process proposed in Definition4 is used to approximate the Wiener increment ΔWn of the numerical methods. This paper analyzes the Heun method equipped with two-point random variables for the driving process. So ΔWn:=ηnh and P(ηn=±1)=1/2, where P denotes the probability.The Heun method applied to (6) and (7), respectively, gives
(12)Xn+1=(1+ah+12a2h2+cΔWn)Xn+12bh(1+ah)Xn-m+12bhXn-m+1,(13)Xn+1=(1+ah+12a2h2+cΔWn)Xn+[12bh(1+ah)+dΔWn]Xn-m+12bhXn-m+1,
where h>0 is a step size with τ=mh for a positive integer m and tn=nh, Xn≈X(tn) if tn≤0, Xn=φ(tn).For the Heun method in (12), we have
(14)|Xn+1|≤(|1+ah+12a2h2+cΔWn|+|12bh(1+ah)|+|12bh|)×max⁡{|Xn|,|Xn-m|,|Xn-m+1|}=(|12(1+ah)2+12+cηnh|+|12bh(1+ah)|+|12bh|)×max⁡{|Xn|,|Xn-m|,|Xn-m+1|}.
Let
(15)R(h,a,b,c)=|12bh(1+ah)|+|12bh|+|12(1+ah)2+12+cηnh|.
It is clear that |Xn|→0,a.s.(n→∞) if P(R(h,a,b,c)<1)=1, and therefore the Heun method in (12) is T-stable. Denote
(16)RT(h,a,b,c)=(|12bh(1+ah)|+|12bh|+|12(1+ah)2+12+ch|)×(|12bh(1+ah)|+|12bh|+|12(1+ah)2+12-ch|)
Since ηn’s follow two-point distribution, we get |Xn|→0,a.s.(n→∞) if RT(h,a,b,c)<1, which means that the Heun method in (12) is T-stable.Similarly, for the Heun method in (13), we can get R~T(h,a,b,c,d) as follows:
(17)R~T(h,a,b,c,d)=(|12bh(1+ah)+dh|+|12bh|+|12(1+ah)2+12+ch|)×(|12bh(1+ah)-dh|+|12bh|+|12(1+ah)2+12-ch|),
and |Xn|→0,a.s.(n→∞) if R~T(h,a,b,c,d)<1, which means that the Heun method in (13) is T-stable.Theorem 5.
Suppose (6) meets the condition in (9). The Heun method in (12) is T-stable if h<H, where
(18)H=min⁡{2|b|-a,14c2}.Proof.
The condition in (9) gives a<0. Denote
(*)R1(h,a,b,c)=|12bh(1+ah)|+|12bh|+|12(1+ah)2+12+ch|,R2(h,a,b,c)=|12bh(1+ah)|+|12bh|+|12(1+ah)2+12-ch|,
and RT(h,a,b,c)=R1(h,a,b,c)R2(h,a,b,c).
Forb≠0, we have a+b<0, a-b<0 from the condition in (9). Let
(19)h1=-1a,h2=min⁡{-1a,14c2}.
Only the conclusion of the theorem whenb<0,c>0 is proven, and that of the theorem when b<0,c<0 or b>0,c>0 or b>0,c<0 can be proven similarly.
Ifh<h2, then h<-1/a and h<1/4c2, and we have 1+ah>0 and (1/2)-ch>0. Hence
(20)R1(h,a,b,c)+R2(h,a,b,c)=-bh(1+ah)-bh+(1+ah)2+1=(a-b)(ah+2)h+2.
Since ah+1>0 and a-b<0, we have ah+2>0 and
(21)0<R1(h,a,b,c)+R2(h,a,b,c)<2.
Consequently,
(22)RT(h,a,b,c)=R1(h,a,b,c)R2(h,a,b,c)≤[R1(h,a,b,c)+R2(h,a,b,c)2]2<1,
which means that the Heun method (12) is T-stable if h<h2.
(a) Whenh1=-1/a<H, if H=1/4c2, then h2=min⁡{-1/a,1/4c2}=-1/a=h1.
IfH=2/(|b|-a), then H≤1/4c2, which yields h1=-1/a<H≤1/4c2, and h2=-1/a=h1. Hence h2=h1 if h1<H.
For obtaining the result of this theorem in this case, the left work is to prove the method is T-stable forh1≤h<H.
Ifh1≤h<H, then -1/a≤h<1/4c2, h<2/(|b|-a); that is,
(23)1+ah≤0,12-ch>0,(a-|b|)h+2>0.
Since a-|b|≤a+b, we get (a+b)h+2≥(a-|b|)h+2>0. Since
(24)R1(h,a,b,c)+R2(h,a,b,c)=bh(1+ah)-bh+(1+ah)2+1=ah[(a+b)h+2]+2,
we have 0<R1(h,a,b,c)+R2(h,a,b,c)<2 from a<0 and (*), and therefore
(25)RT(h,a,b,c)=R1(h,a,b,c)R2(h,a,b,c)≤[R1(h,a,b,c)+R2(h,a,b,c)2]2<1,
which means that the Heun method (12) is T-stable if h<H.
(b) Whenh1=H, it is easy to know h2=h1, and the Heun method (12) is T-stable if h<h2=H.
(c) Whenh1=-1/a>H, if H=1/4c2, then h2=min⁡{-1/a,1/4c2}=1/4c2=H. If H=2/(|b|-a), then -1/a>2/(|b|-a). Solving this gives a>-|b|, which contradicts the condition in (9). So h2=H if h1>H. Hence the Heun method in (12) is T-stable if h<h2=H.
The discussion above shows that the Heun method in (12) is T-stable provided that h<H, which completes the proof.Theorem 6.
Suppose (7) meets the condition in (10). The Heun method in (13) is T-stable if H1<h<H2, where
(26)H1=4d2(a+|b|)2,H2=min⁡{-1a,14c2,4d2b2}.Proof.
The condition in (10) gives a<0. Denote
(**)R~1(h,a,b,c,d)=|12bh(1+ah)+dh|+|12bh|+|12(1+ah)2+12+ch|,R~2(h,a,b,c,d)=|12bh(1+ah)-dh|+|12bh|+|12(1+ah)2+12-ch|,
and then R~T(h,a,b,c,d)=R~1(h,a,b,c,d)R~2(h,a,b,c,d) from (17).
(a) Supposeb<0, c>0, d>0, and H1<H2. With h<H2, we have
(27)1+ah>0,12bh+dh>0,12-ch>0,
and R~1(h,a,b,c,d)+R~2(h,a,b,c,d)=(2a-b)h+2dh+a2h2+2 from (**). With h>H1, we have h>4d2/(a-b)2; that is, (2a-b)h+2dh<ah. The inequality 1+ah>0 with a<0 gives ah<-a2h2, and therefore (2a-b)h+2dh<-a2h2. With this and (**), we get 0<R~1(h,a,b,c,d)+R~2(h,a,b,c,d)<2. Consequently,
(28)R~T(h,a,b,c,d)=R~1(h,a,b,c,d)R~2(h,a,b,c,d)<[R~1(h,a,b,c,d)+R~2(h,a,b,c,d)2]2<1,
which means the Heun method in (13) is T-stable.
In the same way, we can prove that the Heun method in (13) is T-stable when b<0, c>0,d<0 or b<0, c<0,d>0 or b<0, c<0,d<0.
(b) Supposeb>0, c>0,d>0, and H1<H2. With h<H2, we get
(29)1+ah>0,12bh-dh<0,12-ch>0,
and R~1(h,a,b,c,d)+R~2(h,a,b,c,d)=(2a+b)h+2dh+a2h2+2 from (**). With h>H1, we have h>4d2/(a+b)2; that is, (2a+b)h+2dh<ah. The inequality 1+ah>0 with a<0 gives ah<-a2h2, and therefore (2a+b)h+2dh<-a2h2. With this and (**), we get 0<R~1(h,a,b,c,d)+R~2(h,a,b,c,d)<2. Consequently,
(30)R~T(h,a,b,c,d)=R~1(h,a,b,c,d)R~2(h,a,b,c,d)<[R~1(h,a,b,c,d)+R~2(h,a,b,c,d)2]2<1,
which means that the Heun method in (13) is T-stable.
In the same way, we can prove that the Heun method in (13) is T-stable when b>0, c>0,d<0 or b>0, c<0,d>0 or b>0, c<0,d<0.
The proof of the theorem is complete.


## 4. T-Stability of the Balanced Method

The balanced method applied to (6) and (7), respectively, gives
(31)Xn+1=(1+ah+cΔWn1+C)Xn+(bh1+C)Xn-m,(32)Xn+1=(1+ah+cΔWn1+C)Xn+(bh+dΔWn1+C)Xn-m,
where C=C0h+C1|ΔWn| and C0 and C1 are real numbers. In the analysis of T-stability of balanced method, we also use two-point random variables for the driving process.For the balanced method in (31), we have
(33)|Xn+1|≤(|1+ah+cΔWn1+C0h+C1|ΔWn||+|bh1+C0h+C1|ΔWn||)max⁡{|Xn|,|Xn-m|}.
Denote
(34)R(h,a,b,c)=|1+ah+cηnh1+C0h+C1|ηn|h|+|bh1+C0h+C1|ηn|h|,RT(h,a,b,c)=(|1+ah+ch1+C0h+C1h|+|bh1+C0h+C1h|)×(|1+ah-ch1+C0h+C1h|+|bh1+C0h+C1h|).
The discussion about the Heun method in Section 3 implies |Xn|→0,a.s.(n→∞) if RT(h,a,b,c)<1, which means that the balanced method in (31) is T-stable if RT(h,a,b,c)<1.Similarly, for the balanced method in (32), we get
(35)R~T(h,a,b,c,d)=(|1+ah+ch1+C0h+C1h|+|bh+dh1+C0h+C1h|)×(|1+ah-ch1+C0h+C1h|+|bh-dh1+C0h+C1h|),
and |Xn|→0,a.s.(n→∞) if R~T(h,a,b,c,d)<1, which means that the Heun method in (32) is T-stable if R~T(h,a,b,c,d)<1.Theorem 7.
Suppose (6) meets the condition in (9). The balanced method in (31) is T-stable if F(h)≥0, where
(36)F(h)=(C0+a)h+(C1-|c|)h.Proof.
The condition in (9) gives a<0. Denote
(***)R1(h,a,b,c)=|1+ah+ch1+C0h+C1h|+|bh1+C0h+C1h|,R2(h,a,b,c)=|1+ah-ch1+C0h+C1h|+|bh1+C0h+C1h|,
and RT(h,a,b,c)=R1(h,a,b,c)R2(h,a,b,c) from (34). It is easy to know a+b<0,a-b<0 for b≠0.
Ifc>0,b>0, we have
(37)C0h+C1h+ah-ch≥0,C0h+C1h+ah+ch≥0,
from the assumption F(h)≥0, which applies 1+C0h+C1h>0. Then R1(h,a,b,c) and R2(h,a,b,c) in (***) become
(38)R1(h,a,b,c)=1+C0h+C1h+ah+ch+bh1+C0h+C1h,R2(h,a,b,c)=1+C0h+C1h+ah-ch+bh1+C0h+C1h,R1(h,a,b,c)+R2(h,a,b,c)=2+2(a+b)h1+C0h+C1h.
Since a+b<0,1+C0h+C1h>0, we have 0<R1(h,a,b,c)+R2(h,a,b,c)<2 and
(39)RT(h,a,b,c)=R1(h,a,b,c)R2(h,a,b,c)<[R1(h,a,b,c)+R2(h,a,b,c)2]2<1,
which means that balanced method in (31) is T-stable.
In the same way, we can prove that the balanced method in (31) is T-stable when
(40)c>0,b<0orc<0,b>0,orc<0,b<0.
The proof of the theorem is complete.Theorem 8.
Suppose (7) meets the condition in (10). The balanced method in (32) is T-stable if h>H and F(h)≥0, where
(41)H=d2b2,F(h)=(C0+a)h+(C1-|c|)h.Proof.
The condition in (10) gives a<0. Denote
(****)R~1(h,a,b,c,d)=|1+ah+ch1+C0h+C1h|+|bh+dh1+C0h+C1h|,R~2(h,a,b,c,d)=|1+ah-ch1+C0h+C1h|+|bh-dh1+C0h+C1h|,
and then R~T(h,a,b,c,d)=R~1(h,a,b,c,d)R~2(h,a,b,c,d) from (35). It is easy to know a+b<0,a-b<0 for b≠0.
Ifc>0,b>0,d>0, we have bh-dh>0 from h>H and
(42)C0h+C1h+ah-ch≥0,C0h+C1h+ah+ch≥0,
from the assumption F(h)≥0, which yields 1+C0h+C1h>0. Then R~1(h,a,b,c,d) and R~2(h,a,b,c,d) in (****) become
(43)R~1(h,a,b,c,d)=1+C0h+C1h+ah+ch+bh+dh1+C0h+C1h,R~2(h,a,b,c,d)=1+C0h+C1h+ah-ch+bh-dh1+C0h+C1h,R~1(h,a,b,c,d)+R~2(h,a,b,c,d)=2+2(a+b)h1+C0h+C1h.
Since a+b<0,1+C0h+C1h>0, we have 0<R~1(h,a,b,c,d)+R~2(h,a,b,c,d)<2, and R~T(h,a,b,c,d)=R~1(h,a,b,c,d)R~2(h,a,b,c,d)<1, which means that the balanced method in (32) is T-stable.
In the same way, we can prove that the balanced method in (32) is T-stable when
(44)c>0,b>0,d<0orc>0,b<0,d>0orc>0,b<0,d<0.
Ifc<0,b>0,d>0, we get bh-dh>0 from h>H and
(45)C0h+C1h+ah+ch≥0,C0h+C1h+ah-ch≥0
from the assumption F(h)≥0, which applies 1+C0h+C1h>0. Then
(46)R~1(h,a,b,c,d)+R~2(h,a,b,c,d)=2+2(a+b)h1+C0h+C1h
from (****). Since a+b<0,1+C0h+C1h>0, we have
(47)0<R~1(h,a,b,c,d)+R~2(h,a,b,c,d)<2;
consequently, R~T(h,a,b,c,d)<1, which means that the balanced method in (32) is T-stable.
In the same way, we can prove that the balanced method in (32) is T-stable when
(48)c<0,b>0,d<0orc<0,b<0,d>0orc<0,b<0,d<0.
The proof of Theorem8 is complete.


## 5. Numerical Examples

Consider test equations (6) and (7) with τ=1,φ(t)=t+1,t∈[-1,0]. In the following figures tn’s are nodes, and Xn denotes the numerical solution at t=tn.Takea=-7,b=1, and c=1 in (6). From (18), we get H=1/4, which means that the Heun method in (12) is T-stable if h<1/4. Figure 1 shows that the Heun method in (12) is T-stable when h=1/9, h=1/7, and h=1/5 but is unstable when h=1/3, since h=1/3 exceeds the range of h in Theorem 5, which verifies Theorem 5.Figure 1
Simulations with the Heun method for (6) with a=-7,b=1, and c=1. (a1)h=19, (b1)h=17, (c1)h=15, and (d1)h=1/3.Takea=-9,b=-1,c=1.1, and d=0.9 in (7). From (26), we get H1≈1/20,  H2=1/9, which means that the Heun method in (13) is T-stable if 1/20<h<1/9. Figure 2 shows that the Heun method in (13) is T-stable when h=1/19, h=1/15, and h=1/11, but it is unstable when h=1/3, since h=1/3 exceeds the range of h in Theorem 6, which verifies Theorem 6.Figure 2
Simulations with the Heun method for (7) with a=-9,b=-1,c=1.1, and d=0.9. (a2)h=1/19, (b2)h=1/15, (c2)h=1/11, and (d2)h=1/3.Takea=-3,b=2, and c=1 in (6) and C0=1,C1=2 in the balanced method in (31). Then the balanced method in (31) is T-stable if h≤1/4 from Theorem 7. Figure 3 shows that the balanced method in (31) is T-stable when h=1/22, h=1/16, h=1/10, and h=1/4, since these h’s are in the range of h in Theorem 7, which verifies Theorem 7.Figure 3
Simulations with the balanced method for (6) with a=-3,b=2, and c=1. (a3)h=1/22, (b3)h=1/16, (c3)h=1/10, and (d3)h=1/4.Takea=-5, b=1, c=1, and d=0.2 in (7) and C0=2, C1=2 in the balanced method (32). Then the balanced method in (32) is T-stable if 1/25<h≤1/9 from Theorem 8. Figure 4 shows that the balanced method in (32) is T-stable when h=1/15,h=1/13, h=1/11, and h=1/9, since these h’s are in the range of h in Theorem 8, which verifies Theorem 8.Figure 4
Simulations with the balanced method for (7) with a=-5,b=1,c=1, and d=0.2. (a4)h=1/15, (b4)h=1/13, (c4)h=1/11, and (d4)h=1/9.


## 6. Conclusion

In this paper, T-stability of the Heun methods and the balanced methods for two kinds of linear SDDEs is studied. The Wiener increment of the numerical methods in this paper is approximated by a discrete random variable with two-point distribution in the process of the study of T-stability. The T-stable conditions for the Heun methods and balanced methods are given, respectively, and are verified by some numerical examples.



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*Source: 545830-2014-06-11.xml*