# The Semimartingale Approach to Almost Sure Stability Analysis of a Two-Stage Numerical Method for Stochastic Delay Differential Equation

Authors: Qian Guo; Xueyin Tao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Almost sure exponential stability of the split-step backward Euler (SSBE) method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. Numerical experiments confirm the theoretical analysis.


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## Body

## 1. Introduction

In this paper we study the following nonlinear SDDE:(1)dX(t)=f(X(t),X(t-τ(t)))dt+g(X(t),X(t-τ(t)))dW(t),
for every t≥0. Here τ(t) is a time-varying delay satisfying τ>0 and -τ~:=inf⁡{t-τ(t):t≥0}. The initial function X(t)=ψ(t) when t∈[-τ~,0]. We further assume that the initial data is independent of Wiener measure driving the equation and W(t) is a scalar Brownian motion on the complete probability space (Ω,F,Ft≥0,P) with a filtration satisfying the usual conditions. Moreover, f,g:Rd×Rd→Rd are Borel-measurable functions.Stability theory for numerical methods applied to stochastic differential equation (SDE) typically deals with mean-square behavior [1]. The mean-square stability analysis of numerical methods for SDDE has received a great deal of attention (see, e.g., [2, 3] and the references therein). Recently, the almost sure (a.s.) stability (or the trajectory stability) is becoming prevalent in the science literature [4–11]. However, the prior works concerned with SDDE are [7, 8, 10]. Rodkina et al. [7] studied almost sure stability of a drift-implicit θ-method applied to an SDE with memory. Using the martingale techniques, Wu and his coauthors [8, 10] discussed almost sure exponential stability of the Euler-Maruyama (EM) method for the SDE with a constant delay and stochastic functional differential equation. We note that the two above schemes are all single-stage method; this paper studies the almost sure stability of a two-stage scheme named split-step backward Euler (SSBE) method [12, 13] applied to the nonlinear SDDE (1) with time-varying delay.Applying the SSBE method (see [12, 13]) to (1) yields(2a)xn*=xn+hf(xn*,x~n),(2b)xn+1=xn*+g(xn*,x~n)Δwn,where Δwn:=W(tn+1)-W(tn) and for 0≤μ≤1, qn∈Z+,
(3)x~n={ψ(tn-τ(tn)),tn-τ(tn)<0;μxn-qn+1+(1-μ)xn-qn,0≤tn-τ(tn)0∈[tn-qn,tn-qn+1).
Here h is the step size and xn denotes the approximation of X(t) at time tn=nh(n=0,1,…). We remark that μ in (3) depends on how memory values are handled on nongrid points. The almost sure convergence of SSBE method has been investigated by Guo and Tao [14]; the main aim of this paper is to study the almost sure stability of the SSBE method applied to (1).


## 2. Preliminary Results

Before stating the main results, we present the essential notation and definitions which are necessary for further consideration. Let|·| be the Euclidean norm in Rd and C([-τ~,0];Rd) the family of continuous functions φ from [-τ~,0] to Rd, equipped with the supremum norm ∥φ∥=sup⁡-τ~≤θ≤0⁡|φ(θ)|. Also, denote by CF0b([-τ~,0];Rd) the family of bounded, F0-measurable, C([-τ~,0];Rd)-valued random variables. If A is a vector or matrix, its transpose is denoted by AT. The inner product of X,Y∈Rd is denoted by 〈X,Y〉 or XTY.Now we give some definitions on the almost sure exponential stability of SDDEs and its numerical approximation.Definition 1.
The solutionX(t,ψ) to (1) is said to be almost surely exponentially stable if there exists a constant η>0 such that
(4)limsup⁡t→∞1tlog⁡|X(t,ψ)|≤-ηa.s
for any initial data ψ∈CF0b([-τ~,0];Rd).Definition 2.
The solutionxn to (2a) and (2b) is said to be almost surely exponentially stable if there exists a constant γ>0 such that
(5)limsup⁡n→∞1nhlog⁡|xn|≤-γa.s
for any bounded variables ψ(kh) when kh∈[-τ~,0].For the purpose of stability, we assume thatf(0,0,t)=g(0,0,t)=0, which implies that (1) admits the equilibrium solution X(t)=0 corresponding to the initial condition ψ(t)=0 for t∈[-τ~,0]. As a standing hypothesis, we will impose the following local Lipschitz condition (cf. [11, 12, 14]) on the coefficients f and g.(A1)
For each integerD, there exists a positive constant KD such that, for all x1,x2,y1,y2∈Rd with |x1|∨|x2|∨|y1|∨|y2|≤D, and all t≥0, |f(x1,y1)-f(x2,y2)|2∨|g(x1,y1)-g(x2,y2)|2≤KD(|x1-x2|2+|y1-y2|2), where ∨ is the maximal operator.To guarantee the almost sure stability of the unique solution to (1), we need the following assumption for the time-varying delay τ(t).(A2)
Let the delay functionτ(t):[0,+∞)→[0,τ¯] be Borel measurable and bounded.In what follows we introduce the result of almost sure stability of SDDEs (1). The proof of the following lemma can be found in [15].Lemma 3.
Let Assumptions (A1) and (A2) hold. Assume that there are four nonnegative constantsλ1–λ4 such that
(6)2xTf(x,0)≤-λ1|x|2,(7)|f(x,y)-f(x,0)|≤λ2|y|,(8)|g(x,y)|2≤λ3|x|2+λ4|y|2
for all t≥t0 and x,y∈Rd. If
(9)λ1>2λ2+λ3+λ4,
then the trivial solution of (1) is almost surely exponentially stable.To explain our idea, we cite the discrete semimartingale convergence theorem as follows.Theorem 4 (see [8, 9]).
LetZ=(Zn)n∈N be an almost sure nonnegative stochastic sequence of (Fn,B)-measurable random variables Zn on probability space (Ω,F,(Fn)n∈N,P). Assume that Z permits the decomposition
(10)Zn≤Z0+An1-An2+Mn,n∈N,
where A1=(An1)n∈N and A2=(An2)n∈N are two nondecreasing, predictable processes with A0i=0(i=1,2); M=(Mn)n∈N is local (Fn)n∈N-martingale with M0=0 on (Ω,F,(Fn)n∈N,P). Then, the requirement of lim⁡n→+∞⁡An1<+∞ (a.s.) implies that
(11)lim⁡n→+∞sup⁡Zn<+∞,lim⁡n→+∞An2<+∞
for almost all ω∈Ω.


## 3. Almost Sure Asymptotic Exponential Stability of Numerical Solution

In this section, our aim is to examine if the SSBE method can reproduce the almost sure exponential stability of the exact solution of (1). Comparing to the existing results of single-stage methods [8, 10], we need to appropriately estimate the intermediate solution xn*, which also leads to more complex structure of the inner product of xn+1, so that the discrete semimartingale convergence theorem is still valid for this case.Now we give the main result of almost sure stability of the SSBE approximate solutionxn.Theorem 5.
Suppose that conditions of Lemma3 are satisfied and the drift coefficient f satisfies the linear growth condition; namely, there exists a constant K>0 such that
(12)|f(x,y)|2≤K(|x|2+|y|2)
for all x,y∈Rn and t≥0. Then there exists an h0 such that if h<h0, the SSBE approximate solution xn is almost surely exponentially stable.Proof.
Note that(13)|xn+1|2=|xn|2+2hxnTf(xn*,x~n)+|hf(xn*,x~n)|2+|g(xn*,x~n)|(Δwn)2+2〈xn+hf(xn*,x~n),g(xn*,x~n)Δwn〉
from the SSBE method (2a) and (2b).
By using (6) and (7), we have
(14)xnTf(xn*,x~n)=(xnT-xn*T)f(xn*,x~n)+xn*Tf(xn*,x~n)≤h|f(xn*,x~n)|2+xn*Tf(xn*,0)+xn*T(f(xn*,x~n)-f(xn*,0))≤h|f(xn*,x~n)|2-λ12|xn*|2+|xn*||f(xn*,x~n)-f(xn*,0)|.
Equation (13), together with (14), shows that
(15)|xn+1|2≤|xn|2+3h2|f(xn*,x~n)|2-λ1h|xn*|2+2λ2h|xn*||x~n|+|g(xn*,x~n)|(Δwn)2+2〈xn+hf(xn*,x~n),g(xn*,x~n)Δwn〉.
Therefore, by conditions (8) and (12), we have
(16)|xn+1|2≤|xn|2+3h2(K|xn*|2+K|x~n|2)-λ1h|xn*|2+2λ2h|xn*||x~n|+λ3(Δwn)2|xn*|2+λ4(Δwn)2|x~n|2+2〈xn+hf(xn*,x~n),g(xn*,x~n)Δwn〉≤|xn|2+(3Kh2-λ1h+λ2h+λ3(Δwn)2)|xn*|2+(3Kh2+λ2h+λ4(Δwn)2)|x~n|2+2〈xn+hf(xn*,x~n),g(xn*,x~n)Δwn〉.
Similarly, under conditions (6), (7), and (12),
(17)|xn*|2=〈xn+hf(xn*,x~n),xn+hf(xn*,x~n)〉=|xn|2+2hxnTf(xn*,x~n)+|hf(xn*,x~n)|2≤|xn|2+3h2|f(xn*,x~n)|2-λ1h|xn*|2+2λ2h|xn*||x~n|≤|xn|2+3Kh2(|xn*|2+|x~n|2)-λ1h|xn*|2+λ2h(|xn*|2+|x~n|2),
which implies that
(18)(1-3Kh2+λ1h-λ2h)|xn*|2≤|xn|2+(3Kh2+λ2h)|x~n|2.
By Vieta theorem, because the discriminant of the quadratic equation 1-3Kh2+λ1h-λ2h=0 is positive and -3K<0, there must exist an h1>0 such that 1-3Kh2+λ1h-λ2h>0 for any 0<h<h1; then
(19)|xn*|2≤11-3Kh2+λ1h-λ2h|xn|2+3Kh2+λ2h1-3Kh2+λ1h-λ2h|x~n|2.
For simplicity, in what follows, the formula 1-3Kh2+λ1h-λ2h is denoted by G. Combining (16) and (19) leads us to
(20)|xn+1|2≤|xn|2+1-G+λ3(Δwn)2G|xn|2+(1-G+λ3(Δwn)2)(3Kh2+λ2h)G|x~n|2+(3Kh2+λ2h+λ4(Δwn)2)|x~n|2+2〈xn+hf(xn*,x~n),g(xn*,x~n)Δwn〉.
For any positive constant C>1, we have
(21)C(i+1)h|xi+1|2-Cih|xi|2=C(i+1)h(|xi+1|2-|xi|2)+(C(i+1)h-Cih)|xi|2,
which yields
(22)C(i+1)h|xi+1|2-Cih|xi|2≤C(i+1)h[1-C-h+1-G+λ3(Δwi)2G]|xi|2+C(i+1)h[(1-G+λ3(Δwi)2)(3Kh2+λ2h)G+3Kh2+λ2h+λ4(Δwi)2(1-G+λ3(Δwi)2)(3Kh2+λ2h)G]|x~i|2+2C(i+1)h〈xi+hf(xi*,x~i),g(xi*,x~i)Δwi〉
by using (20). Summing up both sides of inequality (22) from i=0 to n-1 (n≥1), we get
(23)Cnh|xn|2≤|x0|2+[1-C-h+1-GG]∑i=0n-1‍C(i+1)h|xi|2+(λ3G)∑i=0n-1‍C(i+1)h(Δwi)2|xi|2+3Kh2+λ2hG∑i=0n-1‍C(i+1)h|x~i|2+[(3Kh2+λ2h)λ3G+λ4]∑i=0n-1‍C(i+1)h(Δwi)2|x~i|2+2∑i=0n-1‍C(i+1)h〈xi+hf(xi*,x~i),g(xi*,x~i)Δwi〉.
Let Mn(1)=∑i=0n-1C(i+1)h|xi|2((Δwi)2-h). Since E((Δwn)2-h)=0 and xn is Fnh-measurable, we obtain
(24)E[Mn(1)∣F(n-1)h]=Mn-1(1)+E[Cnh|xn-1|2((Δwn-1)2-h)∣F(n-1)h]=Mn-1(1)+Cnh|xn-1|2E[((Δwn-1)2-h)∣F(n-1)h]=Mn-1(1),
which implies that Mn(1) is a martingale.
Similarly,(25)Mn(2)=∑i=0n-1C(i+1)h|x~i|2((Δwi)2-h),Mn(3)=2∑i=0n-1C(i+1)h〈xi+hf(xi*,x~i),g(xi*,x~i)Δwi〉
are also martingales. Therefore,
(26)Mn=λ3GMn(1)+[λ3(3Kh2+λ2h)G+λ4]Mn(2)+Mn(3)
is a martingale with M0=0. Then we have
(27)Cnh|xn|2≤|x0|2+[-C-h+1+λ3hG]∑i=0n-1C(i+1)h|xi|2+[(1+λ3h)(3Kh2+λ2h)G+λ4h]×∑i=0n-1C(i+1)h|x~i|2+Mn.
Noting that there are two approximating cases of the time dependent delay term X(tn-τ(tn)) in (3), the following analysis will be divided into two situations. First, we have
(28)Cnh|xn|2≤|x0|2+[-C-h+1+λ3hG]∑i=0n-1C(i+1)h|xi|2+[(1+λ3h)(3Kh2+λ2h)G+λ4h]×∑i=0n-1C(i+1)h|ψ(ti-τ(ti))|2+Mn
under condition tn<τ(tn). There exists an h2 such that, for any 0<h<h1∧h2, C-h-(1+λ3h)/G>0, where ∧ is the minimal operator. Further, we set An1=0 for any nonnegative integer n, A02=0,
(29)An2=[C-h-1+λ3hG]∑i=0n-1‍C(i+1)h|xi|2
for n≥1, and
(30)Zn=Cnh|xn|2,Z0=|x0|2+[(1+λ3h)(3Kh2+λ2h)G+λ4h]×∑i=0n-1C(i+1)h|ψ(ti-τ(ti))|2.
Therefore, a direct application of Theorem 4 to the sequence Zn yields that
(31)limsup⁡n→∞Cnh|xn|2≤+∞.
Choose the γ>0, such that C=eγ and hence
(32)limsup⁡n→∞eγnh|xn|2≤+∞.
We therefore obtain that, for any 0<h<h1∧h2,
(33)limsup⁡n→∞1nhlog⁡|xn|≤-γ2,a.s
as required.
Now, let us discuss the second situation:tn≥τ(tn). Inequality (27) gives
(34)Cnh|xn|2≤|x0|2+[-C-h+1+λ3hG]∑i=0n-1C(i+1)h|xi|2+2[(1+λ3h)(3Kh2+λ2h)G+λ4h]×∑i=0n-1C(i+1)h(μ2|xi-qi+1|2+(1-μ)2|xi-qi|2)+Mn.
Since
(35)∑i=0n-1C(i+1)h|xi-qi|2=∑i=-qi-1C(i+qi+1)h|xi|2+∑i=0n-1C(i+qi+1)h|xi|2-∑i=n-qin-1C(i+qi+1)h|xi|2,∑i=0n-1C(i+1)h|xi-qi+1|2=∑i=-qi+1-1C(i+qi)h|xi|2+∑i=0n-1C(i+qi)h|xi|2-∑i=n-qi+1n-1C(i+qi)h|xi|2,
we have
(36)Zn≤Z0-An2+Mn,
where
(37)Zn=Cnh|xn|2+2μ2[(1+λ3h)(3Kh2+λ2h)G+λ4h]×∑i=n-qi+1n-1C(i+qi)h|xi|2+2(1-μ)2[(1+λ3h)(3Kh2+λ2h)G+λ4h]×∑i=n-qin-1C(i+qi+1)h|xi|2,An2=[((1+λ3h)(3Kh2+λ2h)G+λ4h)C-h-1+λ3hG-2μ2×((1+λ3h)(3Kh2+λ2h)G+λ4h)Cqi-1-2(1-μ)2((1+λ3h)(3Kh2+λ2h)G+λ4h)Cqi]×∑i=0n-1‍C(i+1)h|xi|2.
There exists an h3 such that, for any 0<h<h1∧h3,
(38)C-h-1+λ3hG-2μ2((1+λ3h)(3Kh2+λ2h)G+λ4h)Cqi-1-2(1-μ)2((1+λ3h)(3Kh2+λ2h)G+λ4h)Cqi>0.
Similarly, the solution xn is almost surely exponentially stable by using Theorem 4.
Consequently, we conclude that, for any0<h<h1∧h2∧h3, the SSBE approximate solution xn is almost surely exponentially stable.


## 4. Numerical Experiments

In this section, we present some numerical examples to illustrate our theoretical analysis. We calculated 500 sample paths of the approximate solution and plotted them along the timet (see, e.g., Figure 1(a)). Figures 1 to 2 depict the results by SSBE method in the log-scaled vertical axis. Here we set d=1, drift coefficient f=αX(t)+σsin(X(t-τ)), diffusion coefficient g=λX(t)+βsin(X(t-τ)), initial function ψ(t)=t+1, and delay function τ(t)=3(cos⁡(t/2))2+3.Almost sure stability of SSBE method applied to (1) with α=-10, σ=1, λ=1, and β=2.
(a)
h
=
0.4; μ=0.0
(b)
h
=
0.4; μ=0.5
(c)
h
=
0.4; μ=1.0
(d)
h
=
0.2; μ=1.0Almost sure stability of SSBE method applied to (1) with α=-10, σ=1, λ=3, and β=1.
(a)
h
=
1.0; μ=0.0
(b)
h
=
1.0; μ=0.5
(c)
h
=
1.0; μ=1.0
(d)
h
=
0.9; μ=1.0Figures1 to 2 show that the SSBE approximate solution xn has better almost sure stability in the case of choosing the parameter μ=0.5 in (3). Comparing Figures 1(c) and 1(d), the almost sure stability of approximate solution can be obtained by reducing the step size h.



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