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

and Applied Analysis 3 Proof. Note that 󵄨󵄨󵄨󵄨xn+1 󵄨󵄨󵄨󵄨 2 = 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 2hx T n f (x ∗ n , x n ) + 󵄨󵄨󵄨󵄨hf (x ∗ n , x n ) 󵄨󵄨󵄨󵄨 2 + 󵄨󵄨󵄨󵄨g (x ∗ n , x n ) 󵄨󵄨󵄨󵄨 (Δwn) 2 + 2 ⟨x n + hf (x ∗ n , x n ) , g (x ∗ n , x n ) Δw n ⟩ (13) from the SSBE method (2a) and (2b). By using (6) and (7), we have x T n f (x ∗ n , x n ) = (x T n − x ∗T n ) f (x ∗ n , x n ) + x ∗T n f (x ∗ n , x n ) ≤ h 󵄨󵄨󵄨󵄨f (x ∗ n , x n ) 󵄨󵄨󵄨󵄨 2 + x ∗T n f (x ∗ n , 0) + x ∗T n (f (x ∗ n , x n ) − f (x ∗ n , 0)) ≤ h 󵄨󵄨󵄨󵄨f (x ∗ n , x n ) 󵄨󵄨󵄨󵄨 2 − λ 1 2 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 + 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨f (x ∗ n , x n ) − f (x ∗ n , 0) 󵄨󵄨󵄨󵄨 . (14) Equation (13), together with (14), shows that 󵄨󵄨󵄨󵄨xn+1 󵄨󵄨󵄨󵄨 2 ≤ 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 3h 2󵄨󵄨󵄨󵄨f (x ∗ n , x n ) 󵄨󵄨󵄨󵄨 2 − λ 1 h 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 + 2λ 2 h 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨g (x ∗ n , x n ) 󵄨󵄨󵄨󵄨 (Δwn) 2 + 2 ⟨x n + hf (x ∗ n , x n ) , g (x ∗ n , x n ) Δw n ⟩ . (15) Therefore, by conditions (8) and (12), we have 󵄨󵄨󵄨󵄨xn+1 󵄨󵄨󵄨󵄨 2 ≤ 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 3h 2 (K 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 + K 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 ) − λ 1 h 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 + 2λ 2 h 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 + λ 3 (Δw n ) 2󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 + λ 4 (Δw n ) 2󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 2 ⟨x n + hf (x ∗ n , x n ) , g (x ∗ n , x n ) Δw n ⟩ ≤ 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + (3Kh 2 − λ 1 h + λ 2 h + λ 3 (Δw n ) 2 ) 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 + (3Kh 2 + λ 2 h + λ 4 (Δw n ) 2 ) 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 2 ⟨x n + hf (x ∗ n , x n ) , g (x ∗ n , x n ) Δw n ⟩ . (16) Similarly, under conditions (6), (7), and (12), 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 = ⟨x n + hf (x ∗ n , x n ) , x n + hf (x ∗ n , x n )⟩ = 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 2hx T n f (x ∗ n , x n ) + 󵄨󵄨󵄨󵄨hf (x ∗ n , x n ) 󵄨󵄨󵄨󵄨 2 ≤ 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 3h 2󵄨󵄨󵄨󵄨f (x ∗ n , x n ) 󵄨󵄨󵄨󵄨 2 − λ 1 h 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 + 2λ 2 h 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 3Kh 2 ( 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 + 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 ) − λ 1 h 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 + λ 2 h ( 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 + 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 ) , (17) which implies that (1 − 3Kh 2 + λ 1 h − λ 2 h) 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 ≤ 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + (3Kh 2 + λ 2 h) 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 . (18) By Vieta theorem, because the discriminant of the quadratic equation 1 − 3Kh2 + λ 1 h − λ 2 h = 0 is positive and −3K < 0, there must exist an h 1 > 0 such that 1−3Kh2 +λ 1 h−λ 2 h > 0 for any 0 < h < h 1 ; then 󵄨󵄨󵄨󵄨x ∗ n 󵄨󵄨󵄨󵄨 2 ≤ 1 1 − 3Kh + λ 1 h − λ 2 h 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 3Kh 2 + λ 2 h 1 − 3Kh + λ 1 h − λ 2 h 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 . (19) For simplicity, in what follows, the formula 1 − 3Kh2 + λ 1 h − λ 2 h is denoted by G. Combining (16) and (19) leads us to 󵄨󵄨󵄨󵄨xn+1 󵄨󵄨󵄨󵄨 2 ≤ 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 1 − G + λ 3 (Δw n ) 2 G 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + (1 − G + λ 3 (Δw n ) 2 ) (3Kh 2 + λ 2 h) G 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + (3Kh 2 + λ 2 h + λ 4 (Δw n ) 2 ) 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 2 + 2 ⟨x n + hf (x ∗ n , x n ) , g (x ∗ n , x n ) Δw n ⟩ . (20) For any positive constant C > 1, we have C (i+1)h󵄨󵄨󵄨󵄨xi+1 󵄨󵄨󵄨󵄨 2 − C ih󵄨󵄨󵄨󵄨xi 󵄨󵄨󵄨󵄨 2 = C (i+1)h ( 󵄨󵄨󵄨󵄨xi+1 󵄨󵄨󵄨󵄨 2 − 󵄨󵄨󵄨󵄨xi 󵄨󵄨󵄨󵄨 2 ) + (C (i+1)h − C ih ) 󵄨󵄨󵄨󵄨xi 󵄨󵄨󵄨󵄨 2 , (21)

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][5][6][7][8][9][10][11].However, the prior works concerned with SDDE are [7,8,10].Rodkina et al. [7] studied almost sure stability of a driftimplicit -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 where Here ℎ is the step size and   denotes the approximation of () at time   = ℎ ( = 0, 1, . ..).We remark that  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).

Preliminary Results
(A1) For each integer , there exists a positive constant   such that, for all 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 ().
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 constants  1 - 4 such that for all  ≥  0 and ,  ∈ R  .If then the trivial solution of (1) is almost surely exponentially stable.
To explain our idea, we cite the discrete semimartingale convergence theorem as follows.
Before stating the main results, we present the essential notation and definitions which are necessary for further consideration.Let | ⋅ | be the Euclidean norm in R  and ([−τ, 0]; R  ) the family of continuous functions  from [−τ, 0] to R  , equipped with the supremum norm ‖‖ = sup −τ≤≤0 |()|.Also, denote by   F 0 ([−τ, 0]; R  ) the family of bounded, F 0 -measurable, ([−τ, 0]; R  )-valued random variables.If  is a vector or matrix, its transpose is denoted by   .The inner product of ,  ∈ R  is denoted by ⟨, ⟩ or   .Now we give some definitions on the almost sure exponential stability of SDDEs and its numerical approximation.
[8,10]paring to the existing results of singlestage methods[8,10], we need to appropriately estimate the intermediate solution  *  , which also leads to more complex structure of the inner product of  +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 solution   .