Strong Convergence of the Split-Step Theta Method for Stochastic Delay Differential Equations with Nonglobally Lipschitz Continuous Coefficients

and Applied Analysis 3 Using the Gronwall inequality yields E [ 󵄩󵄩󵄩󵄩x (T ∧ ρR) 󵄩󵄩󵄩󵄩 2 ] ≤ (E [ 󵄩󵄩󵄩󵄩φ (0) 󵄩󵄩󵄩󵄩 2 ] + 2αT +Mτ) exp (4βT) , (12) P (ρ R ≤ T) ≤ E[ 󵄩󵄩󵄩󵄩x (T ∧ ρR) 󵄩󵄩󵄩󵄩 2 + 󵄩󵄩󵄩󵄩x (T ∧ ρR − τ) 󵄩󵄩󵄩󵄩 2 R 1 {ρR≤T} ] ≤ (E [ 󵄩󵄩󵄩󵄩φ (0) 󵄩󵄩󵄩󵄩 2 ] + 2αT +Mτ) exp (4βT) +M R . (13) Applying Fatou’s lemma to (12), we obtain E [‖x (T)‖ 2 ] ≤ (E [ 󵄩󵄩󵄩󵄩φ (0) 󵄩󵄩󵄩󵄩 2 ] + 2αT +Mτ) exp (4βT) . (14) The proof is completed. 3. Moment Properties of SST Before proving the strong convergence of the SST method (3), it is necessary to show that the SST method (3) has a unique solution. So we introduce the following assumption and lemma. Assumption 3. There exists a positive constant L, such that ⟨x 1 − x 2 , f (t, x 1 , y) − f (t, x 2 , y)⟩ ≤ L 󵄩󵄩󵄩󵄩x1 − x2 󵄩󵄩󵄩󵄩 2 , (15) for x 1 , x 2 , y ∈ R and t ∈ [0, T]. From [19] we easily obtain that the SST method (3) has a unique solution under 0 < θLh < 1.We now show that, under Assumptions 1 and 3, the 2nd moment of numerical solution y n and Y n is bounded. Lemma 4. Assume that f(t, x(t), x(t− τ)) and g(t, x(t), x(t− τ)) in (2) satisfy Assumptions 1 and 3; then for 1/2 ≤ θ ≤ 1 and h < h ∗ < min{1/θL, 1/2θβ}, the following moment bounds hold: sup 0≤t≤T E [‖x (t)‖ 2 ] ∨ sup h≤h ∗ sup 0≤nh≤T E [ 󵄩󵄩󵄩󵄩yn 󵄩󵄩󵄩󵄩 2 ] ∨ sup h≤h ∗ sup 0≤nh≤T E [ 󵄩󵄩󵄩󵄩Yn−m 󵄩󵄩󵄩󵄩 2 ] ∨ sup h≤h ∗ sup 0≤nh≤T E [ 󵄩󵄩󵄩󵄩Yn 󵄩󵄩󵄩󵄩 2 ] ≤ A < ∞, (16) where A is a positive constant independent ofN. Proof. First, by Lemma 2, we know that sup 0≤t≤T E[‖x(t)‖ 2 ] is bounded. Denoting F (Y n+1 ) := Y n+1 − θhf (t n+1 + θh, Y n+1 , Y n+1−m ) (17) and then inserting (17) into (3), we have 󵄩󵄩󵄩󵄩F(Yn+1) 󵄩󵄩󵄩󵄩 2 = 󵄩󵄩󵄩󵄩F(Yn) 󵄩󵄩󵄩󵄩 2 + 2h ⟨Y n , f (t n , Y n , Y n−m )⟩ + 󵄩󵄩󵄩󵄩g(tn + θh, Yn, Yn−m)ΔWn 󵄩󵄩󵄩󵄩 2 + (1 − 2θ) h 󵄩󵄩󵄩󵄩f(tn + θh, Yn, Yn−m) 󵄩󵄩󵄩󵄩 2 +M n , (18) where M n = 2⟨F(Y n ), g(t n + θh, Y n , Y n−m )ΔW n ⟩ + 2⟨hf(t n + θh, Y n , Y n−m ), g(t n + θh, Y n , Y n−m )ΔW n ⟩. By recursive calculation, we obtain 󵄩󵄩󵄩󵄩F (Yn+1) 󵄩󵄩󵄩󵄩 2 = 󵄩󵄩󵄩󵄩F (Y0) 󵄩󵄩󵄩󵄩 2 + n ∑ j=0 (2h ⟨Y j , f (t j + θh, Y j , Y j−m )⟩ + 󵄩󵄩󵄩󵄩 g (t j + θh, Y j , Y j−m ) ΔW j 󵄩󵄩󵄩󵄩 2 )


Introduction
Stochastic delay differential equations (SDDEs) play an important role in modeling some real-world phenomena in many scientific areas, such as economics [1], biology [2,3], and medicine [4,5].However, many SDDEs arising in applications cannot be solved analytically; hence one needs to develop effective numerical methods to solve them.
In recent years, the numerical solution of SDDEs has attracted much attention and a number of numerical methods have been constructed (see, e.g., [6][7][8]).An important topic in this context is the investigation of the convergence of numerical methods and a number of interesting results have been found (see, e.g., [9][10][11][12][13]).In the analysis of strong convergence, a widely used assumption is that the drift and diffusion coefficients satisfy global Lipschitz and linear growth conditions [9][10][11].In order to weaken this assumption, Mao and Sabanis [14] proved strong convergence of Euler-Maruyama type methods with local Lipschitz conditions and the bounded th moments ( > 2) for solving SDDEs.Wang and Gan [12] showed that the improved split-step backward Euler method is convergent in the mean square sense under the condition that the diffusion coefficient (, ) is globally Lipschitz, and the drift coefficient (, ) satisfies a one-sided Lipschitz condition in the nondelay variable  and a global Lipschitz condition in the delay variable .Bao and Yuan [15] proved the convergence rate of the Euler-Maruyama (EM) scheme for a class of SDDEs, where the corresponding coefficients may be highly nonlinear with respect to the delay variables.The strong convergence was also studied in [16,17].Nevertheless, all the above results are derived for SDDEs of which the diffusion coefficient with respect to the nondelay variables satisfies a linear growth or global Lipschitz condition.For example, they cannot be applied to some highly nonlinear problems such as  () = (− () −  3 () +  ( − 1))  +  2 ()  () . ( In this paper, we study the strong convergence of the splitstep theta method [8] under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients.These conditions admit that the diffusion coefficient with respect to the nondelay variables is highly nonlinear; that is, it does not necessarily satisfy a linear growth or global Lipschitz condition. The structure of this paper is organized as follows.First, the existence of a unique solution of SDDEs under weaker conditions is recalled in Section 2.Then, some moment properties of the split-step theta method (3) are investigated in Section 3, while its strong convergence is derived in Section 4. Finally, some numerical results to support our theorems are presented.
Huang [8] studied the exponential mean square stability of the SST method (3) under the following condition: where  is a real symmetric, positive definite matrix.It is proved that when α + β < 0 and  > 0.5, SST method (3) is exponentially mean square stable for all positive stepsizes.In this paper, we further study its strong convergence property under weaker conditions.
To ensure the existence of a unique solution on [−, ] to SDDEs (2), we introduce the following assumption. where and  ∈ (0, ]. From Theorem 1.2 in [18], it follows that for any given initial value {() : − ≤  ≤ 0} ∈ ([−, 0]; R  ) there exists a unique solution () to SDDEs (2).Now, we state the following lemma which will play an important role in proving the strong convergence of the SST method (3).Lemma 2. Under Assumption 1, the solution of the SDDEs (2) on [0, ] has the properties that where Proof.Let (, ) = ‖‖ 2 .Then by using Itô formula, we infer that According to Assumption 1 and taking mathematical expectation on both sides of (10), we have Using the Gronwall inequality yields Applying Fatou's lemma to (12), we obtain The proof is completed.

Moment Properties of SST
Before proving the strong convergence of the SST method (3), it is necessary to show that the SST method (3) has a unique solution.So we introduce the following assumption and lemma.
Applying the discrete Gronwall's inequality, we have Therefore, there exists a positive constant The proof is completed.

Numerical Results
In this section we consider the following numerical experiments that confirm the conclusions obtained in the previous sections.