CONVERGENCE RATE AND STABILITY OF THE SPLIT-STEP THETA METHOD FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH PIECEWISE CONTINUOUS ARGUMENTS

. In this paper, we investigate the strong convergence rate of the split-step theta (SST) method for a kind of stochastic diﬀerential equations with piecewise continuous arguments (SDEPCAs) under some polynomially growing conditions. It is shown that the SST method with θ ∈ [ 12 , 1] is strongly convergent with order 12 in p th( p ≥ 2) moment if both drift and diﬀusion coeﬃcients are polynomially growing with regard to the delay terms, while the diﬀusion coeﬃcients are globally Lipschitz continuous in non-delay arguments. The exponential mean square stability of the improved split-step theta (ISST) method is also studied without the linear growth condition. With some relaxed restrictions on the step-size, it is proved that the ISST method with θ ∈ ( 12 , 1] is exponentially mean square stable under the monotone condition. Without any restriction on the step-size, there exists θ ∗ ∈ ( 12 , 1] such that the ISST method with θ ∈ ( θ ∗ , 1] is exponentially stable in mean square. Some numerical simulations are presented to illustrate the analytical theory.

[t] denotes the greatest-integer part of t. Our aim is to investigate the convergence order and the exponential stability of numerical methods for SDEPCAs without linear growth conditions. An SDEPCA belongs to the stochastic differential delay equations (SDDEs), but the delay term [t] is different from t − τ . On one hand, t − τ is continuous while [t] is discontinuous. On the other hand, the delays t − [t] vary with t, which implies that Eq. (1) is nonautonomous. In fact, SDEPCAs are with properties of both differential and difference equations. Based on these characteristics, SDEPCAs play an important role in many fields, especially in control theory (see [16,18,28]).
Motivated by the influential paper [6], lots of scholars attend to study the strong convergence of the numerical methods for stochastic differential equations (SDEs) under non-global Lipschitz conditions. According to [10], the classical explicit Euler method fails to converge when the SDEs are superlinearly growing. Therefore, some modified explicit Euler methods have been proposed to investigate the SDEs which behave superlinearly growing. For example, there are the tamed Euler method [11], the tamed Milstein method [26], the stopped Euler method [14], the truncated Euler method [19,20], and some other methods (see [2,3,5]) et al. The convergence order are all proved to be standard 1 2 or close to 1 2 . Although there are some advantages of cheap computational cost, it is a challenge to construct the explicit methods.
As a result, the implicit methods attract a great deal of attention in numerical analysis because of their better convergence and stability compared with the explicit methods. For SDEs under local Lipschitz conditions, several achievements on the convergence analysis of the implicit methods are obtained, such as [7,12,17,25]. There are also many conclusions on the stability of implicit methods, see [8,23]. Moreover, for SDDEs with non-global Lipschitz conditions, the implicit numerical methods are investigated in [9,22,30,31] and the related references therein. Although there are some conclusions on the strong convergence and stability of numerical methods for SDEPCAs, they are all under the linear growth conditions [21,29]. Without the linear growth condition, Song and Zhang [24] prove that the Euler method is convergent in probability under the Khasminskii-type conditions. We prove that the SST method is strongly convergent to SDEPCAs under the monotone condition, but the convergence rate is not given. In this paper, we give the convergence order of the SST method under some polynomial conditions, this is novel for SDEPCAs. Meanwhile, the convergence order of SST method is standard 1 2 , which is better than that of some explicit methods. Moreover, we improve the SST method to obtain the exponential mean square stability. The restrictions on h is more relaxed for the ISST method than the SST method.
Bao and Yuan [1] obtained the strong convergence rate of the Euler method for SDDEs in which the coefficients are highly nonlinear with respect to delay variables. They proved that the convergence order is 1 2 for SDDEs, but it is close to 1 2 for SDDEs with jumps. Kumar and Sabanis [13] obtained the convergence order of the Euler method for SDDEs when the drift coefficients are one-sided Lipschitz continuous, and both the drift and diffusion coefficients satisfy the polynomial Lipschitz conditions with respect to the delay arguments. However, both the two papers require that the drift terms are linearly growing on the non-delay variables. In this paper, we prove the convergence order of the SST method for SDEPCAs in which the drift coefficients are polynomially growing. This condition is much more relaxed than the linearly growing conditions. The conclusions are as follows.
• Both the underlying system and the SST method with θ ∈ [ 1 2 , 1] are pth (p ≥ 2) moment bounded under milder conditions, namely, the drift coefficients are one-sided and polynomially growing in non-delay variables as well as both the drift and diffusion coefficients are both polynomially Lipschitz continuous with respect to delay arguments.
• Based on the pth moment boundedness, we show that the SST method with θ ∈ [ 1 2 , 1] is strongly convergent in pth (p ≥ 2) moment with order 1 2 under the conditions above.
• When considering the stability, we improve the SST method. It is obtained that the improved split-step theta(ISST) method with θ ∈ ( 1 2 , 1] reproduces the exponential mean square stability of the underlying systems under the monotone condition. • There exists θ * ∈ ( 1 2 , 1] such that the ISST method with θ ∈ (θ * , 1] is exponentially stable in mean square for all step-sizes. An outline of this paper is as follows. In Section 2, some preliminary notations and the SST method for SDEPCAs are introduced. In Section 3, strong convergence of the SST method in pth (p ≥ 2) moment is obtained under the one-sided and polynomial Lipschitz conditions. In addition, the order is 1 2 . In Section 4, the ISST method is proved to be exponentially mean square stable with some weaker request on the step-size. Simulations are provided to verify the analytical theory in Section 5.
2. Theoretical analysis and the SST method. Throughout this paper, we use the following notations unless otherwise specified. Let (Ω, F, P) be a completed probability space with a filtration {F t } t≥0 satisfying the usual conditions. Let x denotes the Euclidean vector norm in R n and x, y denotes the inner product of vectors x, y. If A ∈ R d×r , then its trace norm is defined as A := trace(A T A). For arbitrary a, b ∈ R, we denote max(a, b) and min(a, b) by a ∨ b and a ∧ b, respectively. We define inf ∅ = ∞. {t} denotes the fractional part of t.
Eq. (1) is equivalent to the following stochastic integral equation We impose the following hypotheses in this paper.
It can be seen from (7) and (8) that the drift coefficients of SDEPCAs (1) grow polynomially on both delay and non-delay terms, while the diffusion coefficients are polynomially growing only on delay terms. Using Theorem 4.1 in [4], the existence and uniqueness of the solution to (1) is obtained. Define stopping times ρ R = inf{t ≥ 0 : x(t) ≥ R}. We establish the following lemma.
Proof. Using Itô's formula, we obtain Taking supremum and expectation to (10), we have According to (6) and (7), we have and Using Young inequality, we obtain and where . Using Burkholder-Davis-Gundy inequality and (7), F 3 yields Due to 2ab ≤ δa 2 + 1 δ b 2 , δ > 0, we have Substituting (14), (15) and (17) where The following argument is similar to the proof of Theorem 2.1 in [27], while we will prove in detail because of the property of [t]. Let It can be verified p k ≥ 2 because of p ≥ 2 and l2+2 2 ≥ 1. We can also compute that l2+2 2 p k+1 < p k and p [T ]+1 = p, k = 1, 2, · · · , [T ] + 1. According to Assumption 2.1, Using Gronwall inequality, we have . We assume that for each k ∈ {1, 2, · · · , [T ]} there exists a constant a k such that the following formula holds Then for any Taking R → ∞ and which implies (9). The proof is completed.
3. Convergence rate of the SST method. Let h = 1 m be given step-size with integer m ≥ 1. Grid points t n are defined as t n = nh, n = 0, 1, · · · . Denote n = km + l, k ∈ N and l = 0, 1, 2, · · · , m − 1 for any t n ∈ [0, T ]. Then the SST method to (1) is given by 1], and x km+l is the approximation to x(t km+l ) at t = t km+l . For convenience, we extend the discrete approximations y km+l and x km+l to continuous time approximations.
, then the general continuous form of (19) is given byx It is required thatx(t) is F t -adapted, which satisfies the fundamental requirement in the classical stochastic analysis. We observe thatx(t km+l ) = x km+l , that is,x(t) coincides with the numerical approximations at grid points.
In this section, we consider the convergence of the SST method under the assumptions in Section 2. To guarantee the implicit method (19) has a unique solution x km+l+1 for given x km+l , we assume Kθh < 1 in the following. Firstly, we show the boundedness of the SST method in pth moment.

CONVERGENCE RATE AND STABILITY OF THE SST METHOD FOR SDEPCAS 705
Step 2. For k = 1, l = 0, 1, · · · , m − 1, by (32) and Gronwall inequality, we get Let R → ∞ and applying Fatou's lemma, we obtain Moreover, for any fixed T , there is also a constant C 1 independent of h such that Step 3. For any k ∈ {2, 3, · · · , [T ]}, l = 0, 1, · · · , m − 1, repeating the same steps as above, for any fixed T , there exists a constant C k independent of h such that, Combining steps 1-3, we have E sup t km+l ∈[0,T ] x km+l p ≤ max{C 0 , C 1 , · · · , C [T ] } for any fixed T . According to the relation of x km+l and y km+l , there exists C independent of h such that E sup t km+l ∈[0,T ] y km+l p ≤ C . Then we have where C = max{C 0 , C 1 , · · · , C [T ] , C }. The proof is completed.
where C may be different from that in Lemma 3.1.
Using Hölder inequality and the property of martingale, we obtain where C = 2 and E sup where B 10 (p) = 3 . For any k ∈ {1, 2, · · · , [T ]}, we assume that there exists a constant b k dependent on p k but independent of h such that If By Gronwall inequality, we obtain Taking ) . Therefore, by using mathematical induction method, we can conclude that there exists a , then we complete the proof.
Remark 2. From Theorem 3.3, it can be seen that the convergence rate of the SST method reaches 1 2 , while it is just close to 1 2 for some explicit numerical methods, such as the stopped Euler method [14], the truncated Euler method [20]. 4. Stability of the improved split-step theta(ISST) method. Having obtained the convergence rate of the SST method, we will proceed to the stability of the improved split-step theta(ISST) method for SDEPCAs (1). The ISST method to (1) is given by y km+l = x km+l + θhµ(y km+l , y km ), x km+l+1 = x km+l + hµ(y km+l , y km ) + σ(y km+l , y km )∆B km+l .
Throughout this section, we shall assume that Eq. (1) has a unique global solution for any given value ξ, we also assume that the ISST method (51) has unique solution.
Assumption 4.1. There exist two positive constants λ 1 , λ 2 such that for any x, y ∈ R d , x, µ(x, y) In the following, we always assume that both the equation (1) and the ISST method have unique solutions. The following theorem shows the exponential mean square stability of system (1) (see [15]).

Numerical simulations. This section is devoted to two examples and their
numerical simulations to illustrate the convergence rate of the SST method and the exponential stability of the ISST method, respectively.
Example 1. We consider the following scalar SDEPCA (71) We set ξ = −0.5 and T = 5. The coefficients of (71) satisfy Assumptions 2.1-2.4. We use the split-step backward Euler method with very small step-size h = 2 −14 to approximate Eq. (71). The Brownian paths are 1000. The approximate solution is regarded as the "exact solution" of (71). The numerical solutions of the SST method are computed using 6 different step-sizes h = 2 −6 , 2 −7 , 2 −8 , 2 −9 , 2 −10 , 2 −11 on the same Brownian path and the corresponding errors are obtained in Tables 1 and  2. We use x(t km+l , ω i ) to denote the "exact solution" of (71) in the ith Brownian path, and x km+l (ω i ) denotes the SST approximation in the ith Brownian path at t = t km+l . The errors in pth moment are calculated as follows There are three numerical tests in each experiment with θ = 0.5, 0.75, 1. The mean square errors are presented in Table 1, and Table 2 shows the 3th moment errors. From these tables, we observe that the errors are smaller and smaller as the stepsize decreasing when θ is fixed, which verify the convergence of the SST method. We can also observe that, with the same step-size, the errors are descending as θ increasing at the same time point. The mean square errors and the 3th moment errors are plotted in Figure 1. It shows that the convergence order of the SST method is 0.5 for all θ ∈ ( 1 2 , 1]. Example 2. We consider the following two dimensional SDEPCA on t ∈ [0, T ]. The initial value x(0) = (0.5, −1) T . Let T = 3, and other parameters are as same as Example 1. It can be verified that the coefficients of (73) satisfy Assumptions 2.1-2.4. Table 3 and Table 4 show the mean square errors and the 3th moment errors for (73). The convergence order of the SST method is presented in Figure 2. From the tables and Figure 2, we observe that the SST method is convergent and the order is 0.5.
Example 3. We consider the following scalar SDEPCA (74) We use two sets of parameters to illustrate the mean square exponential stability of the ISST method. The Monte Carlo Brownian paths are taken 1000.