Convergence rates of theta-method for neutral SDDEs under non-globally Lipschitz continuous coefficients

This paper is concerned with strong convergence and almost sure convergence for neutral stochastic differential delay equations under non-globally Lipschitz continuous coefficients. Convergence rates of $\theta$-EM schemes are given for these equations driven by Brownian motion and pure jumps respectively, where the drift terms satisfy locally one-sided Lipschitz conditions, and diffusion coefficients obey locally Lipschitz conditions, and the corresponding coefficients are highly nonlinear with respect to the delay terms.

Throughout the paper, we shall assume that C is a positive constant, which may change line by line.
Then there exists a unique global solution to (2.1), moreover, the solution has the properties that for any p ≥ 2, T > 0, where C = C(ξ, p, T ) is a positive constant depending on the initial data ξ, p and T .
Proof. With assumptions (A1)-(A3) and Remark 2.2, it is easy to see that (2.1) has a unique local solution. To verify that (2.1) admits a unique global solution, it is sufficient to show (2.5). Applying the Itô formula and using (2.4), we have (2.6) Application of the Burkholder-Davis-Gundy(BDG) inequality, the Young inequality and (2.7) where l = l 1 ∨ l 2 ∨ l 3 . Then, with (2.3), we derive from (2.8) that where in the last step we have used the Young inequality. The Gronwall inequality then leads to For t ∈ [0, τ ], the above inequality implies Finally, the desired result can be obtained with induction.
We now introduce θ-EM scheme for (2.1). Given any time T > τ > 0, without loss of generality, assume that T and τ are rational numbers, and there exist two positive integers such that ∆ = τ m = T M , where ∆ ∈ (0, 1) is the step size. For k = −m, · · · , 0, set y t k = ξ(k∆), for k = 0, 1, · · · , M − 1, we form Here θ ∈ [0, 1] is an additional parameter that allows us to control the implicitness of the numerical scheme. For θ = 0, the θ-EM scheme reduces to the EM scheme, and for θ = 1, it is exactly the backward EM scheme. For given y t k , in order to guarantee a unique solution y t k+1 to (2.9), the step size is required to satisfy ∆ < 1 4K 2 1 θ according to a fixed point theorem (see Mao and Szpruch [11] for more information), where K 1 is defined as in assumption (A1). In order for simplicity, we introduce the corresponding split-step theta scheme to (2.1) as follows: For k = −m, · · · , −1, set z t k = y t k = ξ(k∆), and for k = 0, · · · , M − 1, Through computation, we can easily deduce that y t k+1 in (2.10) can be rewritten as the form of (2.9). Due to the implicitness of θ-EM scheme, we also require ∆ < 1 2Kθ , where K is defined as in Remark 2.2. Thus, throughout this paper, we set ∆ * ∈ (0, (2K ∨ 4K 2 1 ) −1 θ −1 ), and 0 < ∆ ≤ ∆ * .

Moment Bounds
Lemma 2.2. Let (A1)-(A3) hold. Then for θ ∈ [ 1 2 , 1] there exists a positive constant C independent of ∆ such that for p ≥ 2, Proof. By (2.10), we see Noting that θ ≥ 1 2 and substituting b( ] into the last term, and using (2.4) yields Summing both sides from 0 to k, we get (2.11) Using the elementary inequality we then have By assumption (A2), we compute With (A2)-(A3), the Hölder inequality and the BDG inequality, we get Similarly, with (A2) and the BDG inequality again Sorting this inequalities together yields The discrete Gronwall inequality then leads to (2.14) this implies By the elementary inequality (2.12) again, we derive from (2.13) that Further, for j ≤ 2m − 1, it follows by the Gronwall inequality that The desired assertion follows by the method of induction.
, besides assumptions (A1)-(A3), if we further assume that there exists a positive constant K such that for any x ∈ R n , we can also show that p-th moment of θ-EM scheme is bounded by a positive constant independent of ∆.

Convergence Rates
We find it is convenient to work with a continuous form of a numerical method. Noting that the split-step θ-EM scheme (2.10) can be rewritten as Hence, we define the corresponding continuous-time split-step θ-EM solution Z(t) as follows: We now define the continuous θ-EM solution Y (t) as follows: It can be verified that Y (t k ) = y t k , k = −m, · · · , M. In order to obtain convergence rate, we impose another assumption as follows: Remark 2.4. From assumptions (A1) and (A4), one sees that and further, where C is a constant independent of ∆.
Proof. For any p ≥ 2, by the elementary inequality (2.12), we have Using the Hölder inequality, the BDG inequality, and together with (A2)-(A4), Lemma 2.2 yields (2.17) With the relationship (2.16), similar to (2.14), we get We then derive from (2.17) that Following the procedure of Lemma 2.1, we can show that the p-th moment of Y (t) is bounded by a positive constant C. Denote by Z(t) := z t k for t ∈ [t k , t k+1 ), we see from (2.15) that With (A2), (A4), Lemma 2.2, the Hölder inequality, and the BDG inequality, we get (2.18) On the other hand, we have the following relationship between Y (t) and Z(t), Combing (2.16) and (2.19) gives Using similar skills of (2.14), we derive from (A1) and (A3) Obviously, due to (2.18), The desired result follows by repeating the techniques of Lemma 2.1.

By (A4), Lemma 2.3 and the Hölder inequality,
By (A1), Lemma 2.3 and the Hölder inequality, Due to (A1)-(A2), Lemma 2.3 and the Hölder inequality, In the same way to estimate H 1 (t) and H 2 (t), we get Furthermore, by (A3), Lemma 2.3, the BDG inequality and the Hölder inequality, we compute Consequently, by sorting H 1 (t) − H 7 (t) together, we arrive at By the definition of e(t), we derive from (A3) that Taking (A1) and Lemma 2.3 into consideration, The Gronwall inequality yields Again, the desired result follows by the induction.
With strong convergence rate given in Theorem 2.4, we can easily show the following result on almost sure convergence.
Proof. We omit the proof here since it is similar to that of Lemma 2.1.

Moment Bounds
Firstly, we introduce an important lemma coming from [13]. Lemma 3.2. Let φ : R + × U → R n be progressively measurable and assume that the right side is finite. Then there exists a positive constant C such that Proof. It is easy to see from (3.4) Applying (3.3) to the last term and using assumption (A1) lead to Summing both sides from 0 to k, we deduce that Consequently, With assumption (A5), we find that for 0 < j < M, Using (A5), Lemma 3.2 and the Hölder inequality, we compute Similarly, by (A5) and Lemma 3.2 again This implies that By the discrete Gronwall inequality we find that E|y t i−m | 2p(l+1) .
Following the steps of (2.13), the desired assertion can be derived by similar skills.
Remark 3.2. We see from Theorems 2.4 and 3.5 that the strong convergence rate of θ-EM scheme for neutral SDDEs is 1 2 for the Brownian motion case, while for the pure jumps case, the order is 1 2p , that is to say, lower moment has a better convergence rate for neutral SDDEs with jumps, whence it is better to use the mean-square convergence for jump case.