Convergence rates of theta-method for neutral SDDEs under non-globally Lipschitz continuous coefficients
aa r X i v : . [ m a t h . P R ] J a n Convergence rates of theta-method forneutral SDDEs under non-globally Lipschitzcontinuous coefficients ∗ Li Tan a,b and Chenggui Yuan c a School of Statistics, Jiangxi University of Finance and Economics, Nanchang, Jiangxi, 330013, P. R. China b Research Center of Applied Statistics, Jiangxi University of Finance and Economics,Nanchang, Jiangxi, 330013, P. R. China c Department of Mathematics, Swansea University, Swansea, SA2 8PP, U. K.
[email protected] 21, 2018
Abstract
This paper is concerned with strong convergence and almost sure convergence forneutral stochastic differential delay equations under non-globally Lipschitz continuouscoefficients. Convergence rates of θ -EM schemes are given for these equations driven byBrownian 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. AMS Subject Classification : 65C30, 65L20
Keywords : stochastic differential delay equations; θ -EM scheme; strong convergence; al-most sure convergence; highly nonlinear With the development of computer technology, numerical analyses have been witnessed rapidgrowth since most equations can not be solved explicitly. There is an extensive literature con-cerned with numerical solutions for stochastic differential equations (SDEs) and stochastic ∗ Supported by NSFC(No., 11561027, 11661039), NSF of Jiangxi(No., 20161BAB211018), Scientific Re-search Fund of Jiangxi Provincial Education Department(No., GJJ150444). p -th moment and almost sure exponential stability of the exact and EM-scheme solutions ofneutral SDDEs; Ji et al. [7] generalized the results of [1] to neutral SDDEs; Tan and Yuan[14] proposed a tamed θ -EM scheme and gave convergence rate for neutral SDDEs driven byBrownian motion and pure jumps under one-sided Lipschitz condition. Motivated by Baoand Yuan [1] and Zong et al. [17], the drift and diffusion coefficients may be highly nonlinearwith respect to delay variables. Will the θ -EM scheme converges to exact solutions stronglyand almost surely for neutral SDDEs if the drift terms satisfy locally one-sided Lipschitzcondition and diffusion coefficients obey locally Lipschitz conditions, and the correspondingcoefficients is highly nonlinear? In this paper, we shall give a positive answer when thecorresponding coefficients are highly nonlinear with respect to the delay terms.The rest of paper is organized as follows: in Section 2, strong convergence rate and almostsure convergence rate are given for neutral SDDEs driven by Brownian motion under non-globally Lipschitz condition, while in Section 3, the Brownian motion is replaced by purejumps, the convergence rates are also provided under similar conditions. Let (Ω , F , { F t } t ≥ , P ) be a complete probability space with { F t } t ≥ satisfying the usualconditions (i.e., it is right continuous and increasing while F contains all P -null sets).2 R n , h· , ·i , | · | ) is an n -dimensional Euclidean space. Denote R n × d by the set of all n × d matrices A with trace norm k A k = p trace( A T A ), where A T is the transpose of matrix A . Fora given τ ∈ (0 , ∞ ), define the uniform norm k ζ k ∞ := sup − τ ≤ θ ≤ | ζ ( θ ) | for ζ ∈ C ([ − τ, R n )which denotes all continuous functions from [ − τ,
0] to R n . W ( t ) is a d -dimensional Brownianmotion defined on (Ω , F , { F t } t ≥ , P ). In this section, we consider the following neutralSDDE on R n :d[ X ( t ) − D ( X ( t − τ ))] = b ( X ( t ) , X ( t − τ ))d t + σ ( X ( t ) , X ( t − τ ))d W ( t ) , t ≥ X ( t ) = ξ ( t ) ∈ L p F ([ − τ, R n ) for t ∈ [ − τ, ξ is an F -measurable C ([ − τ, R n )-valued random variable with E k ξ k p ∞ < ∞ for p ≥
2. Here, D : R n → R n , and b : R n × R n → R n , σ : R n × R n → R n × d are continuous functions. In order to guarantee theexistence and uniqueness of solutions to (2.1), we firstly introduce functions V i , i = 1 , , x, y ∈ R n ,(2.2) 0 ≤ V i ( x, y ) ≤ L i (1 + | x | l i + | y | l i ) , i = 1 , , L i > , l i ≥
1. Furthermore, in the sequel, for any x, y, x, y ∈ R n , we shall assumethat (A1) There exists a positive constant K such that h x − D ( y ) − x + D ( y ) , b ( x, y ) − b ( x, y ) i ≤ K | x − x | + | V ( y, y ) | | y − y | , and | b ( x, y ) − b ( x, y ) | ≤ V ( y, y ) | y − y | . (A2) There exists a positive constant K such that k σ ( x, y ) − σ ( x, y ) k ≤ K | x − x | + V ( y, y ) | y − y | . (A3) D (0) = 0 and | D ( y ) − D ( y ) | ≤ V ( y, y ) | y − y | . Remark 2.1.
There are some examples such that (A1)-(A3) hold. For example, set D ( y ) = − y , b ( x, y ) = x − x + y , σ ( x, y ) = x + y for any x, y ∈ R . It is to easy to check that (A1)-(A3) is satisfied with V i ( x, y ) = 1 + | x | + | y | , i = 1 , V ( x, y ) = 1 + | x | + | y | for arbitrary x, y ∈ R . Remark 2.2.
With assumption (A3), we immediately arrive at(2.3) | D ( y ) | ≤ V ( y, | y | ≤ L (1 + | y | + | y | l +1 ) . With assumptions (A1)-(A3), we have h x − D ( y ) , b ( x, y ) i = h x − D ( y ) − D (0) , b ( x, y ) − b (0 , i + h x − D ( y ) , b (0 , i≤ K | x | + | V ( y, | | y | + 12 | x − D ( y ) | + 12 | b (0 , | ≤ ( K + 1) | x | + | V ( y, | | y | + | V ( y, | | y | + 12 | b (0 , | , k σ ( x, y ) k ≤ k σ ( x, y ) − σ (0 , k + 2 k σ (0 , k ≤ K | x | + 4 | V ( y, | | y | + 2 k σ (0 , k . Denote K = max { K +1) , K , | b (0 , | , k σ (0 , k } , and | V ( y, | = 2 max {| V ( y, | + | V ( y, | , | V ( y, | } , we can rewrite the above inequalities as(2.4) 2 h x − D ( y ) , b ( x, y ) i ∨ k σ ( x, y ) k ≤ K (1 + | x | ) + | V ( y, | | y | . Throughout the paper, we shall assume that C is a positive constant, which may changeline by line. Lemma 2.1.
Let (A1)-(A3) hold. Then there exists a unique global solution to (2.1),moreover, the solution has the properties that for any p ≥ T > E (cid:18) sup ≤ t ≤ T | X ( t ) | p (cid:19) ≤ C, 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 aunique local solution. To verify that (2.1) admits a unique global solution, it is sufficient toshow (2.5). Applying the Itˆo formula and using (2.4), we have | X ( t ) − D ( X ( t − τ )) | p = | ξ (0) − D ( ξ ( − τ )) | p + p Z t | X ( s ) − D ( X ( s − τ )) | p − h X ( s ) − D ( X ( s − τ )) , b ( X ( s ) , X ( s − τ )) i d s + p ( p − Z t | X ( s ) − D ( X ( s − τ )) | p − k σ ( X ( s ) , X ( s − τ )) k d s + p Z t | X ( s ) − D ( X ( s − τ )) | p − h X ( s ) − D ( X ( s − τ )) , σ ( X ( s ) , X ( s − τ ))d W ( s ) i≤| ξ (0) − D ( ξ ( − τ )) | p + p K Z t | X ( s ) − D ( X ( s − τ )) | p − (1 + | X ( s ) | )d s + p Z t | X ( s ) − D ( X ( s − τ )) | p − | V ( X ( s − τ ) , | | X ( s − τ ) | d s + p Z t | X ( s ) − D ( X ( s − τ )) | p − h X ( s ) − D ( X ( s − τ )) , σ ( X ( s ) , X ( s − τ ))d W ( s ) i =: | ξ (0) − D ( ξ ( − τ )) | p + I ( t ) + I ( t ) + I ( t ) . (2.6)Application of the Burkholder-Davis-Gundy(BDG) inequality, the Young inequality and42.4) yields E (cid:18) sup ≤ u ≤ t | I ( u ) | (cid:19) ≤ C E (cid:18)Z t | X ( s ) − D ( X ( s − τ )) | p − k σ ( X ( s ) , X ( s − τ )) k d s (cid:19) ≤ C E (cid:18) sup ≤ u ≤ t | X ( u ) − D ( X ( u − τ )) | p − Z t k σ ( X ( s ) , X ( s − τ )) k d s (cid:19) ≤ E (cid:18) sup ≤ u ≤ t | X ( u ) − D ( X ( u − τ )) | p (cid:19) + C E (cid:18)Z t k σ ( X ( s ) , X ( s − τ )) k d s (cid:19) p ≤ E (cid:18) sup ≤ u ≤ t | X ( u ) − D ( X ( u − τ )) | p (cid:19) + C E Z t [1 + | X ( s ) | p + | V ( X ( s − τ ) , | p | X ( s − τ ) | p ]d s. (2.7)Substituting (2.7) into (2.6), we obtain E (cid:18) sup ≤ u ≤ t | X ( u ) − D ( X ( u − τ )) | p (cid:19) ≤ C + C E Z t | X ( s ) | p d s + C E Z t | V ( X ( s − τ ) , | p | X ( s − τ ) | p d s + C E Z t | V ( X ( s − τ ) , | p | X ( s − τ ) | p d s. By (2.2), we see that E (cid:18) sup ≤ u ≤ t | X ( u ) − D ( X ( u − τ )) | p (cid:19) ≤ C + C E Z t | X ( s ) | p d s + C E Z t | X ( s − τ ) | ( l +1) p d s, (2.8)where l = l ∨ l ∨ l . Then, with (2.3), we derive from (2.8) that E (cid:18) sup ≤ u ≤ t | X ( u ) | p (cid:19) ≤ C E (cid:18) sup ≤ u ≤ t | D ( X ( u − τ )) | p (cid:19) + C E (cid:18) sup ≤ u ≤ t | X ( u ) − D ( X ( u − τ )) | p (cid:19) ≤ C + C E (cid:18) sup − τ ≤ u ≤ t − τ | X ( u ) | ( l +1) p (cid:19) + C E Z t | X ( s ) | p d s + C E Z t | X ( s − τ ) | ( l +1) p d s ≤ C ( k ξ k ( l +1) p ∞ , p ) + C E (cid:18) sup − τ ≤ u ≤ t − τ | X ( u ) | ( l +1) p (cid:19) + C Z t E (cid:18) sup ≤ u ≤ s | X ( u ) | p (cid:19) d s, where in the last step we have used the Young inequality. The Gronwall inequality thenleads to E (cid:18) sup ≤ u ≤ t | X ( u ) | p (cid:19) ≤ C + C E sup ≤ u ≤ ( t − τ ) ∨ | X ( u ) | ( l +1) p ! . t ∈ [0 , τ ], the above inequality implies E (cid:18) sup ≤ t ≤ τ | X ( t ) | p (cid:19) ≤ C, this further gives E (cid:18) sup ≤ t ≤ τ | X ( t ) | p (cid:19) ≤ C + C E (cid:18) sup ≤ t ≤ τ | X ( t ) | ( l +1) p (cid:19) ≤ C. 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 ofgenerality, assume that T and τ are rational numbers, and there exist two positive integerssuch that ∆ = τm = TM , where ∆ ∈ (0 ,
1) is the step size. For k = − m, · · · ,
0, set y t k = ξ ( k ∆),for k = 0 , , · · · , M −
1, we form y t k +1 − D ( y t k +1 − m ) = y t k − D ( y t k − m ) + θb ( y t k +1 , y t k +1 − m )∆+ (1 − θ ) b ( y t k , y t k − m )∆ + σ ( y t k , y t k − m )∆ W t k , (2.9)where t k = k ∆, ∆ W t k = W ( t k +1 ) − W ( t k ). Here θ ∈ [0 ,
1] is an additional parameter thatallows us to control the implicitness of the numerical scheme. For θ = 0, the θ -EM schemereduces to the EM scheme, and for θ = 1, it is exactly the backward EM scheme. Forgiven y t k , in order to guarantee a unique solution y t k +1 to (2.9), the step size is requiredto satisfy ∆ < K θ according to a fixed point theorem (see Mao and Szpruch [11] formore information), where K is defined as in assumption (A1). In order for simplicity, weintroduce the corresponding split-step theta scheme to (2.1) as follows: For k = − m, · · · , − z t k = y t k = ξ ( k ∆), and for k = 0 , · · · , M − ( y t k = D ( y t k − m ) + z t k − D ( z t k − m ) + θb ( y t k , y t k − m )∆ ,z t k +1 = D ( z t k +1 − m ) + z t k − D ( z t k − m ) + b ( y t k , y t k − m )∆ + σ ( y t k , y t k − m )∆ W t k . Through computation, we can easily deduce that y t k +1 in (2.10) can be rewritten as the formof (2.9). Due to the implicitness of θ -EM scheme, we also require ∆ < Kθ , where K isdefined as in Remark 2.2. Thus, throughout this paper, we set ∆ ∗ ∈ (0 , (2 K ∨ K ) − θ − ),and 0 < ∆ ≤ ∆ ∗ . Lemma 2.2.
Let (A1)-(A3) hold. Then for θ ∈ [ ,
1] there exists a positive constant C independent of ∆ such that for p ≥ E (cid:18) sup ≤ k ≤ M | y t k | p (cid:19) ≤ C. roof. By (2.10), we see | z t k +1 − D ( z t k +1 − m ) | = | z t k − D ( z t k − m ) | + 2 h z t k − D ( z t k − m ) , b ( y t k , y t k − m )∆ i + | b ( y t k , y t k − m ) | ∆ + | σ ( y t k , y t k − m )∆ W t k | + 2 h z t k − D ( z t k − m ) + b ( y t k , y t k − m )∆ , σ ( y t k , y t k − m )∆ W t k i = | z t k − D ( z t k − m ) | + 2 h y t k − D ( y t k − m ) , b ( y t k , y t k − m )∆ i + (1 − θ ) | b ( y t k , y t k − m ) | ∆ + | σ ( y t k , y t k − m )∆ W t k | + 2 h y t k − D ( y t k − m ) + (1 − θ ) b ( y t k , y t k − m )∆ , σ ( y t k , y t k − m )∆ W t k i . Noting that θ ≥ and substituting b ( y t k , y t k − m )∆ = θ [ y t k − D ( y t k − m ) − z t k + D ( z t k − m )] intothe last term, and using (2.4) yields | z t k +1 − D ( z t k +1 − m ) | ≤| z t k − D ( z t k − m ) | + 2∆ h y t k − D ( y t k − m ) , b ( y t k , y t k − m ) i + | σ ( y t k , y t k − m )∆ W t k | + 2 θ h y t k − D ( y t k − m ) , σ ( y t k , y t k − m )∆ W t k i− − θθ h z t k − D ( z t k − m ) , σ ( y t k , y t k − m )∆ W t k i≤| z t k − D ( z t k − m ) | + ∆ K (1 + | y t k | ) + ∆ | V ( y t k − m , | | y t k − m | + | σ ( y t k , y t k − m )∆ W t k | + 2 θ h y t k − D ( y t k − m ) , σ ( y t k , y t k − m )∆ W t k i− − θθ h z t k − D ( z t k − m ) , σ ( y t k , y t k − m )∆ W t k i . Summing both sides from 0 to k , we get | z t k +1 − D ( z t k +1 − m ) | ≤| z t − D ( z t − m ) | + KT + ∆ K k X i =0 | y t i | + ∆ k X i =0 | V ( y t i − m , | | y t i − m | + k X i =0 | σ ( y t i , y t i − m )∆ W t i | + 2 θ k X i =0 h y t i − D ( y t i − m ) , σ ( y t i , y t i − m )∆ W t i i− − θθ k X i =0 h z t i − D ( z t i − m ) , σ ( y t i , y t i − m )∆ W t i i . (2.11)Using the elementary inequality(2.12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ n p − n X i =1 | x i | p , p ≥ ,
7e then have | z t k +1 − D ( z t k +1 − m ) | p ≤ p − ( | z t − D ( z t − m ) | + KT ) p + 6 p − K p ∆ p k X i =0 | y t i | ! p + 6 p − ∆ p k X i =0 | V ( y t i − m , | | y t i − m | ! p + 6 p − k X i =0 | σ ( y t i , y t i − m )∆ W t i | ! p + 6 p − p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i =0 h y t i − D ( y t i − m ) , σ ( y t i , y t i − m )∆ W t i i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p + 6 p − p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i =0 h z t i − D ( z t i − m ) , σ ( y t i , y t i − m )∆ W t i i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p . For 0 < j < M , it is easy to observe that E " sup ≤ k ≤ j k X i =0 | y t i | ! p ≤ M p − j X i =0 E | y t i | p , and E " sup ≤ k ≤ j k X i =0 | V ( y t i − m , | | y t i − m | ! p ≤ M p − j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) . By assumption (A2), we compute E " sup ≤ k ≤ j k X i =0 | σ ( y t i , y t i − m )∆ W t i | ! p ≤ M p − E j X i =0 | σ ( y t i , y t i − m ) | p | ∆ W t i | p ! ≤ M p − j X i =0 E | σ ( y t i , y t i − m ) | p E | ∆ W t i | p ≤ M p − (2 p − p j X i =0 E [ K (1 + | y t i | ) + | V ( y t i − m , | | y t i − m | ] p ≤ C + C j X i =0 E | y t i | p + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) . E " sup ≤ k ≤ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i =0 h y t i − D ( y t i − m ) , σ ( y t i , y t i − m )∆ W t i i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ C E j X i =0 | y t i − D ( y t i − m ) | | σ ( y t i , y t i − m ) | ∆ ! p ≤ C ∆ p ( j + 1) p − E j X i =0 | y t i − D ( y t i − m ) | p [ K (1 + | y t i | ) + | V ( y t i − m , | | y t i − m | ] p ≤ C + C j X i =0 E | y t i | p + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) . Similarly, with (A2) and the BDG inequality again E " sup ≤ k ≤ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i =0 h z t i − D ( z t i − m ) , σ ( y t i , y t i − m )∆ W t i i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ C E j X i =0 | z t i − D ( z t i − m ) | | σ ( y t i , y t i − m ) | ∆ ! p ≤ C ∆ p ( j + 1) p − E j X i =0 | z t i − D ( z t i − m ) | p [ K (1 + | y t i | ) + | V ( y t i − m , | | y t i − m | ] p ≤ C + C j X i =0 E ( | z t i − D ( z t i − m ) | p ) + C j X i =0 E | y t i | p + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) . Sorting this inequalities together yields E (cid:18) sup ≤ k ≤ j +1 | z t k − D ( z t k − m ) | p (cid:19) ≤ C + C j X i =0 E ( | z t i − D ( z t i − m ) | p ) + C j X i =0 E | y t i | p + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) ≤ C + C j X i =0 E (cid:18) sup ≤ k ≤ i | z t k − D ( z t k − m ) | p (cid:19) + C j X i =0 E | y t i | p + C j X i =0 E | y t i − m | p ( l +1) . The discrete Gronwall inequality then leads to E (cid:18) sup ≤ k ≤ j +1 | z t k − D ( z t k − m ) | p (cid:19) ≤ C + C j X i =0 E | y t i | p + C j X i =0 E | y t i − m | p ( l +1) . (2.13) 9ince y t k − D ( y t k − m ) = z t k − D ( z t k − m ) + θb ( y t k , y t k − m )∆, we deduce from (A1)-(A3) that | z t k − D ( z t k − m ) | = | y t k − D ( y t k − m ) | + θ ∆ | b ( y t k , y t k − m ) | − θ ∆ h y t k − D ( y t k − m ) , b ( y t k , y t k − m ) i≥ | y t k | − | V ( y t k − m , | | y t k − m | − θ ∆[ K (1 + | y t k | ) + | V ( y t k − m , | | y t k − m | ]= (cid:18) − θK ∆ (cid:19) | y t k | − [ | V ( y t k − m , | + θ ∆ | V ( y t k − m , | ] | y t k − m | − θK ∆ , (2.14)this implies | y t k | ≤ (cid:18) − θK ∆ (cid:19) − (cid:2) | z t k − D ( z t k − m ) | + [ | V ( y t k − m , | + θ ∆ | V ( y t k − m , | ] | y t k − m | + θK ∆ (cid:3) ≤ (cid:18) − θK ∆ (cid:19) − [ | z t k − D ( z t k − m ) | + 2 | V ( y t k − m , | | y t k − m | + θK ∆] . By the elementary inequality (2.12) again, we derive from (2.13) that E (cid:18) sup ≤ k ≤ j +1 | y t k | p (cid:19) ≤ (cid:18) − θ ∆ K (cid:19) − p p − (cid:20) E (cid:18) sup ≤ k ≤ j +1 | z t k − D ( z t k − m ) | p (cid:19) + 2 p E (cid:18) sup ≤ k ≤ j +1 | V ( y t k − m , | p | y t k − m | p (cid:19) + ( θ ∆ K ) p (cid:21) ≤ C + C j X i =0 E | y t i | p + C j − m X i = − m E | y t i | p ( l +1) + C E (cid:18) sup ≤ k ≤ j +1 | y t k − m | p ( l +1) (cid:19) ≤ C + C j X i =0 E (cid:18) sup ≤ k ≤ i | y t k | p (cid:19) + C E sup ≤ k ≤ ( j +1 − m ) ∨ | y t k | p ( l +1) ! . In case of j ≤ m −
1, it is obvious that E (cid:18) sup ≤ k ≤ m | y t k | p (cid:19) ≤ C. Further, for j ≤ m −
1, it follows by the Gronwall inequality that E (cid:18) sup ≤ k ≤ m | y t k | p (cid:19) ≤ C + C E (cid:18) sup ≤ k ≤ m | y t k | p ( l +1) (cid:19) ≤ C. The desired assertion follows by the method of induction.
Remark 2.3.
For θ ∈ [0 , ), besides assumptions (A1)-(A3), if we further assume that thereexists a positive constant K such that for any x ∈ R n , | b ( x, | ≤ K (1 + | x | ) , we can also show that p -th moment of θ -EM scheme is bounded by a positive constantindependent of ∆. 10 .3 Convergence Rates We find it is convenient to work with a continuous form of a numerical method. Noting thatthe split-step θ -EM scheme (2.10) can be rewritten as z t k +1 − D ( z t k +1 − m ) = z t − D ( z t − m ) + k X i =0 b ( y t i , y t i − m )∆ + k X i =0 σ ( y t i , y t i − m )∆ W t i = ξ (0) − D ( ξ ( − τ )) − θb ( ξ (0) , ξ ( − τ ))∆ + k X i =0 b ( y t i , y t i − m )∆ + k X i =0 σ ( y t i , y t i − m )∆ W t i . Hence, we define the corresponding continuous-time split-step θ -EM solution Z ( t ) as follows:For any t ∈ [ − τ, Z ( t ) = ξ ( t ), Z (0) = ξ (0) − θb ( ξ (0) , ξ ( − τ ))∆, For any t ∈ [0 , T ],(2.15) d[ Z ( t ) − D ( Z ( t − τ ))] = b ( Y ( t ) , Y ( t − τ ))d t + σ ( Y ( t ) , Y ( t − τ ))d W ( t ) , where Y ( t ) is defined by Y ( t ) := y t k for t ∈ [ t k , t k +1 ) , thus Y ( t − τ ) = y t k − m . We now define the continuous θ -EM solution Y ( t ) as follows:(2.16) Y ( t ) − D ( Y ( t − τ )) = Z ( t ) − D ( Z ( t − τ )) + θb ( Y ( t ) , Y ( t − τ ))∆ . 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: (A4) For x, x, y ∈ R n , | b ( x, y ) − b ( x, y ) | ≤ V ( x, x ) | x − x | . Remark 2.4.
From assumptions (A1) and (A4), one sees that | b ( x, y ) | ≤ | b ( x, y ) − b ( x, | + | b ( x, − b (0 , | + | b (0 , | ≤ V ( x, | x | + V ( y, | y | + | b (0 , | , and further, | b ( x, y ) − b ( x, y ) | ≤ | b ( x, y ) − b ( x, y ) | + | b ( x, y ) − b ( x, y ) | ≤ V ( x, x ) | x − x | + V ( y, y ) | y − y | . Lemma 2.3.
Consider the θ -EM scheme (2.9), and let (A1)-(A4) hold. Then, for any p ≥ θ -EM scheme solution Y ( t ) has the following properties, E (cid:18) sup ≤ t ≤ T | Y ( t ) | p (cid:19) ≤ C, and E (cid:18) sup ≤ t ≤ T | Y ( t ) − Y ( t ) | p (cid:19) ≤ C ∆ p , where C is a constant independent of ∆. 11 roof. For any p ≥
2, by the elementary inequality (2.12), we have E (cid:18) sup ≤ u ≤ t | Z ( u ) − D ( Z ( u − τ )) | p (cid:19) ≤ p − | Z (0) − D ( Z ( − τ )) | p + 3 p − E (cid:18) sup ≤ u ≤ t (cid:12)(cid:12)(cid:12)(cid:12)Z u b ( Y ( s ) , Y ( s − τ ))d s (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) + 3 p − E (cid:18) sup ≤ u ≤ t (cid:12)(cid:12)(cid:12)(cid:12)Z u σ ( Y ( s ) , Y ( s − τ ))d W ( s ) (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) . Using the H¨older inequality, the BDG inequality, and together with (A2)-(A4), Lemma 2.2yields E (cid:18) sup ≤ u ≤ t | Z ( u ) − D ( Z ( u − τ )) | p (cid:19) ≤ p − | Z (0) − D ( Z ( − τ )) | p + 3 p − t p − E Z t (cid:12)(cid:12) b ( Y ( s ) , Y ( s − τ )) (cid:12)(cid:12) p d s + C E (cid:18)Z t k σ ( Y ( s ) , Y ( s − τ )) k d s (cid:19) p ≤ C + C E Z t (cid:2) | V ( Y ( s ) , | p | Y ( s ) | p + | V ( Y ( s − τ ) , | p | Y ( s − τ ) | p + | b (0 , | p (cid:3) d s + C E Z t [ | Y ( s ) | p + | V ( Y ( s − τ )) | p | Y ( s − τ ) | p ]d s ≤ C + C E Z t | Y ( s ) | p d s + C E Z t | Y ( s ) | ( l +1) p d s ≤ C. (2.17)With the relationship (2.16), similar to (2.14), we get | Y ( t ) | ≤ (cid:18) − θK ∆ (cid:19) − [ | Z ( t ) − D ( Z ( t − τ )) | + 2 | V ( Y ( t − τ ) , | | Y ( t − τ ) | + θK ∆] . We then derive from (2.17) that E (cid:18) sup ≤ u ≤ t | Y ( u ) | p (cid:19) ≤ C + C E (cid:18) sup ≤ u ≤ t | Z ( u ) − D ( Z ( u − τ )) | p (cid:19) + C E (cid:18) sup ≤ u ≤ t | Y ( u − τ ) | ( l +1) p (cid:19) ≤ C + C E sup ≤ u ≤ ( t − τ ) ∨ | Y ( u ) | ( l +1) p ! . Following the procedure of Lemma 2.1, we can show that the p -th moment of Y ( t ) is boundedby a positive constant C . Denote by Z ( t ) := z t k for t ∈ [ t k , t k +1 ), we see from (2.15) that Z ( t ) − D ( Z ( t − τ )) − Z ( t )+ D ( Z ( t − τ )) = Z tt k b ( Y ( s ) , Y ( s − τ ))d s + Z tt k σ ( Y ( s ) , Y ( s − τ ))d W ( s ) , Z ( t ) , Z ( t )) := Z ( t ) − D ( Z ( t − τ )) − Z ( t ) + D ( Z ( t − τ )), then E sup t k ≤ t Let assumptions (A1)-(A4) hold and θ ∈ (cid:2) , (cid:3) . Then it holds that the θ -EMsolution Y ( t ) converges to the exact solution X ( t ) in L p sense with order , i.e., E (cid:18) sup ≤ t ≤ T | Y ( t ) − X ( t ) | p (cid:19) ≤ C ∆ p for p ≥ Proof. Denote by e ( t ) := Z ( t ) − D ( Z ( t − τ )) − X ( t ) + D ( X ( t − τ )), then e ( t ) = e (0) + Z t [ b ( Y ( s ) , Y ( s − τ )) − b ( X ( s ) , X ( s − τ ))]d s + Z t [ σ ( Y ( s ) , Y ( s − τ )) − σ ( X ( s ) , X ( s − τ ))]d W ( s ) , where e (0) = − θb ( ξ (0) , ξ ( − τ ))∆. Application of the Itˆo formula yields | e ( t ) | p = | e (0) | p + p Z t | e ( s ) | p − h e ( s ) , b ( Y ( s ) , Y ( s − τ )) − b ( X ( s ) , X ( s − τ )) i d s + 12 p ( p − Z t | e ( s ) | p − k σ ( Y ( s ) , Y ( s − τ )) − σ ( X ( s ) , X ( s − τ )) k d s + p Z t | e ( s ) | p − h e ( s ) , σ ( Y ( s ) , Y ( s − τ )) − σ ( X ( s ) , X ( s − τ ))d W ( s ) i . Rewrite | e ( t ) | p as | e ( t ) | p ≤| e (0) | p + p Z t | e ( s ) | p − h e ( s ) , b ( Y ( s ) , Y ( s − τ )) − b ( Y ( s ) , Y ( s − τ )) i d s + p Z t | e ( s ) | p − h e ( s ) , b ( Y ( s ) , Y ( s − τ )) − b ( Y ( s ) , Y ( s − τ )) i d s + p Z t | e ( s ) | p − h e ( s ) , b ( Y ( s ) , Y ( s − τ )) − b ( X ( s ) , X ( s − τ )) i d s + 32 p ( p − Z t | e ( s ) | p − k σ ( Y ( s ) , Y ( s − τ )) − σ ( Y ( s ) , Y ( s − τ )) k d s + 32 p ( p − Z t | e ( s ) | p − k σ ( Y ( s ) , Y ( s − τ )) − σ ( Y ( s ) , Y ( s − τ )) k d s + 32 p ( p − Z t | e ( s ) | p − k σ ( Y ( s ) , Y ( s − τ )) − σ ( X ( s ) , X ( s − τ )) k d s + p Z t | e ( s ) | p − h e ( s ) , σ ( Y ( s ) , Y ( s − τ )) − σ ( X ( s ) , X ( s − τ ))d W ( s ) i =: | e (0) | p + H ( t ) + H ( t ) + H ( t ) + H ( t ) + H ( t ) + H ( t ) + H ( t ) . 14y (A4), Lemma 2.3 and the H¨older inequality, E (cid:18) sup ≤ u ≤ t | H ( u ) | (cid:19) ≤ C E Z t | e ( s ) | p d s + C E Z t | b ( Y ( s ) , Y ( s − τ )) − b ( Y ( s ) , Y ( s − τ )) | p d s ≤ C E Z t [ | Y ( s ) − X ( s ) | p + | V ( Y ( s − τ ) , X ( s − τ )) | p | Y ( s − τ ) − X ( s − τ ) | p + θ p ∆ p | b ( Y ( s ) , Y ( s − τ )) | p ]d s + C E Z t | V ( Y ( s ) , Y ( s )) | p | Y ( s ) − Y ( s ) | p d s ≤ C E Z t | Y ( s ) − X ( s ) | p d s + C Z t [ E | V ( Y ( s − τ ) , X ( s − τ )) | p ] [ E | Y ( s − τ ) − X ( s − τ ) | p ] d s + C ∆ p E Z t [ V ( Y ( s ) , | Y ( s ) | + V ( Y ( s − τ ) , | Y ( s − τ ) | + | b (0 , | ] p d s + C Z t [ E | V ( Y ( s ) , Y ( s )) | p ] [ E | Y ( s ) − Y ( s ) | p ] d s ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p + C ∆ p . By (A1), Lemma 2.3 and the H¨older inequality, E (cid:18) sup ≤ u ≤ t | H ( u ) | (cid:19) ≤ C E Z t | e ( s ) | p d s + C E Z t | b ( Y ( s ) , Y ( s − τ )) − b ( Y ( s ) , Y ( s − τ )) | p d s ≤ C E Z t | e ( s ) | p d s + C E Z t | V ( Y ( s − τ ) , Y ( s − τ )) | p | Y ( s − τ ) − Y ( s − τ ) | p d s ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p + C ∆ p . Due to (A1)-(A2), Lemma 2.3 and the H¨older inequality, E (cid:18) sup ≤ u ≤ t | H ( u ) + H ( u ) | (cid:19) ≤ C E Z t | e ( s ) | p − | Y ( s ) − X ( s ) | d s + C E Z t | e ( s ) | p − | V ( Y ( s − τ ) , X ( s − τ )) | | Y ( s − τ ) − X ( s − τ ) | d s + C E Z t | e ( s ) | p − | V ( Y ( s − τ ) , X ( s − τ )) | | Y ( s − τ ) − X ( s − τ ) | d s + C E Z t | e ( s ) | p − | θb ( Y ( s ) , Y ( s − τ ))∆ || b ( Y ( s ) , Y ( s − τ )) − b ( X ( s ) , X ( s − τ )) | d s ≤ C E Z t | Y ( s ) − X ( s ) | p d s + C ∆ p ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p . 15n the same way to estimate H ( t ) and H ( t ), we get E (cid:18) sup ≤ u ≤ t | H ( u ) | (cid:19) ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p + C ∆ p , and E (cid:18) sup ≤ u ≤ t | H ( u ) | (cid:19) ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p + C ∆ p . Furthermore, by (A3), Lemma 2.3, the BDG inequality and the H¨older inequality, we com-pute E (cid:18) sup ≤ u ≤ t | H ( u ) | (cid:19) ≤ C E (cid:18)Z t | e ( s ) | p − k σ ( Y ( s ) , Y ( s − τ )) − σ ( X ( s ) , X ( s − τ )) k d s (cid:19) ≤ E (cid:18) sup ≤ u ≤ t | e ( u ) | p (cid:19) + C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p + C ∆ p . Consequently, by sorting H ( t ) − H ( t ) together, we arrive at E (cid:18) sup ≤ u ≤ t | e ( u ) | p (cid:19) ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p . By the definition of e ( t ), we derive from (A3) that | Y ( t ) − X ( t ) | p ≤ p − | e ( t ) | p + 3 p − | θb ( Y ( t ) , Y ( t − τ ))∆ | p + 3 p − | D ( Y ( t − τ )) − D ( X ( t − τ )) | p ≤ p − | e ( t ) | p + 3 p − θ p ∆ p | b ( Y ( t ) , Y ( t − τ )) | p + 3 p − | V ( Y ( t − τ ) , X ( t − τ )) | p | Y ( t − τ ) − X ( t − τ ) | p . Taking (A1) and Lemma 2.3 into consideration, E (cid:18) sup ≤ u ≤ t | Y ( u ) − X ( u ) | p (cid:19) ≤ C E (cid:18) sup ≤ u ≤ t | e ( u ) | p (cid:19) + C ∆ p + C E (cid:18) sup ≤ u ≤ t | Y ( u − τ ) − X ( u − τ ) | p (cid:19) ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p + C ∆ p + C E sup ≤ u ≤ ( t − τ ) ∨ | Y ( u ) − X ( u ) | p ! . The Gronwall inequality yields E (cid:18) sup ≤ u ≤ t | Y ( u ) − X ( u ) | p (cid:19) ≤ C ∆ p + C E sup ≤ u ≤ ( t − τ ) ∨ | Y ( u ) − X ( u ) | p ! . Again, the desired result follows by the induction.16ith strong convergence rate given in Theorem 2.4, we can easily show the followingresult on almost sure convergence. Theorem 2.5. Let the conditions of Theorem 2.4 hold. Then the continuous form of θ -EMscheme (2.9) converges to the exact solution of (2.1) almost surely with order α < , i.e.,there exists a finite random variable ζ α such thatsup ≤ t ≤ T | Y ( t ) − X ( t ) | ≤ ζ α ∆ α for α ∈ (0 , ). Proof. Define a sequence ∆ k , k = 1 , , · · · such that ∆ = ∆ > ∆ > · · · and P k ∆( − α ) pk < ∞ , p ≥ 2. By the Chebyshev inequality and Theorem 2.4, for α < X k P (cid:18) sup ≤ t ≤ T | Y ( t ) − X ( t ) | > ∆ αk (cid:19) ≤ X k E (cid:18) sup ≤ t ≤ T | Y ( t ) − X ( t ) | p (cid:19) ∆ − αpk ≤ C X k ∆( − α ) pk < ∞ . The Borel-Cantelli lemma implies that there exists a finite random variable ζ α such thatsup ≤ t ≤ T | Y ( t ) − X ( t ) | ≤ ζ α ∆ α . In this section, we further introduce some notation. Let N ( · , · ) be a Poisson random processwith characteristic measure λ on a measurable subset U of [0 , ∞ ) such that λ ( U ) < ∞ , then e N (d u, d t ) = N (d u, d t ) − λ (d u )d t is a compensated martingale process. We consider thefollowing neutral SDDE with jumps on R n :d[ X ( t ) − D ( X ( t − τ ))] = b ( X ( t ) , X ( t − τ ))d t + Z U h ( X ( t ) , X ( t − τ ) , u ) e N (d u, d t ) , t ≥ X ( θ ) = ξ ( θ ) ∈ L p F ([ − τ, R n ) for θ ∈ [ − τ, ξ is an F -measurable D ([ − τ, R n )-valued random variable such that E k ξ k p ∞ < ∞ for p ≥ 2, where D ([ − τ, R n )denotes the space of all c´adl´ag paths ζ : [ − τ, → R n with uniform norm k ζ k ∞ :=sup − τ ≤ θ ≤ | ζ ( θ ) | . Here, D : R n → R n , and b : R n × R n → R n , h : R n × R n × U → R n are mea-surable functions. We further assume that b is a continuous function and R U | u | p λ (d u ) < ∞ for p ≥ 2. Similar to Brownian motion case, for x, y, x, y ∈ R n , we shall assume that:17 A5) There exist positive constants K and r ≥ | h ( x, y, u ) − h ( x, y, u ) | ≤ [ K | x − x | + V ( y, y ) | y − y | ] | u | r , and | h (0 , , u ) | ≤ | u | r . Remark 3.1. With assumption (A5), we have | h ( x, y, u ) | ≤| h ( x, y, u ) − h (0 , , u ) | + | h (0 , , u ) | ≤ [1 + K | x | + V ( y, | y | ] | u | r . Lemma 3.1. Let (A1), (A3) and (A5) hold. Then there exists a unique global solution to(3.1), moreover, the solution has the property that for any p ≥ T > E (cid:18) sup ≤ t ≤ T | X ( t ) | p (cid:19) ≤ C, where C = C ( ξ, p, T ) is a positive constant which only depends on the initial data ξ and p, T . Proof. We omit the proof here since it is similar to that of Lemma 2.1.We now introduce the θ -EM scheme for (3.1). Given any time T > τ > 0, assumethat T and τ are rational numbers, and there exists two positive integers such that ∆ = τm = TM , where ∆ ∈ (0 , 1) is the step size. For k = − m, · · · , 0, set y t k = ξ ( k ∆); For k = 0 , , · · · , M − 1, we form y t k +1 − D ( y t k +1 − m ) = y t k − D ( y t k − m ) + θb ( y t k +1 , y t k +1 − m )∆+ (1 − θ ) b ( y t k , y t k − m )∆ + Z U h ( y t k , y t k − m , u )∆ e N k (d u ) , (3.3)where t k = k ∆, and ∆ e N k (d u ) = e N ( t k +1 , d u ) − e N ( t k , d u ). Here θ ∈ [0 , 1] is an additionalparameter 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 the backward EM scheme. Herewe always assume θ ≥ / 2. The corresponding split-step θ -EM scheme to (3.1) is defined asfollows: For k = − m, · · · , − 1, set z t k = y t k = ξ ( k ∆); For k = 0 , , · · · , M − ( y t k = D ( y t k − m ) + z t k − D ( z t k − m ) + θb ( y t k , y t k − m )∆ ,z t k +1 = D ( z t k +1 − m ) + z t k − D ( z t k − m ) + b ( y t k , y t k − m )∆ + R U h ( y t k , y t k − m , u )∆ e N k (d u ) . It is easy to see y t k +1 in (3.4) can be rewritten as the form of (3.3). Due to the implicitnessof θ -EM scheme, we require 0 < ∆ ≤ ∆ ∗ , where ∆ ∗ ∈ (0 , (2 K ∨ K ) − θ − ), K and K aredefined as in (A1) and Remark 2.2 with σ ≡ respectively. Firstly, we introduce an important lemma coming from [13].18 emma 3.2. Let φ : R + × U → R n be progressively measurable and assume that the rightside is finite. Then there exists a positive constant C such that E (cid:18) sup ≤ s ≤ t (cid:12)(cid:12)(cid:12)(cid:12)Z s Z U φ ( r − , u ) e N (d u, d r ) (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) ≤ C E Z t Z U | φ ( s, u ) | p λ (d u )d s for p ≥ Lemma 3.3. Let (A1), (A3) and (A5) hold. Then, there exists a positive constant C independent of ∆ such that E (cid:18) sup ≤ k ≤ M | y t k | p (cid:19) ≤ C for p ≥ Proof. It is easy to see from (3.4) | z t k +1 − D ( z t k +1 − m ) | = | z t k − D ( z t k − m ) | + 2 h z t k − D ( z t k − m ) , b ( y t k , y t k − m )∆ i + | b ( y t k , y t k − m ) | ∆ + (cid:12)(cid:12)(cid:12)(cid:12)Z U h ( y t k , y t k − m , u )∆ e N k (d u ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:28) z t k − D ( z t k − m ) + b ( y t k , y t k − m )∆ , Z U h ( y t k , y t k − m , u )∆ e N k (d u ) (cid:29) = | z t k − D ( z t k − m ) | + 2 h y t k − D ( y t k − m ) , b ( y t k , y t k − m )∆ i + (1 − θ ) | b ( y t k , y t k − m ) | ∆ + (cid:12)(cid:12)(cid:12)(cid:12)Z U h ( y t k , y t k − m , u )∆ e N k (d u ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:28) y t k − D ( y t k − m ) + (1 − θ ) b ( y t k , y t k − m )∆ , Z U h ( y t k , y t k − m , u )∆ e N k (d u ) (cid:29) . Applying (3.3) to the last term and using assumption (A1) lead to | z t k +1 − D ( z t k +1 − m ) | ≤ | z t k − D ( z t k − m ) | + 2∆ h y t k − D ( y t k − m ) , b ( y t k , y t k − m ) i + (cid:12)(cid:12)(cid:12)(cid:12)Z U h ( y t k , y t k − m , u )∆ e N k (d u ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 θ (cid:28) y t k − D ( y t k − m ) , Z U h ( y t k , y t k − m , u )∆ e N k (d u ) (cid:29) − − θθ (cid:28) z t k − D ( z t k − m ) , Z U h ( y t k , y t k − m , u )∆ e N k (d u ) (cid:29) ≤| z t k − D ( z t k − m ) | + ∆ K (1 + | y t k | ) + ∆ | V ( y t k − m , | | y t k − m | + (cid:12)(cid:12)(cid:12)(cid:12)Z U h ( y t k , y t k − m , u )∆ e N k (d u ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 θ (cid:28) y t k − D ( y t k − m ) , Z U h ( y t k , y t k − m , u )∆ e N k (d u ) (cid:29) − − θθ (cid:28) z t k − D ( z t k − m ) , Z U h ( y t k , y t k − m , u )∆ e N k (d u ) (cid:29) . k , we deduce that | z t k +1 − D ( z t k +1 − m ) | ≤ | z t − D ( z t − m ) | + KT + ∆ K k X i =0 | y t i | + ∆ k X i =0 | V ( y t i − m , | | y t i − m | + k X i =0 (cid:12)(cid:12)(cid:12)(cid:12)Z U h ( y t i , y t i − m , u )∆ e N i (d u ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 θ k X i =0 (cid:28) y t i − D ( y t i − m ) , Z U h ( y t i , y t i − m , u )∆ e N i (d u ) (cid:29) − − θθ k X i =0 (cid:28) z t i − D ( z t i − m ) , Z U h ( y t i , y t i − m , u )∆ e N i (d u ) (cid:29) . Consequently, | z t k +1 − D ( z t k +1 − m ) | p ≤ p − ( | z t − D ( z t − m ) | + KT ) p + 6 p − K p ∆ p k X i =0 | y t i | ! p + 6 p − ∆ p k X i =0 | V ( y t i − m , | | y t i − m | ! p + 6 p − k X i =0 (cid:12)(cid:12)(cid:12)(cid:12)Z U h ( y t i , y t i − m , u )∆ e N i (d u ) (cid:12)(cid:12)(cid:12)(cid:12) ! p + 6 p − p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i =0 (cid:28) y t i − D ( y t i − m ) , Z U h ( y t i , y t i − m , u )∆ e N i (d u ) (cid:29)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p + 6 p − p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i =0 (cid:28) z t i − D ( z t i − m ) , Z U h ( y t i , y t i − m , u )∆ e N i (d u ) (cid:29)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p . With assumption (A5), we find that for 0 < j < M , E " sup ≤ k ≤ j k X i =0 (cid:12)(cid:12)(cid:12)(cid:12)Z U h ( y t i , y t i − m , u )∆ e N i (d u ) (cid:12)(cid:12)(cid:12)(cid:12) ! p ≤ M p − C E j X i =0 Z U | h ( y t i , y t i − m , u ) | p λ (d u ) ! ≤ C j X i =0 E Z U ([1 + K | y t i | + V ( y t i − m , | y t i − m | ] p | u | pr ) λ (d u ) ≤ C + C j X i =0 E | y t i | p + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) . E " sup ≤ k ≤ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i =0 (cid:28) y t i − D ( y t i − m ) , Z U h ( y t i , y t i − m , u )∆ e N i (d u ) (cid:29)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ C E j X i =0 | y t i − D ( y t i − m ) | Z U | h ( y t i , y t i − m , u ) | λ (d u ) ! p ≤ C E j X i =0 | y t i − D ( y t i − m ) | p Z U [1 + K | y t i | + V ( y t i − m , | y t i − m | ] p | u | pr λ (d u ) ≤ C + C j X i =0 E | y t i | p + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) . Similarly, by (A5) and Lemma 3.2 again E " sup ≤ k ≤ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i =0 (cid:28) z t i − D ( z t i − m ) , Z U h ( y t i , y t i − m , u )∆ e N i (d u ) (cid:29)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ C + C j X i =0 E | z t i − D ( z t i − m ) | p + C j X i =0 E | y t i | p + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) . This implies that E (cid:20) sup ≤ k ≤ j +1 | z t k − D ( z t k − m ) | p (cid:21) ≤ C + C j X i =0 E | z t i − D ( z t i − m ) | p + C j X i =0 E | y t i | p + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) + C j X i =0 E ( | V ( y t i − m , | p | y t i − m | p ) ≤ C + C j X i =0 E (cid:20) sup ≤ k ≤ i | z t k − D ( z t k − m ) | p (cid:21) + C j X i =0 E | y t i | p + C j X i =0 E | y t i − m | p ( l +1) . By the discrete Gronwall inequality we find that E (cid:20) sup ≤ k ≤ j +1 | z t k − D ( z t k − m ) | p (cid:21) ≤ C + C j X i =0 E | y t i | p + C j X i =0 E | y t i − m | p ( l +1) . Following the steps of (2.13), the desired assertion can be derived by similar skills.21 .2 Convergence Rates Firstly, we define the corresponding continuous-time split-step θ -EM solution Z ( t ) as follows:For any t ∈ [ − τ, Z ( t ) = ξ ( t ), Z (0) = ξ (0) − θb ( ξ (0) , ξ ( − τ ))∆; For any t ∈ [0 , T ],(3.5) d[ Z ( t ) − D ( Z ( t − τ ))] = b ( Y ( t ) , Y ( t − τ ))d t + Z U h ( Y ( t ) , Y ( t − τ ) , u ) e N (d u, d t ) , where Y ( t ) is defined by Y ( t ) := y t k for t ∈ [ t k , t k +1 ) , thus Y ( t − τ ) = y t k − m . The continuous form of θ -EM solution Y ( t ) is defined by(3.6) Y ( t ) − D ( Y ( t − τ )) = Z ( t ) − D ( Z ( t − τ )) + θb ( Y ( t ) , Y ( t − τ ))∆ . Lemma 3.4. Consider the θ -EM scheme (3.3), and let (A1), (A3)-(A5) hold. Then, for any p ≥ 2, the continuous form Y ( t ) of θ -EM scheme has the following properties: E (cid:18) sup ≤ t ≤ T | Y ( t ) | p (cid:19) ≤ C, and E (cid:18) sup ≤ t ≤ T | Y ( t ) − Y ( t ) | p (cid:19) ≤ C ∆ , where C is a constant independent of ∆. Proof. The proof is similar to that of Lemma 2.3, here we only give the most critical partto show the differences between the Brownian motion case. For t ∈ [ t k , t k +1 ), (3.5) gives that Z ( t ) − D ( Z ( t − τ )) − Z ( t k ) + D ( Z ( t k − m ))= Z tt k b ( Y ( s ) , Y ( s − τ ))d s + Z tt k Z U h ( Y ( s ) , Y ( s − τ ) , u ) e N (d u, d s ) . Denote by Φ( Z ( t ) , Z ( t k )) = Z ( t ) − D ( Z ( t − τ )) − Z ( t k ) + D ( Z ( t k − m )), then E sup t k ≤ t Let assumptions (A1), (A3)-(A5) hold, then the θ -EM solution Y ( t ) con-verges to the exact solution X ( t ) in L p sense, i.e., E (cid:18) sup ≤ t ≤ T | Y ( t ) − X ( t ) | p (cid:19) ≤ C ∆ for p ≥ Proof. Let e ( t ) = Z ( t ) − D ( Z ( t − τ )) − X ( t ) + D ( X ( t − τ )), it is obvious that e ( t ) = e (0) + Z t [ b ( Y ( s ) , Y ( s − τ )) − b ( X ( s ) , X ( s − τ ))]d s + Z t Z U [ h ( Y ( s ) , Y ( s − τ ) , u ) − h ( X ( s ) , X ( s − τ ) , u )] e N (d u, d s ) , where e (0) = − θb ( ξ (0) , ξ ( − τ ))∆. Define µ ( t ) = b ( Y ( t ) , Y ( t − τ )) − b ( X ( t ) , X ( t − τ )) , and υ ( t ) = h ( Y ( t ) , Y ( t − τ ) , u ) − h ( X ( t ) , X ( t − τ ) , u ) . Application of the Itˆo formula yields | e ( t ) | p = | e (0) | p + p Z t | e ( s ) | p − h e ( s ) , µ ( s ) i d s + Z t Z U [ | e ( s ) + υ ( s ) | p − | e ( s ) | p − p | e ( s ) | p − h e ( s ) , υ ( s ) i ] λ (d u )d s + Z t Z U [ | e ( s ) + υ ( s ) | p − | e ( s ) | p ] e N (d u, d s ) ≤| e (0) | p + p Z t | e ( s ) | p − h e ( s ) , µ ( s ) i d s + C Z t Z U | e ( s ) | p − | υ ( s ) | λ (d u )d s + C Z t Z U | υ ( s ) | p λ (d u )d s + Z t Z U [ | e ( s ) + υ ( s ) | p − | e ( s ) | p ] e N (d u, d s )=: | e (0) | p + H ( t ) + H ( t ) + H ( t ) + H ( t ) . E (cid:18) sup ≤ u ≤ t | H ( u ) | (cid:19) ≤ C E Z t | e ( s ) | p d s + C E Z t | b ( Y ( s ) , Y ( s − τ )) − b ( Y ( s ) , Y ( s − τ )) | p d s + C E Z t | b ( Y ( s ) , Y ( s − τ )) − b ( Y ( s ) , Y ( s − τ )) | p d s + C E Z t | b ( Y ( s ) , Y ( s − τ )) − b ( X ( s ) , X ( s − τ )) | p d s ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p + C Z t [ E (1 + | Y ( s ) | l + | Y ( s ) | l ) p ] [ E | Y ( s ) − Y ( s ) | p ] d s + C Z t [ E (1 + | Y ( s − τ ) | l + | Y ( s − τ ) | l ) p ] [ E | Y ( s − τ ) − Y ( s − τ ) | p ] d s + C Z t [ E (1 + | Y ( s ) | l + | X ( s ) | l ) p ] [ E | Y ( s ) − X ( s ) | p ] d s + C Z t [ E (1 + | Y ( s − τ ) | l + | X ( s − τ ) | l ) p ] [ E | Y ( s − τ ) − X ( s − τ ) | p ] d s ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p + C ∆ . Similarly, we obtain E (cid:18) sup ≤ u ≤ t | H ( u ) | (cid:19) + E (cid:18) sup ≤ u ≤ t | H ( u ) | (cid:19) ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p + C ∆ . Furthermore, by Lemmas 3.2-3.4 and the H¨older inequality, we compute E (cid:18) sup ≤ u ≤ t | H ( u ) | (cid:19) ≤ E (cid:18) sup ≤ u ≤ t | e ( u ) | p (cid:19) + C E (cid:18)Z t Z U | υ ( s ) | p λ (d u )d s (cid:19) ≤ E (cid:18) sup ≤ u ≤ t | e ( u ) | p (cid:19) + C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ p + C ∆ . Putting H ( t ) − H ( t ) together, we arrive at E (cid:18) sup ≤ u ≤ t | e ( u ) | p (cid:19) ≤ C Z t E (cid:18) sup ≤ u ≤ s | Y ( u ) − X ( u ) | p (cid:19) d s + C ∆ . Consequently, following the process of Theorem 2.4, the desired result will be obtained.24 emark 3.2. We see from Theorems 2.4 and 3.5 that the strong convergence rate of θ -EMscheme for neutral SDDEs is for the Brownian motion case, while for the pure jumps case,the order is p , that is to say, lower moment has a better convergence rate for neutral SDDEswith jumps, whence it is better to use the mean-square convergence for jump case. Theorem 3.6. Let (A1), (A3)-(A5) hold, then the continuous form of θ -EM scheme (3.3)converges to the exact solution of (3.1) almost surely with order α < p , i.e., there exists afinite random variable ζ α such thatsup ≤ t ≤ T | Y ( t ) − X ( t ) | ≤ ζ α ∆ α for α ∈ (0 , p ). Proof. The desired result can be obtained with Theorem 3.5 similar to the process of The-orem 2.5. References [1] Bao, J.H., Yuan, C., Convergence rate of EM scheme for SDDEs, Proc. Amer. 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