Convergence Rate of EM Scheme for SDDEs

In this paper we investigate the convergence rate of Euler-Maruyama scheme for a class of stochastic differential delay equations, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. In particular, we reveal that the convergence rate of Euler-Maruyama scheme is 1/2$ for the Brownian motion case, while show that it is best to use the mean-square convergence for the pure jump case, and that the order of mean-square convergence is close to 1/2.


Introduction
Since most stochastic differential equations (SDEs) can not be solved explicitly, numerical methods have become essential. Recently, there is extensive literature in investigating the strong convergence, weak convergence or sample path convergence of numerical schemes for SDEs, e.g., in [3] for SDEs with a monotone condition, in [4,7,12] for SDEs with jumps, in [6,7,9,10] for stochastic differential delay equations (SDDEs) and in [4,5] for SDEs with a one-side Lipschitz condition. For the comprehensive monographs on numerical approximate methods of SDEs, we can also refer to [8,12,13]. Although the results on convergence of Euler-Maruyama (EM) schemes are substantial, there are limited ones on convergence rate under weaker conditions than global Lipschitz condition and linear growth condition. For example, a recent work in [2] reveals the convergence rate of EM schemes for a class of SDEs under a Hölder condition, and, with local Lipschitz constants satisfying a logarithm growth condition, [15] and [1,7] discuss the convergence rate of EM approximate methods for SDEs and stochastic functional differential equations with jumps, respectively. We should also point out that the strong convergence of EM schemes for SDDEs is, in general, discussed under a linear growth condition or bounded moments of analytic and numerical solutions, e.g., [7,9,10], and that the convergence rate [1,7] is also revealed under a linear growth condition.
To further motivate our work, we first consider an SDDE on R where a, b, c ∈ R, τ > 0, are constant, and W (t) is a scalar Brownian motion. It is easy to observe that both the drfit coefficient and the diffusion coefficient are highly nonlinear especially with respect to the delay arguments. Therefore, the existing convergence results, e.g., [7,9,10], can not cover Eq. (1.1), and the convergence rate of the corresponding EM scheme can not also be revealed by the techniques of [1,7] as we have explained in the end of the last paragraph. On the other hand, our work is also enlightened by the recent work in [2] such that consider SDE on R and discuss the convergence rate of the associated EM method, where g is Hölder continuous, of linear growth, and monotone decreasing with respect to the second variable. Motivated by the previous literature, in this paper we not only study the strong convergence of EM schemes for a class of SDDEs, which may be highly nonlinear with respect to the delay variables, but also reveal the convergence rate of the corresponding EM numerical methods. The rest of the paper are organized as follows: under highly nonlinear growth conditions with respect to the delay arguments, in Section 2 we reveal the convergence rate of EM schemes for SDDEs driven by Brownian motion is 1 2 , while in Section 3 we show that it is best to use the mean-square convergence for the pure jump case, and that the rate of mean-square convergence is close to 1 2 .

Convergence Rate for Brownian Motion Case
For integer n > 0, let (R n , ·, · , | · |) be the Euclidean space and A := trace(A * A) the Hilbert-Schmidt norm for a matrix A, where A * is its transpose. Let W (t) be an mdimensional Brownian motion defined on some complete probability space (Ω, F , P, {F t } t≥0 ). Throughout the paper, C > 0 denotes a generic constant whose values may change from lines to lines. For fixed T > 0, in this section we consider SDDE on R n To guarantee the existence and uniqueness of solution we introduce the following conditions. Let V i : R n × R n → R + such that for some K i > 0, q i ≥ 1 and arbitrary x, y ∈ R n . We further assume that (A1) b : R n × R n → R n and there exists L 1 > 0 such that (A2) σ : R n × R n → R n×m and there exists L 2 > 0 such that We now introduce an EM method for Eq. (2.1). Without loss of generality, we may assume that there exist sufficiently large integers N, M > 0 such that Define a continuous EM scheme associated with Eq. (2.1) Proof. Note that Eq. (2.1) has a unique local solution due to the fact that both b and σ are locally Lipschitzian. To verify that Eq. (2.1) admits a unique global solution on time interval [0, T ], it is sufficient to show that By a straightforward computation, we can deduce from (A1), (A2) and (2.2) that Set γ 1 := q 1 + 1 and γ 2 := q 2 + 1. To show (2.7), by (2.8) and (2.9), the Hölder inequality and the Burkhold-Davis-Gundy inequality, we have that for any p ≥ 2 and t ∈ [0, T ] where we have also used the Young inequality in the last step. This, together with the Gronwall inequality, yields that for t ∈ [0, T ] and p ≥ 2 The following argument is similar to that of [14, Theorem 2.1], however we give a detailed proof, which will also be used in the proof of Theorem 2.2 below. Let β := γ 1 ∨ γ 2 , and where [a] denotes the integer part of real number a. Thus, due to β ≥ 1 and p ≥ 2, it is easy to see that p i ≥ 2 such that which, combining (2.10) with the Hölder inequality, further leads to Repeating the previous procedures gives (2.7) and E sup 0≤t≤T |Y (t)| p ≤ C. Finally, the statement (2.6) can also be obtained by taking into account the Hölder inequality, the Burkhold-Davis-Gundy inequality and (2.5).
Remark 2.2. Lemma 2.1 gives a new result on existence and uniqueness of solutions to SDDEs on finite-time interval, where the coefficients may be polynomial of any degree d ≥ 1 with regard to the delay variables.
We can now state our main result, which not only shows the strong convergence of EM scheme associated with Eq. (2.1) but also reveals its convergence rate, although the drift coefficient and the diffusion coefficient may be highly nonlinear with respect to the delay arguments.
that is, the rate of convergence of EM scheme (2.4) is 1 2 . Proof. The argument is motivated by that of [2, Theorem 2.1]. For fixed δ > 1 and arbitrary ǫ ∈ (0, 1), there exists a continuous nonnegative function Then φ δǫ ∈ C 2 (R + ; R + ) possesses the following properties: By the definition of φ δǫ , it is trivial to note that V δǫ ∈ C 2 (R n ; R + ). For any x ∈ R n set We then have for x ∈ R n , i = 1, 2, · · · , n, where δ ij = 1 if i = j or otherwise 0, and For any t ∈ [0, T ], let Application of the Itô formula yields that By (2.14), (A1) and the Hölder inequality, we derive that and due to (A2) and (2.14) again that (2.16) By virtue of the Burkhold-Davis-Gundy inequality, the Hölder inequality and (2.14), for any p ≥ 2 and t ∈ [0, T ] This, together with the Gronwall inequality, implies It is easy to see that This, together with (2.19) and the Hölder inequality, further gives that The desired assertion then follows by repeating the previous procedures.
Remark 2.3. The strong convergence of EM scheme for SDDEs is generally investigated under local Lipschitz condition and bounded moments of analytic solutions and numerical solutions, or local Lipschitz condition and linear growth condition, e.g., [10]. In this section, for a class of SDDEs, which may be highly nonlinear with respect to the delay variables, we show the strong convergence of EM scheme under rather general conditions. To the best of our knowledge, there are relatively few results in the existing literature.
Remark 2.4. There are only limited results on convergence order of EM scheme for SDEs or SDDEs under weaker condition than global Lipschitz and linear growth condition, For example, under a Hölder continuous condition, [2] reveals the convergence order of EM scheme for a class of SDEs, and, with local Lipschitz constants satisfying a logarithm growth condition, [15] and [1,7] discuss the convergence rate of EM approximate methods for SDEs and stochastic functional differential equations with jumps respectively, where linear growth condition is imposed in [1,7]. While, in this section, under very general conditions we reveal the convergence order of EM scheme for a class of SDDEs although which are highly nonlinear with respect to delay arguments.

Convergence Rate for Pure Jump Case
In the last section we discuss the strong convergence of EM scheme for a class of SDDEs, and reveal the convergence rate is 1 2 although both the drift coefficient and the diffusion coefficient may be highly nonlinear with respect to the delay variables. In this section we turn to the counterpart for SDDEs with jumps. We further need to introduce some notation. Let B(R) be the Borel σ-algebra on R, and λ(dx) a σ-finite measure defined on B(R). Let p = (p(t)), t ∈ D p , be a stationary F t -Poisson point process on R with characteristic measure λ(·). Denote by N(dt, du) the Poisson counting measure associated with p, i.e., N(t, U) = s∈Dp,s≤t I U (p(s)) for U ∈ B(R). LetÑ(dt, du) := N(dt, du) − dtλ(du) be the compensated Poisson measure associated with N(dt, du). In what follows, we further assume that U |u| p λ(du) < ∞ for any p ≥ 2.
Remark 3.1. The jump coefficient may be also highly nonlinear with respect to the delay arguments, e.g., for x, y ∈ R, u ∈ U and q > 1, h(x, y, u) = y q u satisfies (A4).
Fix T > 0 and let the stepsize △ be defined by (2.3). The EM scheme associated with Eq. (3.1) is defined as follows: To reveal the convergence order of EM scheme (3.3), we need two auxiliary lemmas, where the first one is Bichteler-Jacod inequality for Poisson integrals, e.g., [11,Lemma 3.1].
Then there exists D(p) > 0 such that Using the Lemma above and the similar argument of Lemma 2.1, we have Remark 3.2. We remark that for p ≥ 2 all pth-moments of Y (t) −Ȳ (t) are bounded by △ up to a constant, which is completely different from the Brownian motion case (2.6). This is due to the fact that all moments of the incrementÑ((0, (i + 1)△], du) −Ñ((0, i△], du) have order O(△) for △ ∈ (0, 1).
We now state our main result in this section.
This, together with (3.5), (3.7) and the Hölder inequality, yields that where the last step is due to (3.8). Similarly, we have from (3.7)-(3.9) that Following the previous procedures gives that Remark 3.3. By Theorem 3.3, with p ≥ 2 increasing the convergence rate of EM scheme (3.3) is decreasing, which is quite different from the Brownian motion case with a constant order 1 2 , and it is therefore best to use the mean-square convergence for the jump case. On the other hand, we reveal that the order of mean-square convergence is close to 1 2 although the jump diffusion may be highly nonlinear with respect to the delay variables.