Convergence Rate of Euler–Maruyama Scheme for SDEs with Hölder–Dini Continuous Drifts

In this paper, we are concerned with convergence rate of Euler–Maruyama scheme for stochastic differential equations with Hölder–Dini continuous drifts. The key contributions are as follows: (i) by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler–Maruyama scheme for a class of stochastic differential equations which allow the drifts to be Dini continuous and unbounded; (ii) by the aid of regularization properties of degenerate Kolmogrov equation, we discuss convergence rate of Euler–Maruyama scheme for a range of degenerate stochastic differential equations where the drifts are Hölder–Dini continuous of order 23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{2}{3}$$\end{document} with respect to the first component and are merely Dini-continuous concerning the second component.


Introduction and Main Results
In their paper [23], Wang and Zhang studied existence and uniqueness for a class of stochastic differential equations (SDEs) with Hölder-Dini continuous drifts; Wang Supported [22] also investigated the strong Feller property, log-Harnack inequality and gradient estimates for SDEs with Dini-continuous drifts. So far there are no numerical schemes available for SDEs with Hölder-Dini continuous drifts. So the aim of this paper is to prove the convergence of Euler-Maruyama (EM) scheme and obtain the rate of convergence for these equations under reasonable conditions.
It is well-known that convergence rate of EM for SDEs with regular coefficients is one-half, see, e.g., [11]. With regard to convergence rate of EM scheme under various settings, we refer to, e.g., [1] for stochastic differential delay equations (SDDEs) with polynomial growth with respect to (w.r.t.) the delay variables, [4] for SDDEs under local Lipschitz and monotonicity condition, [14] for SDEs with discontinuous coefficients, and [25] for SDEs under log-Lipschitz condition, whereas for SDEs with non-globally Lipschitz continuous coefficients; see, e.g., [2,[6][7][8], to name a few. On the other hand, Hairer et al. [5] have established the first result in the literature that Euler's method converges to the solution of an SDE with smooth coefficients in the strong and numerical weak sense without any arbitrarily small polynomial rate of convergence, and Jentzen et al. [9] have further given a counterexample that no approximation method converges to the true solution in the mean square sense with polynomial rate.
The rate of convergence of EM scheme for SDEs with irregular coefficients has also gained much attention. For instance, adopting the Yamada-Watanabe approximation approach, [3] discussed strong convergence rate in L p -norm sense; using the Yamada-Watanabe approximation trick and heat kernel estimate, [16] studied strong convergence rate in L 1 -norm sense for a class of non-degenerate SDEs, where the bounded drift term satisfies a weak monotonicity and is of bounded variation w.r.t. a Gaussian measure and the diffusion term is Hölder continuous; applying the Zvonkin transformation, [18] discussed strong convergence rate in L p -norm sense for SDEs with additive noises, where the drift coefficient is bounded and Hölder continuous.
It is worth pointing out that [16,18] focused on convergence rate of EM for SDEs with Hölder continuous and bounded drifts, which rules out Hölder-Dini continuous and unbounded drifts. On the other hand, most of the existing literature on convergence rate of EM scheme is concerned with non-degenerate SDEs. Yet the corresponding issue for degenerate SDEs is scarce, to the best of our knowledge. So, in this work, we will not only investigate the convergence of the EM scheme for SDEs with Hölder-Dini continuous drifts, but will also study the degenerate setup. For wellposedness of SDEs with singular coefficients, we refer to, e.g., [13,22,23,27] for more details.
Throughout the paper, the following notation will be used. Let n, m be positive integers, (R n , ·, · , |·|) the n-dimensional Euclidean space, and R n ⊗R m the family of all n×m matrices. Let · and · HS stand for the usual operator norm and the Hilbert-Schmidt norm, respectively. Fix T > 0 and set f T ,∞ := sup t∈[0,T ],x∈R m f (t, x) for an operator-valued map f on [0, T ] × R m . C(R m ; R n ) means the continuous functions f : R m → R n . Let C 2 (R n ; R n ⊗ R n ) be the family of all continuously twice differentiable functions f : R n → R n ⊗ R n . Denote M n non by the collection of all nonsingular n ×n-matrices. Let S 0 be the collection of all slowly varying functions φ : R + → R + at zero in Karamata's sense (i.e., lim t→0 φ(λt) φ(t) = 1 for any λ > 0), which are bounded from 0 and ∞ on [ε, ∞) for any ε > 0. Let D 0 be the family of Dini functions, i.e., for some constants δ > 0 and c ≥ e 3+2δ . Set for some constants c, δ > 0 and α ∈ (0, 1). Before proceeding further, a few words about the notation are in order. Generic constants will be denoted by c; we use the shorthand notation a b to mean a ≤ c b. If the constant c depends on a parameter p, we shall also write c p and a p b. Throughout the paper, for fixed T > 0, C T > 0, dependent on the quantity T , is a generic constant which may change from line to line.

Non-degenerate SDEs with Bounded Coefficients
In this subsection, we consider an SDE on (R n , ·, · , | · |) where b : R + ×R n → R n , σ : R + ×R n → R n ⊗R n , and (W t ) t≥0 is an n-dimensional Brownian motion defined on a complete filtered probability space ( , F , (F t ) t≥0 , P).
With regard to (1.1), we suppose that there exists φ ∈ D such that, for any s, t ∈ [0, T ] and x, y ∈ R n , where ∇ i means the ith order gradient operator; (A2) (Regularity of b w.r.t. spatial variables) Without loss of generality, we take an integer N > 0 sufficiently large such that the stepsize δ := T /N ∈ (0, 1). The continuous-time EM scheme corresponding to (1.1) Herein, t δ := t/δ δ with t/δ the integer part of t/δ. The first contribution in this paper is stated as follows.
for some constant C T ≥ 1.

Non-degenerate SDEs with Unbounded Coefficients
As we see, in Theorem 1.1, the coefficients are uniformly bounded, and that the drift term b satisfies the global Dini-continuous condition [see (A2) above], which seems to be a little bit stringent. Therefore, concerning the coefficients, it is quite natural to replace uniform boundedness by local boundedness and global Dini continuity by local Dini continuity, respectively. In lieu of (A1)-(A3), as for (1.1) we assume that, for any s, t ∈ [0, T ] and k ≥ 1, By employing the cutoff approach, Theorem 1.1 can be extended to include SDEs with local Dini-continuous coefficients, which is presented as below.

Remark 1.3
In fact, in terms of [10, Theorem D], (1.4) holds under (A1')-(A3') as well as the pathwise uniqueness of (1.1), whereas in Sect. 4 we provide an alternative proof of (1.4) in order to reveal the convergence rate of the EM scheme.

Degenerate SDEs
So far, most of the existing literature on convergence of EM scheme for SDEs with irregular coefficients is concerned with non-degenerate SDEs; see, e.g., [16][17][18] for SDEs driven by Brownian motions, and [18] for SDEs driven by jump processes. The issue for the setup of degenerate SDEs has not yet been considered to date to the best of our knowledge. Nevertheless, in this subsection, we make an attempt to discuss the topic for degenerate SDEs with Hölder-Dini continuous drift.
For notation simplicity, we shall write R 2n instead of R n × R n . Consider the fol- is an ndimensional Brownian motion defined on the complete filtered probability space .7) is also called the stochastic Hamiltonian system, which has been investigated extensively in [24,26] on Bismut formulae, in [15] on ergodicity, in [21] on hypercontractivity, and in [23] on wellposedness, to name a few. For applications of the model (1.7), we refer to, e.g., Soize [20].
The continuous-time EM scheme associated with (1.7) is as follows: (1.8) Another contribution in this paper reads as below.
for some constant C T ≥ 1, in which According to [23, Theorem 1.2], (1.7) admits a unique strong solution under the assumptions (C1)-(C3). In fact, Nevertheless, the requirement φ ∈ D ε ∩ S 0 is imposed in order to reveal the order of convergence for the EM scheme above. By applying the cutoff approach and refining the argument of [23, Theorem 2.3] (see also Lemma 5.1 below), the boundedness of coefficients can be removed. We herein do not go into details since the corresponding trick is quite similar to the proof of Theorem 1.2.
The outline of this paper is organized as follows: In Sect. 2, we elaborate regularity of non-degenerate Kolmogorov equation, which plays an important role in dealing with convergence rate of EM scheme for non-degenerate SDEs with Hölder-Dini continuous and unbounded drifts; In Sects. 3, 4 and 5, we complete the proofs of Theorems 1.1, 1.2 and 1.4, respectively.

Regularity of Non-degenerate Kolmogorov Equation
Let (e i ) i≥1 be an orthogonal basis of R n . For any λ > 0, consider the following R n -valued parabolic equation: where ∇ b t u λ t means the directional derivative along the direction b t , 0 n is the zero vector in R n and L t : with σ * t standing for the transpose of σ t . Let (P 0 s,t ) 0≤s≤t be the semigroup generated by (Z s,x t ) 0≤s≤t which solves an SDE below By the chain rule, it follows from (2.1) that Thus, integrating from s to T and taking advantage of u λ T = 0 n , we arrive at For notation simplicity, let The lemma below plays a crucial role in investigating error analysis. (A1) and (A2), for any λ ≥ 9π 2 In what follows, we aim to show (ii) and (iii) hold true, one-by-one. Observe from [12, Theorem 3.1, p.218] that
In the sequel, we intend to verify (iii). Set γ s,t := ∇ η ∇ η Z s,x t for any η, η ∈ R n . Notice from (2.7) that By the Doob submartingale inequality and the Itô isometry, besides the Gronwall inequality and (2.8), we derive that From (2.10) and the Markov property, we have This further gives that Thus, applying Cauchy-Schwartz's inequality and Itô's isometry and taking (2.9), (2.11) and (2.12) into consideration, we derive that for some φ ∈ D. For f ∈ B b (R n ) such that (2.14), (2.13) implies that and utilized Jensen's inequality as well as Itô's isometry. 2 , note from (ii), (2.11) and (2.13) that

Proof of Theorem 1.1
With Lemma 2.1 in hand, we now in the position to complete the Proof of Theorem 1.1 Throughout the whole proof, we assume λ ≥ 9π 2 T ,σ b 2 T ,∞ + 4( b T ,∞ + T ,σ ) 2 so that (i)-(iii) in Lemma 2.1 hold. For any t ∈ [0, T ], applying Itô's formula to x + u λ t (x), x ∈ R n , we deduce from (2.1) that 1) where I n×n is an n × n identity matrix, and that (3.2) For notation simplicity, set Using the elementary inequality: (a +b) 2 ≤ (1+ε)(a 2 +ε −1 b 2 ) for arbitrary ε, a, b > 0, we derive from (ii) that In particular, taking ε = 1 leads to As a consequence, In what follows, our goal is to estimate the term on the right-hand side of (3.4). Observe from the definition of the Hilbert-Schmidt norm that Thus, by Hölder's inequality, Doob's submartingale inequality and Itô's isometry, it follows from (3.1), (3.2) and (3.5) that for some constant C T > 0. Also, applying Hölder's inequality and Itô's isometry, we deduce from (A1) that for some constant β T ≥ 1. By Taylor's expansion, it is obvious to see that From (A3) and due to the fact that φ(·) is increasing and δ ∈ (0, 1), one has In view of (A2), we derive that Thus, taking (3.6)-(3.9) into account and applying Jensen's inequality gives that Owing to φ ∈ D, we conclude that φ(0) = 0, φ > 0 and φ < 0 so that, for any c > 0 and δ ∈ (0, 1), where ξ ∈ (0, cδ). This further implies that Substituting this into (3.4) gives that Thus, Gronwall's inequality implies that there existsC T > 0 such that So the desired assertion holds immediately.

Proof of Theorem 1.2
We shall adopt the cutoff approach to finish the It is easy to see that b (k) and σ (k) satisfy (A1). For fixed k ≥ 1, consider the following SDE dX (k) (4.1) The corresponding continuous-time EM of (4.1) is defined by Applying BDG's inequality, Hölder's inequality and Gronwall's inequality, we deduce from (A1') that For the terms I 1 and I 3 , in terms of the Chebyshev inequality we find from (4.3) that where in the first display we have used the facts that for some c > 0. Next, according to (3.11), by taking λ = e c k 2 there exits C T > 0 such that Herein, C T ,σ (k) ,λ > 0 is defined as in (3.10) with σ and u λ replaced by σ (k) and u λ,k , respectively, where u λ,k solves (2.3) by writing b (k) instead of b. Consequently, we conclude that for somec 0 > 0. For any ε > 0, taking k = 2c 0 /ε and letting δ go to zero implies that Thus, (1.4) follows due to the arbitrariness of ε.