Abstract
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 \(\frac{2}{3}\) with respect to the first component and are merely Dini-continuous concerning the second component.
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1 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 [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, \((\mathbb {R}^n, \left\langle \cdot ,\cdot \right\rangle ,|\cdot |)\) the n-dimensional Euclidean space, and \(\mathbb {R}^n\otimes \mathbb {R}^m\) the family of all \(n\times m\) matrices. Let \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _{\mathrm {HS}}\) stand for the usual operator norm and the Hilbert–Schmidt norm, respectively. Fix \(T>0\) and set \(\Vert f\Vert _{T,\infty }:=\sup _{t\in [0,T],x\in \mathbb {R}^m}\Vert f(t,x)\Vert \) for an operator-valued map f on \([0,T]\times \mathbb {R}^m\). \(C(\mathbb {R}^m;\mathbb {R}^n)\) means the continuous functions \(f:\mathbb {R}^m\rightarrow \mathbb {R}^n.\) Let \(C^2(\mathbb {R}^n;\mathbb {R}^n\otimes \mathbb {R}^n)\) be the family of all continuously twice differentiable functions \(f:\mathbb {R}^n\rightarrow \mathbb {R}^n\otimes \mathbb {R}^n\). Denote \(\mathbb {M}_\mathrm{non}^n\) by the collection of all nonsingular \(n\times n\)-matrices. Let \(\mathscr {S}_0\) be the collection of all slowly varying functions \(\phi :\mathbb {R}_+\rightarrow \mathbb {R}_+\) at zero in Karamata’s sense (i.e., \(\lim _{t\rightarrow 0}\frac{\phi (\lambda t)}{\phi (t)}=1\) for any \(\lambda >0\)), which are bounded from 0 and \(\infty \) on \([\varepsilon ,\infty )\) for any \(\varepsilon >0\). Let \(\mathscr {D}_0\) be the family of Dini functions, i.e.,
A function \(f:\mathbb {R}^m\rightarrow \mathbb {R}^n\) is called Dini continuity if there exists \(\phi \in \mathscr {D}_0\) such that \(|f(x)-f(y)|\le \phi (|x-y|)\) for any \(x,y\in \mathbb {R}^m.\) We remark that every Dini-continuous function is continuous and every Lipschitz continuous function is Dini continuous; Moreover, if f is Hölder continuous, then f is Dini continuous. Nevertheless, there are numerous Dini-continuous functions, which are not Hölder continuous at all, see, e.g.,
for some constants \(\delta >0\) and \(c\ge {\mathrm{e}}^{3+2\delta }\). Set
for some \(\varepsilon \in (0,1)\) sufficiently small. Clearly, \(\phi \) constructed above belongs to \(\mathscr {D}^\varepsilon .\) A function \(f:\mathbb {R}^m\rightarrow \mathbb {R}^n\) is called Hölder–Dini continuity of order \(\alpha \in [0,1)\) if
for some \(\phi \in \mathscr {D}_0;\) see, for instance,
for some constants \(c,\delta >0\) and \(\alpha \in (0,1).\)
Before proceeding further, a few words about the notation are in order. Generic constants will be denoted by c; we use the shorthand notation \(a\lesssim b\) to mean \(a\le c\,b\). If the constant c depends on a parameter p, we shall also write \(c_p\) and \(a\lesssim _p b\). 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.
1.1 Non-degenerate SDEs with Bounded Coefficients
In this subsection, we consider an SDE on \((\mathbb {R}^n,\left\langle \cdot ,\cdot \right\rangle , |\cdot |)\)
where \(b: \mathbb {R}_+\times \mathbb {R}^n\rightarrow \mathbb {R}^n\), \(\sigma : \mathbb {R}_+\times \mathbb {R}^n\rightarrow \mathbb {R}^n\otimes \mathbb {R}^n\), and \((W_t)_{t\ge 0}\) is an n-dimensional Brownian motion defined on a complete filtered probability space \((\Omega , \mathscr {F}, (\mathscr {F}_t)_{t\ge 0}, \mathbb {P})\).
With regard to (1.1), we suppose that there exists \(\phi \in \mathscr {D}\) such that, for any \(s,t\in [0,T]\) and \(x,y\in \mathbb {R}^n,\)
- (A1) :
-
\(\sigma _t\in C^2(\mathbb {R}^n;\mathbb {R}^n\otimes \mathbb {R}^n)\), \(\sigma _t(x)\in \mathbb {M}_\mathrm{non}^n\), and
$$\begin{aligned} \Vert b\Vert _{T,\infty }+\sum _{i=0}^{2}\Vert \nabla ^{i} \sigma \Vert _{T,\infty }+\Vert \nabla \sigma ^{-1}\Vert _{T,\infty }+\Vert \sigma ^{-1}\Vert _{T,\infty }<\infty , \end{aligned}$$(1.2)where \(\nabla ^i\) means the ith order gradient operator;
- (A2) :
-
(Regularity of b w.r.t. spatial variables)
$$\begin{aligned} |b_t(x)-b_t(y)|\le \phi (|x-y|); \end{aligned}$$ - (A3) :
-
(Regularity of b and \(\sigma \) w.r.t. time variables)
$$\begin{aligned} |b_{s}(x)-b_{t}(x)|+\Vert \sigma _{s}(x)-\sigma _{t}(x)\Vert _{\mathrm {HS}}\le \phi (|s-t|). \end{aligned}$$
Without loss of generality, we take an integer \(N>0\) sufficiently large such that the stepsize \(\delta :=T/N\in (0,1)\). The continuous-time EM scheme corresponding to (1.1) is
Herein, \(t_\delta :=\lfloor t/\delta \rfloor \delta \) with \(\lfloor t/\delta \rfloor \) the integer part of \(t/\delta \).
The first contribution in this paper is stated as follows.
Theorem 1.1
Let (A1)–(A3) hold. Then
for some constant \(C_T\ge 1\).
Under (A1) and (A2), (1.1) admits a unique non-explosive strong solution \((X_t)_{t\in [0,T]}\); see, e.g., [22, Theorem 1.1]. In Theorem 1.1, by taking \(\phi (x)=x^\beta \) for \(x\ge 0\) and \(\beta \in (0,1],\) and inspecting closely the argument of Theorem 1.1, the concave property of \(\phi ^2\) can be dropped. Moreover, we have
So, our present result covers [18, Theorem 2.13], where the drift is Hölder continuous. In particular, for the setting \(\beta =1,\) it reduces to the classical result on strong convergence of EM scheme for SDEs with regular coefficients; see, e.g., [11] for more details.
1.2 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 \(s,t\in [0,T]\) and \(k\ge 1\),
- (A1’) :
-
\(\sigma _t\in C^2(\mathbb {R}^n;\mathbb {R}^n\otimes \mathbb {R}^n)\), for every \(x\in \mathbb {R}^n,\)\(\sigma _t(x)\in \mathbb {M}_\mathrm{non}^n\), and
$$\begin{aligned}&|b_t(x)|+\sum _{i=0}^{2}\Vert \nabla ^{i} \sigma _t(x)\Vert _{\mathrm {HS}}+\Vert \nabla \sigma _t^{-1}(x)\Vert _{\mathrm {HS}}\\&\quad +\Vert \sigma _t^{-1}(x)\Vert _{\mathrm {HS}}\le K_T (1+|x|),\quad x\in \mathbb {R}^n \end{aligned}$$for some constant \(K_T>0\);
- (A2’) :
-
(Regularity of b w.r.t. spatial variables) There exists \(\phi _k\in \mathscr {D}\) such that
$$\begin{aligned} |b_{t}(x)-b_{t}(y)|\le \phi _k(|x-y|),\quad |x|\vee |y|\le k; \end{aligned}$$ - (A3’) :
-
(Regularity of b and \(\sigma \) w.r.t. time variables) For \(\phi _k\in \mathscr {D}\) such that (A2’),
$$\begin{aligned} |b_{s}(x)-b_{t}(x)|+\Vert \sigma _{s}(x)-\sigma _{t}(x)\Vert _\mathrm{HS}\le \phi _k(|s-t|),\quad |x| \le k. \end{aligned}$$
By employing the cutoff approach, Theorem 1.1 can be extended to include SDEs with local Dini-continuous coefficients, which is presented as below.
Theorem 1.2
Assume (A1’)–(A3’) hold. Then it holds that
In particular, if \(\phi _k(s)={\mathrm{e}}^{{\mathrm{e}}^{c_0k^4}}s^{\alpha }, s\ge 0,\) for some \(\alpha \in (0,1]\) and \(c_0>0\), then
Moreover, if \(\sigma _{\cdot }(\cdot )\) is uniformly bounded (i.e., \(\Vert \sigma \Vert _{T,\infty }<\infty \)), then
for some constant \(C_T>0\), where \(\Vert \sigma \Vert _{T,\infty }:=\sup _{0\le t\le T,x\in \mathbb {R}^n}\Vert \sigma _t(x)\Vert _\mathrm{HS}\).
Under (A1’) and (A2’), (1.1) enjoys a unique strong solution \((X_t)_{t\in [0,T]}\); see, for instance, [22, Theorem 1.1]. Theorem 1.2 has improved the result in [17] since the drift involved is allowed to be unbounded and local Dini continuous, while the drift in [17] is bounded and Hölder continuous. Furthermore, by comparing (1.5) with (1.6), we infer that the convergence rate of EM scheme is better whenever \(\sigma _{\cdot }(\cdot )\) is uniformly bounded.
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.
1.3 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 \(\mathbb {R}^{2n}\) instead of \(\mathbb {R}^n\times \mathbb {R}^n\). Consider the following degenerate SDE on \(\mathbb {R}^{2n}\)
where \(b^{(1)}_t,b^{(2)}_t:\mathbb {R}^{2n}\rightarrow \mathbb {R}^{n}\), \(\sigma _t: \mathbb {R}^{2n} \rightarrow \mathbb {R}^n\otimes \mathbb {R}^n\), and \((W_t)_{t\ge 0}\) is an n-dimensional Brownian motion defined on the complete filtered probability space \((\Omega , \mathscr {F}, (\mathscr {F}_t)_{t\ge 0}, \mathbb {P})\). (1.7) 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].
Write the gradient operator on \(\mathbb {R}^{2n}\) as \(\nabla =(\nabla ^{(1)},\nabla ^{(2)})\), where \(\nabla ^{(1)}\) and \( \nabla ^{(2)}\) stand for the gradient operators w.r.t. the first and the second components, respectively.
We assume that there exists \(\phi \in \mathscr {D}^\varepsilon \cap \mathscr {S}_0\) such that for any \(x=(x^{(1)},x^{(2)}),y=(y^{(1)},y^{(2)})\in \mathbb {R}^{2n}\) and \(s,t\in [0,T]\),
- (C1) :
-
(Hypoellipticity) \((\nabla ^{(2)}b_t^{(1)})(x), \sigma _t(x)\in \mathbb {M}_\mathrm{non}^{n }\), and
$$\begin{aligned}&\Vert b^{(1)}\Vert _{T,\infty }+\Vert b^{(2)}\Vert _{T,\infty }+ \Vert \nabla ^{(2)}b^{(1)}\Vert _{T,\infty }+\left\| (\nabla ^{(2)}b^{(1)})^{-1}\right\| _{T,\infty }\\&\quad +\Vert \sigma \Vert _{T,\infty }+\Vert \nabla \sigma \Vert _{T,\infty }+\Vert \sigma ^{-1}\Vert _{T,\infty }<\infty ; \end{aligned}$$ - (C2) :
-
(Regularity of \(b^{(1)}\) w.r.t. spatial variables)
$$\begin{aligned} |b_t^{(1)}(x)-b_t^{(1)}(y)|\le |x^{(1)}-y^{(1)}|^{\frac{2}{3}}\phi (|x^{(1)}-y^{(1)}|)&\quad \text{ if } x^{(2)}=y^{(2)},\\ \Vert (\nabla ^{(2)}b_t^{(1)})(x)-(\nabla ^{(2)}b_t^{(1)})(y)\Vert _{\mathrm {HS}}\le \phi (|x^{(2)}-y^{(2)}|)&\quad \text{ if } x^{(1)}=y^{(1)}; \end{aligned}$$ - (C3) :
-
(Regularity of \(b^{(2)}\) w.r.t. spatial variables)
$$\begin{aligned} |b_t^{(2)}(x)-b_t^{(2)}(y)|\le |x^{(1)}-y^{(1)}|^{\frac{2}{3}}\phi (|x^{(1)}-y^{(1)}|)+ \phi ^{\frac{7}{2}}(|x^{(2)}-y^{(2)}|); \end{aligned}$$ - (C4) :
-
(Regularity of \(b^{(1)}, b^{(2)} \) and \(\sigma \) w.r.t. time variables)
$$\begin{aligned} |b_t^{(1)}(x)-b_s^{(1)}(x)| +|b_t^{(2)}(x)-b_s^{(2)}(x)| +\Vert \sigma _t(x)-\sigma _s(x)\Vert _{\mathrm {HS}}\le \phi (|t-s|). \end{aligned}$$
Observe from (C2) and (C3) that \(b^{(1)}(\cdot ,x^{(2)})\) and \(b^{(2)}(\cdot ,x^{(2)})\) with fixed \(x^{(2)}\) are locally Hölder–Dini continuous of order \(\frac{2}{3}\), and \((\nabla ^{(2)}b^{(1)})(x^{(1)},\cdot )\) and \(b^{(2)}(x^{(1)},\cdot )\) with fixed \(x^{(1)}\) are merely Dini-continuous.
The continuous-time EM scheme associated with (1.7) is as follows:
Another contribution in this paper reads as below.
Theorem 1.4
Let (C1)–(C4) hold. Then
for some constant \(C_T\ge 1\), in which
According to [23, Theorem 1.2], (1.7) admits a unique strong solution under the assumptions (C1)–(C3). In fact, (1.7) is wellposed under (C1)–(C3) with \(\phi \in \mathscr {D}_0\cap \mathscr {S}_0\) in lieu of \(\phi \in \mathscr {D}^\varepsilon \cap \mathscr {S}_0\). Nevertheless, the requirement \(\phi \in \mathscr {D}^\varepsilon \cap \mathscr {S}_0\) 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.
2 Regularity of Non-degenerate Kolmogorov Equation
Let \((e_i)_{i\ge 1}\) be an orthogonal basis of \(\mathbb {R}^n.\) For any \(\lambda >0\), consider the following \(\mathbb {R}^n\)-valued parabolic equation:
where \(\nabla _{b_t}u_t^\lambda \) means the directional derivative along the direction \(b_t\), \(\mathbf{0_n}\) is the zero vector in \(\mathbb {R}^n\) and
with \(\sigma _t^*\) standing for the transpose of \(\sigma _t.\) Let \((P_{s,t}^0)_{0\le s\le t}\) be the semigroup generated by \((Z_t^{s,x})_{0\le s\le t}\) 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 \(u_T^\lambda =\mathbf{0_n}\), we arrive at
For notation simplicity, let
and
Moreover, set
The lemma below plays a crucial role in investigating error analysis.
Lemma 2.1
Under (A1) and (A2), for any \( \lambda \ge 9\pi \Lambda _{T,\sigma }^2\Vert b\Vert _{T,\infty }^2+4(\Vert b\Vert _{T,\infty }+\Lambda _{T,\sigma })^2, \)
- (i):
-
(2.1) (i.e., (2.3)) enjoys a unique strong solution \(u^\lambda \in C([0,T];C_b^1(\mathbb {R}^n;\mathbb {R}^n))\);
- (ii):
-
\(\Vert \nabla u^\lambda \Vert _{T,\infty }\le \frac{1}{2}\);
- (iii):
-
\( \Vert \nabla ^2 u^\lambda \Vert _{T,\infty }\le \Upsilon _{T,\sigma }\int _0^T\frac{{\mathrm{e}}^{-\lambda t} }{t}\tilde{\phi }(\Vert \sigma \Vert _{T,\infty }\sqrt{t} ){\mathrm{d}}t, \) where \( \tilde{\phi }(s):=\sqrt{\phi ^2(s)+s},\, s\ge 0. \)
Proof
To show (i)–(iii), it boils down to refine the argument of [22, Lemma 2.1]. (i) holds for any \(\lambda \ge 4(\Vert b\Vert _{T,\infty }+\Lambda _{T,\sigma })^2\) via the Banach fixed-point theorem.
In what follows, we aim to show (ii) and (iii) hold true, one-by-one. Observe from [12, Theorem 3.1, p.218] that
Using Itô’s isometry and Gronwall’s inequality, one has
Utilizing the BDG inequality, we deduce that
which, combining with Gronwall’s inequality, yields that
Recall from [22, (2.8)] that the following Bismut formula
holds. By the Cauchy–Schwartz inequality, the Itô isometry and (2.8), we obtain that
where \( \Lambda _{T,\sigma }>0\) is defined in (2.4). So, one infers from (2.3) and (2.11) that
Thus, (ii) follows by taking \(\lambda \ge 9\pi \Lambda _{T,\sigma }^2\Vert b\Vert _{T,\infty }^2\).
In the sequel, we intend to verify (iii). Set \(\gamma _{s,t}:=\nabla _{\eta }\nabla _{\eta ^{\prime }}Z_t^{s,x}\) for any \(\eta ,\eta '\in \mathbb {R}^n\). 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
where \(\tilde{\Lambda }_{T,\sigma }>0\) is defined as in (2.5).
Set \(\tilde{f}(\cdot ):=f(\cdot )-f(x)\) for fixed \(x\in \mathbb {R}^n\) and \(f\in \mathscr {B}_b(\mathbb {R}^n)\) which verifies
for some \(\phi \in {\mathscr {D}}.\) For \(f\in \mathscr {B}_b(\mathbb {R}^n)\) such that (2.14), (2.13) implies that
where in the second display we have used that
and utilized Jensen’s inequality as well as Itô’s isometry.
Let \(f_t=b_t+\nabla _{b_t}u_t^\lambda \). For any \( \lambda \ge 9\pi \Lambda _{T,\sigma }^2\Vert b\Vert _{T,\infty }^2+4(\Vert b\Vert _{T,\infty }+\Lambda _{T,\sigma })^2, \) note from (ii), (2.11) and (2.13) that
with \( \tilde{\phi }(s):=\sqrt{\phi ^2(s)+s},\,s\ge 0, \) where in the second inequality we have used [22, Lemma 2.2 (1)], and the fact that the function \([0,1]\ni x\mapsto \sqrt{x}\log ({\mathrm{e}}+\frac{1}{x})\) is non-decreasing. As a result, (iii) follows from (2.15). \(\square \)
3 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 \( \lambda \ge 9\pi \Lambda _{T,\sigma }^2\Vert b\Vert _{T,\infty }^2+4(\Vert b\Vert _{T,\infty }+\Lambda _{T,\sigma })^2\) so that (i)–(iii) in Lemma 2.1 hold. For any \(t\in [0,T]\), applying Itô’s formula to \(x+u_t^\lambda (x),x\in \mathbb {R}^n\), we deduce from (2.1) that
where \(\mathbf{I_{n\times n}}\) is an \(n\times n\) identity matrix, and that
For notation simplicity, set
Using the elementary inequality: \( (a+b)^2\le (1+\varepsilon )(a^2+\varepsilon ^{-1}b^2)\) for arbitrary \(\varepsilon ,a,b>0, \) we derive from (ii) that
In particular, taking \(\varepsilon =1\) 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 \(\beta _T\ge 1.\) By Taylor’s expansion, it is obvious to see that
From (A3) and due to the fact that \(\phi (\cdot ) \) is increasing and \(\delta \in (0,1),\) one has
In view of (A2), we derive that
Thus, taking (3.6)–(3.9) into account and applying Jensen’s inequality gives that
where
Owing to \(\phi \in \mathscr {D},\) we conclude that \(\phi (0)=0,\)\(\phi '>0\) and \(\phi ''<0\) so that, for any \(c>0\) and \(\delta \in (0,1),\)
where \(\xi \in (0,c\delta ).\) This further implies that
Substituting this into (3.4) gives that
Thus, Gronwall’s inequality implies that there exists \(\tilde{C}_T>0\) such that
So the desired assertion holds immediately. \(\square \)
4 Proof of Theorem 1.2
We shall adopt the cutoff approach to finish the
Proof of Theorem 1.2
Take \(\psi \in C_b^\infty (\mathbb {R}_+)\) such that \(0\le \psi \le 1\), \(\psi (r)=1\) for \(r\in [0,1]\) and \(\psi (r)=0\) for \(r\ge 2\). For any \(t\in [0,T]\) and \(k\ge 1\), define the cutoff functions
It is easy to see that \(b^{(k)}\) and \(\sigma ^{(k)}\) satisfy (A1). For fixed \(k\ge 1,\) consider the following SDE
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 some constant \(C_T>0\). Note 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 \(\{X_t\ne X_t^{(k)}\}\subset \{\sup _{0\le s\le t}|X_s|\ge k\}\) and \(\{Y_t\ne Y_t^{(k)}\}\subset \{\sup _{0\le s\le t}|Y_s|\ge k\}\). Observe from (A1’) that \( 9\pi \Lambda _{T,\sigma ^{(k)}}^2\Vert b^{(k)}\Vert _{T,\infty }^2+4(\Vert b^{(k)}\Vert _{T,\infty }+\Lambda _{T,\sigma ^{(k)}})^2\le {\mathrm{e}}^{c\,k^2} \) for some \(c>0.\) Next, according to (3.11), by taking \(\lambda ={\mathrm{e}}^{c\,k^2}\) there exits \(C_T>0\) such that
Herein, \( C_{T,\sigma ^{(k)},\lambda }>0\) is defined as in (3.10) with \(\sigma \) and \(u^\lambda \) replaced by \(\sigma ^{(k)}\) and \(u^{\lambda ,k}\), respectively, where \(u^{\lambda ,k}\) solves (2.3) by writing \(b^{(k)}\) instead of b. Consequently, we conclude that
for some \(\bar{c}_0>0.\) For any \(\varepsilon >0\), taking \(k=\lfloor 2\bar{c}_0/\varepsilon \rfloor \) and letting \(\delta \) go to zero implies that
Thus, (1.4) follows due to the arbitrariness of \(\varepsilon \).
For \(\phi _k(s)={\mathrm{e}}^{{\mathrm{e}}^{c_0k^4}}s^{\alpha },s\ge 0,\) with \(\alpha \in (0,1],\) we deduce from Lemma 2.1 (iii) that
whenever
Since the right-hand side of (4.8) can be bounded by \({\mathrm{e}}^{{\mathrm{e}}^{\bar{C}_T\,k^4}}\) for some constant \(\bar{C}_T>0\) due to (A1’), we can take \(\lambda ={\mathrm{e}}^{{\mathrm{e}}^{\bar{C}_T\,k^4}}\) so that (4.7) holds. Thus, (4.6), together with (4.7) and (A1’), yields that
for some constants \(\hat{C}_T,\tilde{C}_T>0\). Thus, (1.5) follows immediately by taking
Next, we aim to show that (1.6) holds true. In view of (4.3) and (4.4), it follows from Hölder’s inequality that
By (A1’), we infer that
where
Thus, Gronwall’s inequality enables us to get that
For any integer \(k\ge 1\) such that
we derive from (4.11) that
This, by taking advantage of [19, Proposition 6.8], yields that
where \(\left\langle N\right\rangle _t\) stands for the quadratic variation process of \(N_t\). Next, by using the inequality: \((a-b)^2\ge \frac{1}{2}a^2-b^2,a,b\in \mathbb {R},\) we deduce from (4.12) that
Similarly, one can obtain that
Inserting (4.13) and (4.14) back into (4.10) leads to
This, together with (4.5), (4.7) and (A1’), gives that
for some constants \(\hat{C}_T,\tilde{C}_T>0\). As a consequence, (1.6) follows by taking k given in (4.9). \(\square \)
5 Proof of Theorem 1.4
For simplicity, for any \(f:\mathbb {R}^{m_1}\rightarrow \mathbb {R}^{m_2}\), let
The proof of Theorem 1.4 relies on regularization properties of the following \(\mathbb {R}^{2n}\)-valued degenerate parabolic equation
where \(\mathbf{0_{2n}}\) is the zero vector in \(\mathbb {R}^{2n}\),
The following lemma on regularity estimate of solution to (5.1) is taken from [23, Theorem 3.10, (4.4)] and is an essential ingredient in analyzing numerical approximation.
Lemma 5.1
Under (C1)–(C3), (5.1) has a unique solution \(u^\lambda \in C([0,T];C_b^1(\mathbb {R}^{2n};\mathbb {R}^{2n}))\) such that for all \(t\in [0,T],\)
where \(C>0\) is a constant.
From now on, we move forward to complete the
Proof of Theorem 1.4
For notation simplicity, set
Then (1.7) and (1.8) can be reformulated, respectively, as
where \(\mathbf{0_{n\times n}}\) is an \(n\times n\) zero matrix, and
Note from (5.2) that there exists \(\lambda _0>0\) sufficiently large such that for any \(t\in [0,T]\),
Applying Itô’s formula to \(x+u_t^\lambda (x)\) for any \( x\in \mathbb {R}^{2n}\), we deduce that
and that
where \(\mathbf{I}_{\mathbf{2n\times 2n}}\) is an \(2n\times 2n\) identity matrix. Thus, using Hölder’s inequality, Doob’s submartingale inequality and Itô’s isometry and taking (3.5) into consideration gives that
for some constant \(C_{0,T}>0\), where \(M_t^\lambda \) is defined as in (3.3). By using Hölder’s inequality and the BDG inequality, (C1) implies that
Utilizing Taylor’s expansion, one gets from (3.6), (5.3) and (5.6) that
Next, (C1), (C5) and (5.3) yield that
where we have also used that \(\phi (\cdot )\) is increasing and \(\delta \in (0,1).\) Additionally, by virtue of (C1), (C2), and (5.3), we infer from (C3) that
for some constant \(C_{1,T}>0\). From (C2), (C3), (5.6) and \(\phi \in \mathscr {D}^\varepsilon \), we derive from Hölder’s inequality and Jensen’s inequality that
for some constant \(C_{2,T}>0.\) With regard to the term \(\Lambda _3(t)\), (C1) and (5.6) lead to
Due to (C3), observe from Jensen’s inequality and (5.6) that
for some constant \( C_{3,T}>0.\) Consequently, we arrive at
for some constant \(C_{4,T}\ge 1\). Thus, the desired assertion follows from the Gronwall inequality. \(\square \)
References
Bao, J., Yuan, C.: Convergence rate of EM scheme for SDDEs. Proc. Am. Math. Soc. 141, 3231–3243 (2013)
Dareiotis, K., Kumar, C., Sabanis, S.: On tamed Euler approximations of SDEs driven by Lévy noise with applications to delay equations. SIAM J. Numer. Anal. 54, 1840–1872 (2016)
Gyöngy, I., Rásonyi, M.: A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients. Stoch. Process. Appl. 121, 2189–2200 (2011)
Gyöngy, I., Sabanis, S.: A note on Euler approximations for stochastic differential equations with delay. Appl. Math. Optim. 68, 391–412 (2013)
Hairer, M., Hutzenthaler, M., Jentzen, A.: Loss of regularity for Kolmogorov equations. Ann. Probab. 43, 468–527 (2015)
Higham, D.J., Mao, X., Yuan, C.: Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM J. Numer. Anal. 45, 592–609 (2007)
Hutzenthaler, M., Jentzen, A.: Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Am. Math. Soc. 236, v+99 (2015)
Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22, 1611–1641 (2012)
Jentzen, A., Müller-Gronbach, T., Yaroslavtseva, L.: On stochastic differential equations with arbitrary slow convergence rates for strong approximation. Commun. Math. Sci. 14, 1477–1500 (2016)
Kaneko, H., Nakao, S.: A note on approximation for stochasitc differential equations. Sémin. Probab. Strasbg. 22, 155–162 (1988)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)
Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. In: École d’été de probabilités de Saint-Flour, XII-1982, Lecture Notes in Math., 1097, pp. 143–303, Springer, Berlin (1984)
Krylov, N.V., Röckner, M.: Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131, 154–196 (2005)
Leobacher, G., Szögyenyi, M.: Convergence of the Euler–Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient. arXiv:1610.07047v2
Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Process. Appl. 101, 185–232 (2002)
Ngo, H.-L., Taguchi, D.: Strong rate of convergence for the Euler–Maruyama approximation of stochastic differential equations with irregular coefficients. Math. Comput. 85, 1793–1819 (2016)
Ngo, H.-L., Taguchi, D.: On the Euler–Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients. arXiv:1509.06532v1
Pamen, O.M., Taguchi, D.: Strong rate of convergence for the Euler–Maruyama approximation of SDEs with Hölder continuous drift coefficient. arXiv: 1508.07513v1
Shigekawa, I.: Stochastic Analysis, Translations of Mathematical Monographs, 224, Iwanami Series in Modern Mathematics. American Mathematical Society, Providence (2004)
Soize, C.: The Fokker–Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions. World Scientific Publishing Co., Inc, River Edge (1994)
Wang, F.-Y.: Hypercontractivity for Stochastic Hamiltonian Systems. arXiv:1409.1995
Wang, F.-Y.: Gradient estimates and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift. J. Differ. Equ. 260, 2792–2829 (2016)
Wang, F.-Y., Zhang, X.: Degenerate SDE with Hölder-Dini Drift and Non-Lipschitz Noise Coefficient. SIAM J. Math. Anal. 48, 2189–2226 (2016)
Wang, F.-Y., Zhang, X.: Derivative formula and applications for degenerate diffusion semigroups. J. Math. Pures Appl. 99, 726–740 (2013)
Yuan, C., Mao, X.: A note on the rate of convergence of the Euler–Maruyama method for stochastic differential equations. Stoch. Anal. Appl. 26, 325–333 (2008)
Zhang, X.: Stochastic flows and Bismut formulas for stochastic Hamiltonian systems. Stoch. Proc. Appl. 120, 1929–1949 (2010)
Zvonkin, A.K.: A transformation of the phase space of a diffusion process that removes the drift. Math. Sb. 93, 129–149 (1974)
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We are indebted to the referee, the associate editor and Professor Feng-Yu Wang for valuable comments and suggestions which have greatly improved our paper.
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Bao, J., Huang, X. & Yuan, C. Convergence Rate of Euler–Maruyama Scheme for SDEs with Hölder–Dini Continuous Drifts. J Theor Probab 32, 848–871 (2019). https://doi.org/10.1007/s10959-018-0854-9
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DOI: https://doi.org/10.1007/s10959-018-0854-9
Keywords
- Euler–Maruyama scheme
- Convergence rate
- Hölder–Dini continuity
- Degenerate stochastic differential equation
- Kolmogorov equation