Citation: Siming Liu, Honglei Gao, Peng Hou, Yong Tan. Risk spillover effects of international crude oil market on China’s major markets[J]. AIMS Energy, 2019, 7(6): 819-840. doi: 10.3934/energy.2019.6.819
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Cloaking using transformation optics (changes of variables) was introduced by Pendry, Schurig and Smith [30] for the Maxwell system and by Leonhardt [16] in the geometric optics setting. These authors used a singular change of variables, which blows up a point into a cloaked region. The same transformation had been used to establish (singular) non-uniqueness in Calderon's problem in [10]. To avoid using the singular structure, various regularized schemes have been proposed. One of them was suggested by Kohn, Shen, Vogelius and Weinstein [11], where instead of a point, a small ball of radius ε is blown up to the cloaked region. Approximate cloaking for acoustic waves has been studied in the quasistatic regime [11,26], the time harmonic regime [12,19,27,20], and the time regime [28,29], and approximate cloaking for electromagnetic waves has been studied in the time harmonic regime [4,14,24], see also the references therein. Finite energy solutions for the singular scheme have been studied extensively [9,32,33]. There are also other ways to achieve cloaking effects, such as the use of plasmonic coating [2], active exterior sources [31], complementary media [13,22], or via localized resonance [23] (see also [17,21]).
The goal of this paper is to investigate approximate cloaking for the heat equation using transformation optics. Thermal cloaking via transformation optics was initiated by Guenneau, Amra and Venante [8]. Craster, Guenneau, Hutridurga and Pavliotis [6] investigate the approximate cloaking for the heat equation using the approximate scheme in the spirit of [11]. They show that for the time large enough, the largeness depends on ε, the degree of visibility is of the order εd (d=2,3) for sources that are independent of time. Their analysis is first based on the fact that as time goes to infinity, the solutions converge to the stationary states and then uses known results on approximate cloaking in the quasistatic regime [11,26].
In this paper, we show that approximate cloaking is achieved at any positive time and established the degree of invisibility of order ε in three dimensions and |lnε|−1 in two dimensions. Our results hold for a general source that depends on both time and space variables, and our estimates depend only on the range of the materials inside the cloaked region. The degree of visibility obtained herein is optimal due to the fact that a finite time interval is considered (compare with [6]). The analysis in this paper is of frequency type via Fourier transform with respect to time. This approach is robust and can be used in different context. A technical issue is on the blow up of the fundamental solution of the Helmholtz type equations in two dimensions in the low frequency regime. We emphasize that even though our setting is in a bounded domain, we employs Fourier transform in time instead of eigenmodes decomposition. This has the advantage that one can put the non-perturbed system and the cloaking system in the same context.
We next describe the problem in more detail and state the main result. Our starting point is the regularization scheme [11] in which a transformation blows up a small ball Bε (0<ε<1/2) instead of a point into the cloaked region B1 in Rd (d=2,3). Here and in what follows, for r>0, Br denotes the ball centered at the origin and of radius r in Rd. Our assumption on the geometry of the cloaked region is mainly to simplify the notations. Concerning the transformation, we consider the map Fε:Rd→Rd defined by
Fε(x)={x in Rd∖B2,(2−2ε2−ε+|x|2−ε)x|x| in B2∖Bε,xε in Bε. | (1.1) |
In what follows, we use the standard notations
F∗A(y)=∇F(x)A(x)∇FT(x)|det∇F(x)|,F∗ρ(y)=ρ(x)|det∇F(x)|,x=F−1(y), | (1.2) |
for the "pushforward" of a symmetric, matrix-valued function A, and a scalar function ρ, by the diffeomorphism F, and I denotes the identity matrix. The cloaking device in the region B2∖B1 constructed from the transformation technique is given by
(Fε∗I,Fε∗1) in B2∖B1, | (1.3) |
a pair of a matrix-valued function and a function that characterize the material properties in B2∖B1. Physically, this is the pair of the thermal diffusivity and the mass density of the material.
Let Ω with B2⋐Ω⊂Rd (d=2,3)* be a bounded region for which the heat flow is considered. Suppose that the medium outside B2 (the cloaking device and the cloaked region) is homogeneous so that it is characterized by the pair (I,1), and the cloaked region B1 is characterized by a pair (aO,ρO) where aO is a matrix-valued function and ρO is a real function, both defined in B1. The medium in Ω is then given by
*The notation D⋐Ω means that the closure of D is a subset of Ω.
(Ac,ρc)={(I,1) in Ω∖B2,(Fε∗I,Fε∗1) in B2∖B1,(aO,ρO) in B1. | (1.4) |
In what follows, we make the usual assumption that aO is symmetric and uniformly elliptic and ρO is a positive function bounded above and below by positive constants, i.e., for a.e. x∈B1,
Λ−1|ξ|2≤⟨aO(x)ξ,ξ⟩≤Λ|ξ|2 for all ξ∈Rd, | (1.5) |
and
Λ−1≤ρO(x)≤Λ, | (1.6) |
for some Λ≥1. Given a function f∈L1((0,+∞),L2(Ω)) and an initial condition u0∈L2(Ω), in the medium characterzied by (Ac,ρc), one obtains a unique weak solution uc∈L2((0,∞);H1(Ω)) ∩C([0,+∞);L2(Ω)) of the system
{∂t(ρcuc)−div(Ac∇uc)=f in (0,+∞)×Ω,uc=0 on (0,+∞)×∂Ω,uc(t=0,⋅)=u0 in Ω, | (1.7) |
and in the homogeneneous medium characterized by (I,1), one gets a unique weak solution u∈L2((0,∞);H1(Ω))∩C([0,+∞);L2(Ω)) of the system
{∂tu−Δu=f in (0,+∞)×Ω,uc=0 on (0,+∞)×∂Ω,uc(t=0,⋅)=u0 in Ω. | (1.8) |
The approximate cloaking meaning of the scheme (1.4) is given in the following result:
Theorem 1.1. Let u0∈L2(Ω) and f∈L1((0,+∞);L2(Ω)) be such that suppu0,suppf(t,⋅)⊂Ω∖B2 for t>0. Assume that uc and u are the solution of (1.7) and (1.8) respectively. Then, for 0<ε<1/2,
‖ |
for some positive constant C depending on \Lambda but independent of f , u_0 , and \varepsilon , where
e(\varepsilon, d) = \left\{ \begin{array}{cl} \varepsilon & \mathit{\mbox{if }} d = 3, \\[6pt] |\ln \varepsilon |^{-1} & \mathit{\mbox{if }} d = 2. \end{array}\right. |
As a consequence of Theorem 1.1, \lim_{\varepsilon \to 0} u_c (t, \cdot) = u (t, \cdot) in (0, + \infty) \times (\Omega \setminus B_2) for all f with compact support outside (0, +\infty) \times B_2 and for all u_0 with compact support outside B_2 . One therefore cannot detect the difference between (A_c, \rho_c) and (I, 1) as \varepsilon \rightarrow 0 by observation of u_c outside B_2 : Cloaking is achieved for observers outside B_2 in the limit as \varepsilon \rightarrow 0 .
We now briefly describe the idea of the proof. The starting point of the analysis is the invariance of the heat equations under a change of variables which we now state.
Lemma 1.1. Let d \ge 2 , T > 0 , \Omega be a bounded open subset of \mathbb{R}^d of class C^1 , and let A be an elliptic symmetric matrix-valued function, and \rho be a bounded, measurable function defined on \Omega bounded above and below by positive constants. Let F: \Omega \mapsto \Omega be bijective such that F and F^{-1} are Lipschitz, \det \nabla F > c for a.e. x \in \Omega for some c > 0 , and F(x) = x near \partial \Omega . Let f \in L^1\big((0, T); L^2(\Omega) \big) and u_0 \in L^2(\Omega) . Then u \in L^2\big((0, T); H^1_0(\Omega) \big) \cap C\big([0, T); L^2(\Omega) \big) is the weak solution of
\begin{equation} \left\{\begin{array}{cl} \partial_{t} (\rho u) - \operatorname{div} (A \nabla u) = f & \mathit{\mbox{in }} \Omega_T, \\[6pt] u = 0 & \mathit{\mbox{on }} (0, T) \times \partial \Omega, \\[6pt] u(0, \cdot) = u_0 & \mathit{\mbox{in }} \Omega, \end{array}\right. \end{equation} | (1.9) |
if and only if v(t, \cdot): = u (t, \cdot) \circ F^{-1} \in L^2\big((0, T); H^1_0(\Omega) \big) \cap C\big([0, T); L^2(\Omega) \big) is the weak solution of
\begin{equation} \left\{\begin{array}{cl} \partial_{t} (F_*\rho \, v) - \operatorname{div} (F_*A \, \nabla v) = F_*f & \mathit{\mbox{in }} \Omega_T, \\[6pt] u = 0 & \mathit{\mbox{on }} (0, T) \times \partial \Omega, \\[6pt] v(0, \cdot) = u_0 \circ F^{-1} & \mathit{\mbox{in }} \Omega. \end{array}\right. \end{equation} | (1.10) |
Recall that F_* is defined in (1.2). In this paper, we use the following standard definition of weak solutions:
Definition 1.1. Let d \ge 2 and T > 0 . We say a function
\begin{equation*} u \in L^2\big((0, T); H^1_0(\Omega) \big) \cap C \big( [0, T); L^2(\Omega) \big) \end{equation*} |
is a weak solution to (1.9) if u(0, \cdot) = u_0 \mbox{ in } \Omega and u satisfies
\begin{align} \frac{d}{dt} \int_{\Omega} \rho u(t, \cdot) \varphi + \int_{\Omega} A \nabla u(t, \cdot) \nabla \varphi = \int_{\Omega} f(t, \cdot) \varphi \mbox{ in } (0, T), \end{align} | (1.11) |
in the distributional sense for all \varphi \in H^1_0(\Omega) .
The existence and uniqueness of weak solutions are standard, see, e.g., [1] (in fact, in [1], f is assumed in L^2\big((0, T); L^2(\Omega) \big) , however, the conclusion holds also for f \in L^1\big((0, T); L^2(\Omega) \big) with a similar proof, see, e.g., [25]). The proof of Lemma 1.1 is similar to that of the Helmholtz equation, see, e.g., [12] (see also [6] for a parabolic version).
We now return to the idea of the proof of Theorem 1.1. Set
u_\varepsilon (t, \cdot) = u_c (t, \cdot) \circ F_\varepsilon^{-1} \mbox{ for } t \in (0, +\infty). |
Then u_\varepsilon is the unique solution of the system
\begin{equation} \left\{\begin{array}{cl} \partial_t (\rho_\varepsilon u_\varepsilon ) - \operatorname{div} (A_\varepsilon \nabla u_\varepsilon) = f & \mbox{ in } (0, + \infty) \times \Omega, \\[6pt] u_\varepsilon = 0 & \mbox{ on } (0, +\infty) \times \partial \Omega, \\[6pt] u_\varepsilon (t = 0, \cdot ) = u_0 & \mbox{ in } \Omega, \end{array}\right. \end{equation} | (1.12) |
where
\begin{equation} (A_\varepsilon, \rho_\varepsilon) = \left\{\begin{array}{cl} (I, 1) & \mbox{ in } \Omega \setminus B_\varepsilon, \\[6pt] \Big( \varepsilon^{2-d} a_O(\cdot/ \varepsilon), \varepsilon^{-d} \rho_O(\cdot / \varepsilon) \Big) & \mbox{ in } B_\varepsilon. \end{array} \right. \end{equation} | (1.13) |
Moreover,
u_c - u = u_\varepsilon - u \mbox{ in } (0, + \infty) \times (\Omega \setminus B_2). |
In comparing the coefficients of the systems verified by u and u_\varepsilon , the analysis can be derived from the study of the effect of a small inclusion B_\varepsilon . The case in which finite isotropic materials contain inside the small inclusion was investigated in [3] (see also [5] for a related context). The analysis in [3] partly involved the polarization tensor information and took the advantage of the fact that the coefficients inside the small inclusion are finite. In the cloaking context, Craster et al. [6] derived an estimate of the order \varepsilon^d for a time larger than a threshold one. Their analysis is based on long time behavior of solutions to parabolic equations and estimates for the degree of visibility of the conducting problem, see [11,26], hence the threshold time goes to infinity as \varepsilon \to 0 .
In this paper, to overcome the blow up of the coefficients inside the small inclusion and to achieve the cloaking effect at any positive time, we follow the approach of Nguyen and Vogelius in [28]. The idea is to derive appropriate estimates for the effect of small inclusions in the time domain from the ones in the frequency domain using the Fourier transform with respect to time. Due to the dissipative nature of the heat equation, the problem in the frequency for the heat equation is more stable than the one corresponding to the acoustic waves, see, e.g., [27,28], and the analysis is somehow easier to handle in the high frequency regime. After using a standard blow-up argument, a technical point in the analysis is to obtain an estimate for the solutions of the equation \Delta v + i \omega \varepsilon^2 v = 0 in \mathbb{R}^d \setminus B_1 ( \omega > 0 ) at the distance of the order 1/\varepsilon in which the dependence on \varepsilon and \omega are explicit (see Lemma 2.2). Due to the blow up of the fundamental solution in two dimensions, the analysis requires new ideas. We emphasize that even though our setting is in a bounded domain with zero Dirichlet boundary condition, we employs Fourier transform in time instead of eigenmodes decomposition as in [6] to put both systems of u_\varepsilon and u in the same context.
To implement the analysis in the frequency domain, let us introduce the Fourier transform with respect to time t :
\begin{equation} \hat \varphi (k, x) = \int_{ \mathbb{R }} \varphi(t, x) e^{i k t} \, dt \mbox{ for } k \in \mathbb{R}, \end{equation} | (2.1) |
for \varphi \in L^2((-\infty, + \infty), L^2(\mathbb{R}^d)) . Extending u, \, u_c , u_\rho and f by 0 for t < 0 , and considering the Fourier with respect to time at the frequency \omega > 0 , we obtain
\begin{equation*} \Delta \hat u + i \omega \hat u = - ( \hat f + u_0) \mbox{ in } \Omega, \end{equation*} |
and
\begin{equation*} \operatorname{div} (A_\varepsilon \nabla \hat u_\varepsilon) + i \omega \rho_\varepsilon \hat u_\varepsilon = - (\hat f + u_0) \mbox{ in } \Omega, \end{equation*} |
where
\begin{equation*} (A_\varepsilon, \rho_\varepsilon) = \left\{ \begin{array}{cl} (I, 1)& \mbox{ in } \Omega \setminus B_\varepsilon, \\[6pt] \big(\varepsilon^{2-d} a_O( \cdot/ \varepsilon) , \varepsilon^{-d}\rho_O( \cdot /\varepsilon) \big) & \mbox{ in } B_\varepsilon. \end{array}\right. \end{equation*} |
The main ingredient in the proof of Theorem 1.1 is the following:
Proposition 2.1. Let \omega > 0 , 0 < \varepsilon < 1/2 , and let g \in L^2(\Omega) with \operatorname{supp} g \subset \Omega \setminus B_2 . Assume that v, \, v_\varepsilon \in H^1(\Omega) are respectively the unique solution of the systems
\begin{equation*} \left\{\begin{array}{cl} \Delta v + i \omega v = g & \mathit{\mbox{in }} \Omega, \\[6pt] v = 0 & \mathit{\mbox{on }} \partial \Omega, \end{array}\right. \end{equation*} |
and
\begin{equation*} \left\{\begin{array}{cl} \operatorname{div} (A_\varepsilon \nabla v_\varepsilon) + i \omega \rho_\varepsilon v_\varepsilon = g & \mathit{\mbox{in }} \Omega, \\[6pt] v_\varepsilon = 0 & \mathit{\mbox{on }} \partial \Omega. \end{array}\right. \end{equation*} |
We have
\begin{equation} \| v_\varepsilon - v\|_{H^1(\Omega \setminus B_2)} \le C e(\varepsilon, \omega, d) (1+\omega^{-1/2}) \| g \|_{L^2(\Omega)}, \end{equation} | (2.2) |
for some positive constant C independent of \varepsilon , \omega and g . Here
\begin{equation} e(\varepsilon, \omega, 3) = \varepsilon e^{-\omega^{1/2} /4}, \end{equation} | (2.3) |
and
\begin{equation} e(\varepsilon, \omega, 2) = \left\{\begin{array}{cl} e^{- \omega^{1/2} /4}/|\ln \varepsilon |& \mathit{\mbox{if }} \omega \ge 1/2, \\[6pt] \ln\omega / \ln (\omega\varepsilon) & \mathit{\mbox{if }} 0 \lt \omega \lt 1/2. \end{array}\right. \end{equation} | (2.4) |
The rest of this section is divided into three subsections. In the first subsection, we present several lemmas used in the proof of Proposition 2.1. The proofs of Proposition 2.1 and Theorem 1.1 are then given in the second and the third subsections, respectively.
In this subsection, we state and prove several useful lemmas used in the proof of Proposition 2.1. Throughout, D \subset B_1 denotes a smooth, bounded, open subset of \mathbb{R}^d such that \mathbb{R}^d \setminus D is connected, and \nu denotes the unit normal vector field on \partial D , directed into \mathbb{R}^d \setminus D .
The first result is the following simple one:
Lemma 2.1. Let d = 2, \, 3 , k > 0 , and let v \in H^1 ({\mathbb{R }}^d \setminus D) be such that \Delta v +i k v = 0 in \mathbb{R}^d \setminus D . We have, for R > 2 ,
\begin{equation} \|v\|_{H^1(B_R\setminus D)} \le C_R (1+k) \|v\|_{H^{1/2}(\partial D)}, \end{equation} | (2.5) |
for some positive constants C_R independent of k and v .
Proof. Multiplying the equation by \bar v (the conjugate of v ) and integrating by parts, we have
\int_{ \mathbb{R}^d \setminus D} |\nabla v|^2 - i k \int_{ \mathbb{R}^d \setminus D} |v|^2 = \int_{\partial D} \partial_\nu v \bar v. |
This implies
\begin{equation} \int_{ \mathbb{R}^d \setminus D} |\nabla v|^2 + k \int_{ \mathbb{R}^d \setminus D} |v|^2 \le C \| \partial_{\nu} v \|_{H^{-1/2}(\partial D)} \| v \|_{H^{1/2}(\partial D)}. \end{equation} | (2.6) |
Here and in what follows, C denotes a positive constant independent of v and k . Since \Delta v = - i k v in B_2 \setminus D , by the trace theory, see, e.g., [7,Theorem 2.5], we have
\begin{equation} \| \partial_{\nu} v \|_{H^{-1/2}(\partial D)} \le C \Big(\|\nabla v \|_{L^2(B_2 \setminus D)} + \| \Delta v\|_{L^2(B_2 \setminus D)} \Big) \le C \Big(\| \nabla v\|_{L^2(B_2 \setminus D)} + k \|v \|_{L^2(B_2 \setminus D)} \Big). \end{equation} | (2.7) |
Combining (2.6) and (2.7) yields
\begin{equation} \int_{ \mathbb{R}^d \setminus D} |\nabla v|^2 + k \int_{ \mathbb{R}^d \setminus D} |v|^2 \le C (1+k) \| v \|_{H^{1/2}(\partial D)}^2 . \end{equation} | (2.8) |
The conclusion follows when k \ge 1 .
Next, consider the case 0 < k < 1 . In the case where d = 3 , the conclusion is a direct consequence of (2.8) and the Hardy inequality (see, e.g., [18,Lemma 2.5.7]):
\begin{equation} \int_{ \mathbb{R}^3 \setminus D} \frac{|v|^2}{|x|^2} \le C \int_{ \mathbb{R}^3\setminus D } |\nabla v|^2. \end{equation} | (2.9) |
We next consider the case where d = 2 . One just needs to show
\begin{equation} \int_{B_R \setminus D} |v|^2 \le C \| v\|^2_{H^{1/2}(\partial D)}. \end{equation} | (2.10) |
By the Hardy inequality (see, e.g., [18,Lemma 2.5.7]),
\begin{equation} \int_{ \mathbb{R}^2 \setminus D} \frac{|v|^2}{|x|^2 \ln (2 + |x|)^2} \le C \left( \int_{ \mathbb{R}^2\setminus D} |\nabla v|^2 + \int_{B_2 \setminus D} |v|^2 \right), \end{equation} | (2.11) |
it suffices to prove (2.10) for R = 2 by contradiction. Suppose that there exists a sequence (k_n) \to 0 and a sequence (v_n) \in H^1(\mathbb{R}^2 \setminus D) such that
\begin{equation*} \Delta v_n + i k_n v_n = 0 \mbox{ in } \mathbb{R}^2 \setminus D, \quad \| v_n\|_{L^2(B_2 \setminus D)} = 1, \quad \mbox{ and } \quad \lim\limits_{n \to + \infty} \|v_n \|_{H^{1/2}(\partial D)} = 0. \end{equation*} |
Denote
\begin{equation*} W^{1}( \mathbb{R}^2 \setminus D) = \left\{ u \in L^1_{ \operatorname{loc }}( \mathbb{R}^2 \setminus D); \frac{u(x)}{\ln(2 + |x|) \sqrt{1 + |x|^2 }} \in L^2( \mathbb{R}^2 \setminus D) \mbox{ and } \nabla u \in L^2( \mathbb{R}^2 \setminus D) \right\}. \end{equation*} |
By (2.8) and (2.11), one might assume that v_n converges to v weakly in H^1_{ \operatorname{loc }}(\mathbb{R}^2 \setminus D) and strongly in L^2(B_2 \setminus D) . Moreover, v \in W^1(\mathbb{R}^2 \setminus D) and v satisfies
\begin{equation} \Delta v = 0 \mbox{ in } \mathbb{R}^2 \setminus D, \quad v = 0 \mbox{ on } \partial D, \end{equation} | (2.12) |
and
\begin{equation} \| v \|_{L^2(B_2 \setminus D)} = 1. \end{equation} | (2.13) |
From (2.12), we have v = 0 in \mathbb{R}^2 \setminus D (see, e.g., [18]) which contradicts (2.13). The proof is complete.
We also have
Lemma 2.2. Let d = 2, 3 , \omega > 0 , 0 < \varepsilon < 1/2 , and let v \in H^1 ({\mathbb{R }}^d \setminus D) be a solution of \Delta v +i\omega \varepsilon^2 v = 0 in \mathbb{R}^d \setminus D . We have, for 3/2 < |x| < R ,
\begin{equation} | v(x/ \varepsilon)| \le C e(\varepsilon, \omega, d) \|v\|_{H^{1/2}(\partial D)}, \end{equation} | (2.14) |
for some positive constant C = C_{R} independent of \varepsilon , \omega and v .
Recall that e(\varepsilon, \omega, d) is given in (2.3) and (2.4).
Proof. By the trace theory and the regularity theory of elliptic equations, we have
\begin{equation} \| v\|_{L^2(\partial B_2)} + \| \nabla v\|_{L^2(\partial B_2)} \le C \| v\|_{H^2(B_{5/2} \setminus B_{3/2})} \le C (1 + \omega^{1/2} \varepsilon) \| v\|_{H^1(B_{3} \setminus B_{1})}. \end{equation} | (2.15) |
It follows from Lemma 2.1 that
\begin{equation} \| v\|_{L^2(\partial B_2)} + \| \nabla v\|_{L^2(\partial B_2)} \le C (1 + \omega^{3/2}) \| v\|_{H^{1/2}(\partial D)}. \end{equation} | (2.16) |
Here and in what follows in this proof, C denotes a positive constant depending only on R and D .
The representation formula gives
\begin{equation} v (x ) = \int_{\partial B_{2 }} \Big(G_{\ell} (x, y) \partial_r v (y) - \partial_{r_y} G_{\ell} (x, y) v(y)\Big) \, dy \mbox{ for } x \in \mathbb{R}^d \setminus \bar B_2, \end{equation} | (2.17) |
where \ell = e^{i \pi/4} \varepsilon \omega^{1/2} , and, for x \neq y ,
G_\ell(x, y) = \frac{e^{i\ell |x-y| }}{4 \pi |x -y|} \mbox{ if } d = 3 \quad \mbox{ and } \quad G_\ell(x, y) = \frac{i}{4}H^{(1)}_0 (\ell|x-y|) \mbox{ if } d = 2. |
Here H^{(1)}_0 is the Hankel function of the first kind of order 0. Recall, see, e.g., [15,Chapter 5], that
\begin{equation} H_0^{(1)} (z) = \frac{2i}{\pi}\ln\frac{|z|}{2} + 1 +\frac{2i\gamma}{\pi}+O(|z|^2\log|z|)\quad \text{ as } z\rightarrow 0, z\notin(-\infty, 0], \end{equation} | (2.18) |
and
\begin{equation} H_0^{(1)} (z) = \sqrt{\frac{2}{\pi z }} e^{i(z+\frac{\pi}{4})} (1+O(|z|^{-1})) \quad z\rightarrow \infty, z\notin(-\infty, 0]. \end{equation} | (2.19) |
We now consider the case d = 3 . We have, for 3/2 < |x| < R and y \in \partial B_{2} ,
|e^{i \ell |x/ \varepsilon - y|}| \le e^{- \frac{\sqrt{2 }}{2} \omega^{1/2} |x - \varepsilon y|} \le e^{- \omega^{1/2} |x|/ 3}. |
It follows that, for 3/2 < |x| < R and y \in \partial B_{2} ,
\begin{equation} |G_\ell(x/ \varepsilon, y)| \le C \varepsilon e^{- 3\omega^{1/2}/ 10}. \end{equation} | (2.20) |
Similarly, one has, for 3/2 < |x| < R and y \in \partial B_{2} ,
\begin{equation} |\partial_{r_y} G_\ell(x/ \varepsilon, y)| \le C \left( \frac{\varepsilon^2 \omega^{1/2 }}{|x|} + \frac{\varepsilon^2}{|x|^2} \right) e^{- \omega^{1/2} |x|/ 3} \le C \varepsilon e^{- 3\omega^{1/2} / 10}. \end{equation} | (2.21) |
Combining (2.17), (2.20) and (2.21) yields
\begin{equation*} |v(x/ \varepsilon)| \le C \varepsilon e^{- 3\omega^{1/2} / 10} (\|v \|_{L^2(\partial B_2)} + \| \nabla v\|_{L^2(\partial B_2)}) \mbox{ for } 3/2 \lt |x| \lt R. \end{equation*} |
We derive from (2.16) that
\begin{equation*} |v(x/ \varepsilon)| \le C \varepsilon e^{- \omega^{1/2} / 4} \| v\|_{H^{1/2}(\partial D)} \mbox{ for } 3/2 \lt |x| \lt R; \end{equation*} |
which is the conclusion in the case d = 3 .
We next deal with the case where d = 2 and \omega > \varepsilon^{-2}/4 , which is equivalent to |\ell| > 1/2 . From (2.19), we derive that, for 3/2 < |x| < R and y \in \partial B_{2} ,
\begin{equation} |G_\ell(x/ \varepsilon, y)| \le C \omega^{-1/4}e^{- 3\omega^{1/2} / 10} \quad \mbox{ and } \quad |\partial_{r_y} G_\ell(x/ \varepsilon, y)| \le C \varepsilon \omega^{1/4} e^{- 3\omega^{1/2} / 10}. \end{equation} | (2.22) |
Using (2.16) and combining (2.17) and (2.22), we obtain, since \omega > \varepsilon^{-2}/4 ,
\begin{equation*} |v(x/ \varepsilon)| \le C \varepsilon e^{- \omega^{1/2} / 4} \| v\|_{H^{1/2}(\partial D)} \mbox{ for } 3/2 \lt |x| \lt R, \end{equation*} |
which gives the conclusion in this case.
We finally deal with the case where d = 2 and 0 < \omega < \varepsilon^{-2}/4 , which is equivalent to |\ell| < 1/2 . From (2.17), we obtain, for x \in \partial B_{4} ,
\begin{equation} v(x) = \int_{\partial B_{2 }} \Big( \big[G_{\ell} (x, y) - G_\ell(x, 0) \big]\partial_r v (y) - \partial_{r_y} G_{\ell} (x, y) v(y)\Big) \, dy + \int_{\partial B_{2 }} G_\ell(x, 0) \partial_r v (y) \, dy. \end{equation} | (2.23) |
Since d = 2 , we have
\begin{equation*} \|v \|_{L^\infty(B_{5} \setminus B_{3})} \le C \|v \|_{H^2(B_{5} \setminus B_{3})} \le C \|v \|_{H^2(B_{5} \setminus B_{2})} \le C (1+\omega^{1/2})\|v \|_{H^1(B_{6} \setminus B_{1})}. \end{equation*} |
It follows from Lemma 2.1 and the trace theory that
\begin{equation} \|v \|_{L^\infty(B_{5} \setminus B_{3})} + \|v \|_{L^2(\partial B_{2})} + \|\nabla v \|_{L^2(\partial B_{2})} \le C (1+\omega^{3/2}) \| v\|_{H^{1/2}(\partial D)}. \end{equation} | (2.24) |
Since, by (2.18),
|\nabla_y G_\ell(x, y)| \le C \mbox{ for } x \in \partial B_{4} \mbox{ and } y \in \partial B_{2} |
and
|G_\ell(x, 0)|\ge C|\ln|\ell|| \mbox{ for } x \in \partial B_4, |
we derive from (2.23) and (2.24) that
\begin{equation} \Big| \int_{\partial B_2} \partial_r v (y) \, dy \Big| \le \frac{C (1+\omega^{3/2})}{ |\ln|\ell| |} \| v \|_{H^{1/2}(\partial D)}. \end{equation} | (2.25) |
Again using (2.17), we get, for 3/2 < |x| < R ,
\begin{equation} v(x/\varepsilon) = \int_{\partial B_2} \Big( \big[G_{\ell} (x/\varepsilon, y) - G_\ell(x/\varepsilon, 0) \big]\partial_r v (y) - \partial_{r_y} G_{\ell} (x/\varepsilon, y) v(y)\Big) \, dy + \int_{\partial B_2} G_\ell(x/ \varepsilon, 0) \partial_r v (y) \, dy. \end{equation} | (2.26) |
Since, by (2.18), for 0 < \omega < 1/2 ,
|G_{\ell} (x/ \varepsilon, 0)| \le C |\ln \omega| \quad \mbox{ and } \quad |\nabla_y G_\ell(x/\varepsilon, y)| \le C \varepsilon \quad \mbox{ for } 3/2 \lt |x| \lt R, \, y \in \partial B_{2}, |
and, by (2.19), for 1/2 < \omega < \varepsilon^{-2}/4 ,
|G_\ell(x/\varepsilon, 0)| \le C \omega^{-1/4} e^{- 3\omega^{1/2}/10} \mbox{ and } |\nabla_y G_\ell(x/\varepsilon, y)| \le C\varepsilon \omega^{1/4} e^{- 3\omega^{1/2}/ 10} \mbox{ for } 3/2 \lt |x| \lt R, \, y \in \partial B_{2}, |
we derive from (2.24), (2.25) and (2.26) that, for 3/2 < |x| < R ,
\begin{equation*} |v(x/ \varepsilon)| \le \left\{ \begin{array}{cl} \frac{C|\ln \omega| }{|\ln |\ell||} \|v \| _{H^{1/2}(\partial D)} & \mbox{ if } 0 \lt \omega \lt 1/2, \\[6pt] \frac{C \omega^{3/2} e^{- 3\omega^{1/2}/10 }}{|\ln |\ell||} \|v \| _{H^{1/2}(\partial D)} & \mbox{ if } 1/2 \lt \omega \lt \varepsilon^{-2}/4, \end{array} \right. \end{equation*} |
which yields the conclusion in the case 0 < \omega < \varepsilon^{-2}/4 . The proof is complete.
In this proof, C denotes a positive constant depending only on \Omega and \Lambda . Multiplying the equation of v_\varepsilon by \bar v_\varepsilon and integrating in \Omega , we derive that
\begin{equation} \int_{\Omega} \langle A_\varepsilon \nabla v_\varepsilon, \nabla v_\varepsilon \rangle + \omega \int_{\Omega} \rho_\varepsilon |v_\varepsilon|^2 \le C \| g\|_{L^2(\Omega)}^2. \end{equation} | (2.27) |
Here we used Poincaré's inequality
\| v_\varepsilon \|_{L^2(\Omega)} \le C \| \nabla v_\varepsilon\|_{L^2(\Omega)}. |
It follows from (2.27) that
\begin{align} \| v_{\varepsilon} (\varepsilon \, \cdot \, ) \|_{H^{1/2}(\partial B_1)}^2 &\le C \| v_{\varepsilon} (\varepsilon \, \cdot \, ) \|_{H^{1}(B_1)}^2 \\[6pt] &\le C \int_{B_\varepsilon} \frac{1}{\varepsilon^{d-2 }}|\nabla v_\varepsilon|^2 + \frac{1}{\varepsilon^d} |v_\varepsilon|^2 \le C (1+\omega^{-1}) \| g\|_{L^2(\Omega)}^2. \end{align} | (2.28) |
Similarly, using the equation for v and Poincaré's inequality, we get
\begin{equation} \|v\|_{H^1(\Omega)} \le C \| g\|_{L^2(\Omega)}. \end{equation} | (2.29) |
Since \Delta v+i\omega v = 0 in B_2 , using Caccioppolli's inequality, we have
\begin{equation} \|v\|_{H^3(B_{1})} \le C\|v\|_{H^2(B_{3/2})} \le C\|v\|_{H^1(B_{2})}\le C \| g\|_{L^2(\Omega)}. \end{equation} | (2.30) |
By Sobolev embedding, as d\le 3 ,
\begin{equation} \| v\|_{W^{1, \infty}(B_1)} \le C \|v\|_{H^3(B_{1})} . \end{equation} | (2.31) |
It follows that
\begin{equation} \| v(\varepsilon \, \cdot \, ) \|_{H^{1/2}(\partial B_1)} \le C \| v(\varepsilon \, \cdot \, )\|_{H^1(B_1)} \le C \| v\|_{W^{1, \infty}(B_1)} \le C \| g\|_{L^2(\Omega)}. \end{equation} | (2.32) |
Set
w_{\varepsilon} = v_{\varepsilon} - v \mbox{ in } \Omega \setminus B_\varepsilon. |
Then w_{\varepsilon} \in H^1(\Omega \setminus B_\varepsilon) and satisfies
\begin{equation} \left\{ \begin{array}{cl} \Delta w_{\varepsilon} + i \omega w_{\varepsilon} = 0 &\mbox{ in } \Omega \setminus B_\varepsilon, \\[6pt] w_{\varepsilon} = v_\varepsilon - v &\mbox{ on } \partial B_{\varepsilon}, \\[6pt] w_{\varepsilon} = 0 &\mbox{ on } \partial \Omega. \end{array}\right. \end{equation} | (2.33) |
Let \widetilde w_{\varepsilon}\in H^1(\mathbb{R}^d \setminus B_{\varepsilon}) be the unique solution of the system
\begin{equation} \left\{ \begin{array}{cl} \Delta \widetilde w_{\varepsilon} + i \omega \widetilde w_{\varepsilon} = 0 & \mbox{ in } \mathbb{R}^d \setminus B_{\varepsilon}, \\[6pt] \widetilde w_{\varepsilon} = w_\varepsilon & \mbox{ on } \partial B_{\varepsilon}, \end{array}\right. \end{equation} | (2.34) |
and set
\widetilde W_{\varepsilon} = \widetilde w_{\varepsilon}(\varepsilon \, \cdot \, ) \mbox{ in } \mathbb{R}^d \setminus B_1. |
Then \widetilde W_{\varepsilon} \in H^1(\mathbb{R}^d \setminus B_{1}) is the unique solution of the system
\begin{equation} \left\{ \begin{array}{cl} \Delta \widetilde W_{\varepsilon} + i \omega \varepsilon^2 \widetilde W_{\varepsilon} = 0 & \mbox{ in } \mathbb{R}^d \setminus B_{1}, \\[6pt] \widetilde W_{\varepsilon} = w_\varepsilon (\varepsilon \, \cdot \, ) & \mbox{ on } \partial B_{1}. \end{array}\right. \end{equation} | (2.35) |
Fix r_0 > 2 such that \Omega \subset B_{r_0} . By Lemma 2.2, we have, for 1 \le |x| < r_0 , that
\begin{equation*} | \widetilde W_{\varepsilon} (x / \varepsilon)| \le C e(\varepsilon, \omega, d) \|w_\varepsilon (\varepsilon \, \cdot \, ) \|_{H^{1/2}(\partial B_1)}, \end{equation*} |
which yields, for x \in B_{r_0} \setminus B_{1} , that
\begin{equation*} | \widetilde w_{\varepsilon}(x)| \le C e(\varepsilon, \omega, d) \|w_\varepsilon (\varepsilon \, \cdot \, ) \|_{H^{1/2}(\partial B_1)}. \end{equation*} |
Since \Delta \widetilde w_{\varepsilon} + i \omega \widetilde w_{\varepsilon} = 0 in B_{r_0} \setminus B_{1} , it follows from Caccioppoli's inequality that
\begin{equation} \| \widetilde w_{\varepsilon} \|_{H^1(B_2 \setminus B_{3/2})} \le C e(\varepsilon, \omega, d) \|w_\varepsilon (\varepsilon \, \cdot \, ) \|_{H^{1/2}(\partial B_1)}. \end{equation} | (2.36) |
Fix \varphi \in C^2(\mathbb{R}^d) such that \varphi = 1 in B_{3/2} and \varphi = 0 in \mathbb{R}^d \setminus B_2 , and set
\chi_{\varepsilon} = w_{\varepsilon} - \varphi \widetilde w_{\varepsilon} \mbox{ in } \Omega \setminus B_\varepsilon. |
Then \chi_{\varepsilon}\in H^1_0(\Omega \setminus B_\varepsilon) and satisfies
\begin{equation*} \Delta \chi_{\varepsilon} +i\omega \chi_{\varepsilon} = -\Delta \varphi \widetilde w_{\varepsilon} - 2\nabla \varphi \cdot \nabla \widetilde w_{\varepsilon} \mbox{ in } \Omega\setminus B_\varepsilon. \end{equation*} |
Multiplying the equation of \chi_{\varepsilon} by \bar \chi_{\varepsilon} and integrating by parts, we obtain, by Poincaré's inequality,
\begin{equation} \| \chi_{\varepsilon} \|_{H^1(\Omega \setminus B_\varepsilon)}\le C \| \widetilde w_{\varepsilon} \|_{H^1(B_2 \setminus B_{3/2})}. \end{equation} | (2.37) |
Combining (2.36) and (2.37) yields
\begin{equation} \| w_{\varepsilon} \|_{H^1(\Omega \setminus B_2)} \le C e(\varepsilon, \omega, d) \|w_\varepsilon (\varepsilon \, \cdot \, ) \|_{H^{1/2}(\partial B_1)}. \end{equation} | (2.38) |
The conclusion now follows from (2.28) and (2.32).
Let v_\varepsilon = u_\varepsilon - u . Using the fact that v_\varepsilon is real, by the inversion theorem and Minkowski's inequality, we have, for t > 0 ,
\begin{align} \|v_\varepsilon(t, \cdot) \|_{L^2(\Omega \setminus B_2)} \le C \int_0^\infty \|\hat{v}_\varepsilon (\omega, \cdot) \|_{L^2(\Omega \setminus B_2) } \, d \omega. \end{align} | (2.39) |
Using Proposition 2.1, we get
\begin{align*} \int_0^\infty \|\hat{v}_\varepsilon (\omega, \cdot) \|_{L^2(\Omega \setminus B_2) } \, d \omega &\le C \int_0^\infty (1+\omega^{-1/2})e(\varepsilon, \omega, d)\| \hat{f}(\omega)+u_0\|_{L^2(\Omega \setminus B_2) } \, d \omega \\[6pt] & \le C \mbox{esssup}_{\omega \gt 0} \|\hat{f}(\omega)+u_0\|_{L^2(\Omega \setminus B_2) } \int_0^\infty (1+\omega^{-1/2})e(\varepsilon, \omega, d) \, d \omega \\[6pt] & \le Ce(\varepsilon, d) \big(\|f\|_{L^1\big( (0, + \infty); L^2(\Omega) \big)} +\|u_0\|_{L^2(\Omega)} \big). \end{align*} |
It follows from (2.39) that, for t > 0 ,
\begin{align*} \|v_\varepsilon(t, \cdot) \|_{L^2(\Omega \setminus B_2)} \le Ce(\varepsilon, d) \big(\|f\|_{L^1\big( (0, + \infty); L^2(\Omega) \big)} +\|u_0\|_{L^2(\Omega)} \big). \end{align*} |
Similarly, we have, for t > 0 ,
\begin{equation*} \|\nabla v_\varepsilon(t, \cdot) \|_{L^2(\Omega \setminus B_2)} \le Ce (\varepsilon, d) \big(\|f\|_{L^1\big( (0, + \infty); L^2(\Omega) \big)} +\|u_0\|_{L^2(\Omega)} \big). \end{equation*} |
The conclusion follows.
The second author is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2015.21.
The authors declare no conflict of interest in this paper.
[1] |
Balcilar M, Hammoudeh S, Toparli E (2018) On the risk spillover across the oil market, stock market, and the oil related CDS sectors: A volatility impulse response approach. Energy Econ 74: 813-827. doi: 10.1016/j.eneco.2018.07.027
![]() |
[2] |
Bouri E (2015) A broadened causality in variance approach to assess the risk dynamics between crude oil prices and the jordanian stock market. Energy Policy 85: 271-279. doi: 10.1016/j.enpol.2015.06.001
![]() |
[3] |
Ji Q, Liu B, Fan Y (2019) Risk dependence of CoVaR and structural change between oil prices and exchange rates: A time-varying copula model. Energy Econ 77: 80-92. doi: 10.1016/j.eneco.2018.07.012
![]() |
[4] |
Li X, Wei Y (2018) The dependence and risk spillover between crude oil market and China stock market: New evidence from a variational mode decomposition-based copula method. Energy Econ 74: 565-581. doi: 10.1016/j.eneco.2018.07.011
![]() |
[5] |
Mensi W, Hammoudeh S, Shahzad S, et al. (2017) Oil and foreign exchange market tail dependence and risk spillovers for MENA, emerging and developed countries: VMD decomposition based copulas. Energy Econ 67: 476-495. doi: 10.1016/j.eneco.2017.08.036
![]() |
[6] |
Mensi W, Hammoudeh S, Shahzad S, et al. (2017) Modeling systemic risk and dependence structure between oil and stock markets using a variational mode decomposition-based copula method. J Bank Financ 75: 258-279. doi: 10.1016/j.jbankfin.2016.11.017
![]() |
[7] |
Pavlova I, De Boyrie ME, Parhizgari AM (2018) A dynamic spillover analysis of crude oil effects on the sovereign credit risk of exporting countries. Q Rev Econ Financ 68: 10-22. doi: 10.1016/j.qref.2018.03.003
![]() |
[8] |
Shahzad S, Mensi W, Hammoudeh S, et al. (2018) Extreme dependence and risk spillovers between oil and Islamic stock markets. Emerg Mark Rev 34: 42-63. doi: 10.1016/j.ememar.2017.10.003
![]() |
[9] |
Wen D, Wang GJ, Ma C, et al. (2019) Risk spillovers between oil and stock markets: A VAR for VaR analysis. Energy Econ 80: 524-535. doi: 10.1016/j.eneco.2019.02.005
![]() |
[10] |
Du L, He Y (2015) Extreme risk spillovers between crude oil and stock markets. Energy Econ 51: 455-465. doi: 10.1016/j.eneco.2015.08.007
![]() |
[11] |
Hong Y, Liu Y, Wang S (2009) Granger causality in risk and detection of extreme risk spillover between financial markets. J Econometrics 150: 271-287. doi: 10.1016/j.jeconom.2008.12.013
![]() |
[12] |
Elyasiani E, Gambarelli L, Muzzioli S (2017) The information content of corridor volatility measures during calm and turmoil periods. Quant Financ Econ 1: 454-473. doi: 10.3934/QFE.2017.4.454
![]() |
[13] |
Hamao Y, Masulis RW, Ng V (1990) Correlations in price changes and volatility across international stock markets. Rev Financ Stud 3: 281-307. doi: 10.1093/rfs/3.2.281
![]() |
[14] |
Hong Y (2001) A test for volatility spillover with applications to exchange rates. J Econometrics 103: 183-224. doi: 10.1016/S0304-4076(01)00043-4
![]() |
[15] |
Madaleno M, Vieira E (2018) Volatility analysis of returns and risk: family versus nonfamily firms. Quant Financ Econ 2: 348-372. doi: 10.3934/QFE.2018.2.348
![]() |
[16] |
Kang W, Ratti RA, Yoon KH (2015) The impact of oil price shocks on the stock market return and volatility relationship. J Int Financ Mark Inst Money 34: 41-54. doi: 10.1016/j.intfin.2014.11.002
![]() |
[17] |
Xu W, Ma F, Chen W, et al. (2019) Asymmetric volatility spillovers between oil and stock markets: Evidence from China and the United States. Energy Econ 80: 310-320. doi: 10.1016/j.eneco.2019.01.014
![]() |
[18] |
Peng C, Zhu H, Guo Y, et al. (2018) Risk spillover of international crude oil to China's firms: Evidence from granger causality across quantile. Energy Econ 72: 188-199. doi: 10.1016/j.eneco.2018.04.007
![]() |
[19] |
Zhang J (2019) Oil and gas trade between China and countries and regions along the 'Belt and Road': A panoramic perspective. Energy Policy 129: 1111-1120. doi: 10.1016/j.enpol.2019.03.020
![]() |
[20] |
Ren YH, Gu WW (2017) The impact on RMB internationalization process: Exchange rate expectation and foreign trade. Quant Financ Econ 1:114-124. doi: 10.3934/QFE.2017.1.114
![]() |
[21] | Babalos V, Stavroyiannis S, Gupta R (2015) Do commodity investors herd? Evidence from a time-varying stochastic volatility model. Resour Policy 46: 281-287. |
[22] |
Irwin SH, Sanders DR (2011) Index funds, financialization, and commodity futures markets. Appl Econ Perspect Policy 33: 1-31. doi: 10.1093/aepp/ppq032
![]() |
[23] |
Tang K, Xiong W (2012) Index investment and the financialization of commodities. Financ Anal J 68: 54-74. doi: 10.2469/faj.v68.n6.5
![]() |
[24] |
Wen S, An H, Huang S, et al. (2019) Dynamic impact of China's stock market on the international commodity market. Resour Policy 61: 564-571. doi: 10.1016/j.resourpol.2018.06.009
![]() |
[25] | Zhang YJ, Chevallier J, Guesmi K (2017) 'De‐financialization' of commodities? Evidence from stock, crude oil and natural gas markets. Energy Econ 68: 228-239. |
[26] |
Adams Z, Glück T (2015) Financialization in commodity markets: A passing trend or the new normal? J Bank Financ 60: 93-111. doi: 10.1016/j.jbankfin.2015.07.008
![]() |
[27] | Cheng IH, Xiong W (2013) The financialization of commodity markets. NBER Working Paper No.19642. |
[28] |
Diebold FX, Yilmaz K (2012) Better to give than to receive: Predictive directional measurement of volatility spillovers. Int J Forecasting 28: 57-66. doi: 10.1016/j.ijforecast.2011.02.006
![]() |
[29] |
Hamilton JD (1983) Oil and the macroeconomy since World War II. J Polit Econ 91: 228-248. doi: 10.1086/261140
![]() |
[30] |
An Y, Sun M, Gao C, et al. (2018) Analysis of the impact of crude oil price fluctuations on China's stock market in different periods-Based on time series network model. Physica A 492: 1016-1031. doi: 10.1016/j.physa.2017.11.032
![]() |
[31] |
Atems B, Kapper D, Lam E (2015) Do exchange rates respond asymmetrically to shocks in the crude oil market? Energy Econ 49: 227-238. doi: 10.1016/j.eneco.2015.01.027
![]() |
[32] |
Bildirici ME, Badu MM (2018) The effects of oil prices on confidence and stock return in China, India and Russia. Quant Financ Econ 2: 884-903. doi: 10.3934/QFE.2018.4.884
![]() |
[33] | Chen H, Liu L, Wang Y, et al. (2016) Oil price shocks and U.S. dollar exchange rates. Energy 112: 1036-1048. |
[34] |
Ghosh S, Kanjilal K (2016) Co-movement of international crude oil price and Indian stock market: evidences from nonlinear cointegration tests. Energy Econ 53: 111-117. doi: 10.1016/j.eneco.2014.11.002
![]() |
[35] |
Kang SH, Mciver R, Yoon SM (2017) Dynamic spillover effects among crude oil, precious metal, and agricultural commodity futures markets. Energy Econ 62: 19-32. doi: 10.1016/j.eneco.2016.12.011
![]() |
[36] |
Khalfaoui R, Boutahar M, Boubaker H (2015) Analyzing volatility spillovers and hedging between oil and stock markets: Evidence from wavelet analysis. Energy Econ 49: 540-549. doi: 10.1016/j.eneco.2015.03.023
![]() |
[37] |
Luo J, Ji Q (2018) High-frequency volatility connectedness between the US crude oil market and China's agricultural commodity markets. Energy Econ 76: 424-438. doi: 10.1016/j.eneco.2018.10.031
![]() |
[38] |
Mensi W, Beljid M, Boubaker A, et al. (2013) Correlations and volatility spillovers across commodity and stock markets: Linking energies, food, and gold. Econ Model 32: 15-22. doi: 10.1016/j.econmod.2013.01.023
![]() |
[39] |
Turhan MI, Sensoy A, Hacihasanoglu E (2014) A comparative analysis of the dynamic relationship between oil prices and exchange rates. J Int Financ Mark Inst Money 32: 397-414. doi: 10.1016/j.intfin.2014.07.003
![]() |
[40] |
Ji Q, Bouri E, Roubaud D, et al. (2018) Risk spillover between energy and agricultural commodity markets: A dependence-switching CoVaR-copula model. Energy Econ 75: 14-27. doi: 10.1016/j.eneco.2018.08.015
![]() |
[41] |
Algieri B, Leccadito A (2017) Assessing contagion risk from energy and non-energy commodity markets. Energy Econ 62: 312-322. doi: 10.1016/j.eneco.2017.01.006
![]() |
[42] |
Wang GJ, Xie C, Jiang ZQ, et al. (2016) Who are the net senders and recipients of volatility spillovers in China's financial markets? Financ Res Lett 18: 255-262. doi: 10.1016/j.frl.2016.04.025
![]() |
[43] |
Demirer M, Diebold FX, Liu L, et al. (2018). Estimating global bank network connectedness. J App Econom 33: 1-15. doi: 10.1002/jae.2585
![]() |
[44] |
Wang GJ, Xie C, Zhao LF, et al. (2018) Volatility connectedness in the Chinese banking system: Do state-owned commercial banks contribute more? J In Financ Mark Inst Money 57: 205-230. doi: 10.1016/j.intfin.2018.07.008
![]() |
[45] |
Yi SY, Xu ZS, Wang GJ (2018) Volatility connectedness in the cryptocurrency market: Is Bitcoin a dominant cryptocurrency? Int Rev Financ Anal 60: 98-114. doi: 10.1016/j.irfa.2018.08.012
![]() |
[46] |
Koop G, Pesaran MH, Potter SM (1996) Impulse response analysis in nonlinear multivariate models. J Econometrics 74:119-147. doi: 10.1016/0304-4076(95)01753-4
![]() |
[47] |
Pesaran MH, Shin Y (1998) Generalised impulse response analysis in linear multivariate models. Econ Lett 58: 17-29. doi: 10.1016/S0165-1765(97)00214-0
![]() |
[48] |
Engle RF, Manganelli S (2004) CAViaR: Conditional autoregressive value at risk by regression quantiles. J Bus Econ Stat 22: 367-381. doi: 10.1198/073500104000000370
![]() |
[49] | Qureshi K (2016) Value-at-Risk: The effect of autoregression in a quantile process. arXiv preprint arXiv: 1605.04940. |
[50] |
Bernardi M, Catania L (2016) Comparison of Value-at-Risk models using the MCS approach. Comput Stat 31: 579-608. doi: 10.1007/s00180-016-0646-6
![]() |
[51] |
Ferraty F, Quintela-Del-Río A (2016) Conditional VAR and expected shortfall: A new functional approach. Econ Rev 35: 263-292. doi: 10.1080/07474938.2013.807107
![]() |
[52] |
Gerlach R, Lu ZD, Huang H (2013) Exponentially smoothing the skewed laplace distribution for Value-at-Risk forecasting. J Forecasting 32: 534-550. doi: 10.1002/for.2255
![]() |
[53] |
Gkillas K, Katsiampa P (2018) An application of extreme value theory to cryptocurrencies. Econ Lett 164: 109-111. doi: 10.1016/j.econlet.2018.01.020
![]() |
[54] |
Li ZH, Dong H, Huang ZH, et al. (2018) Asymmetric effects on risks of Virtual Financial Assets (VFAs) in different regimes: A case of bitcoin. Quant Financ Econ 2: 860-883. doi: 10.3934/QFE.2018.4.860
![]() |
[55] |
Righi MB, Ceretta PS (2015) Forecasting value at risk and expected shortfall based on serial pair-copula constructions. Expert Sys Appl 42: 6380-6390. doi: 10.1016/j.eswa.2015.04.023
![]() |
[56] |
Whitea H, Kimb TH, Manganelli S (2015) VAR for VaR: Measuring tail dependence using multivariate regression quantiles. J Econometrics 187: 169-188. doi: 10.1016/j.jeconom.2015.02.004
![]() |
[57] |
Gong X, Wen F, Xia XH, et al. (2017) Investigating the risk-return trade-off for crude oil futures using high-frequency data. Appl Energy 196: 152-161. doi: 10.1016/j.apenergy.2016.11.112
![]() |
[58] | Chen SL, Liu SM, Cai RJ, et al. (2019) The factors that influence exchange-rate risk: Evidence in China, Emerg Mark Financ Tr. |
[59] |
Dong H, Liu Y, Chang JQ (2019) The heterogeneous linkage of economic policy uncertainty and oil return risks. Green Financ 1: 46-66. doi: 10.3934/GF.2019.1.46
![]() |
[60] |
Liao GK, Li ZH, Du ZQ, et al. (2019) The heterogeneous interconnections between supply or demand side and oil risks. Energies 12: 2226. doi: 10.3390/en12112226
![]() |
[61] |
An Y, Sun M, Gao CX, et al. (2018) Analysis of the impact of crude oil price fluctuations on China's stock market in different periods-based on time series network model. Physica A 492: 1016-1031. doi: 10.1016/j.physa.2017.11.032
![]() |
[62] |
Arouri MEH, Jouini J, Nguyen DK (2012) On the impacts of oil price fluctuations on European equity markets: Volatility spillover and hedging effectiveness. Energy Econ 34: 611-617. doi: 10.1016/j.eneco.2011.08.009
![]() |
[63] |
Zhang YL, Wang J (2019) Linkage influence of energy market on financial market by multiscale complexity synchronization. Physica A 516: 254-266. doi: 10.1016/j.physa.2018.10.038
![]() |
[64] |
Guo J (2018) Co-movement of international copper prices, China's economic activity, and stock returns: Structural breaks and volatility dynamics. Glob Financ J 36: 62-77. doi: 10.1016/j.gfj.2018.01.001
![]() |
[65] |
Zheng X, Chen BM (2014) Identification of stock market forces in the system adaptation framework. Inform Sci 265: 105-122. doi: 10.1016/j.ins.2013.12.028
![]() |
[66] |
Zhang C, Chen X (2014) The impact of global oil price shocks on China's bulk commodity markets and fundamental industries. Energy Policy 66: 32-41. doi: 10.1016/j.enpol.2013.09.067
![]() |
[67] |
Burdekin RCK, Siklos PL (2018) Quantifying the impact of the November 2014 Shanghai-Hong Kong stock connect. Int Rev Econ Financ 57: 156-163. doi: 10.1016/j.iref.2018.01.001
![]() |
[68] |
Ruan Q, Bao J, Zhang M, et al. (2019) The effects of exchange rate regime reform on RMB markets: A new perspective based on MF-DCCA. Physica A 522: 122-134. doi: 10.1016/j.physa.2019.01.110
![]() |
[69] |
Ahmad W, Rais S, Shaik AR (2018) Modelling the directional spillovers from DJIM Index to conventional benchmarks: Different this time? Q Rev Econ Financ 67: 14-27. doi: 10.1016/j.qref.2017.04.012
![]() |
[70] |
Antonakakis N, Kizys R (2015) Dynamic spillovers between commodity and currency markets. Int Rev Financ Anal 41: 303-319. doi: 10.1016/j.irfa.2015.01.016
![]() |
[71] |
Gillaizeau M, Jayasekera R, Maaitah A, et al. (2019) Giver and the receiver: Understanding spillover effects and predictive power in cross-market Bitcoin prices. Int Rev Financ Anal 63: 86-104. doi: 10.1016/j.irfa.2019.03.005
![]() |
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