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Acta Mathematicae Applicatae
Sinica, Chinese Series
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Current Issue
2019 Vol.35 Issue.4,
Published 20190915
ARTICLES
ARTICLES
695
L
^{p}
Solutions of BSDEs with a New Kind of NonLipschitz Coefficients
Shengjun FAN, Long JIANG
In this paper, we are interested in solving multidimensional backward stochastic differential equations (BSDEs) with a new kind of nonLipschitz coefficients. We establish an existence and uniqueness result of the
L
^{p}
(
p
> 1) solutions, which includes some known results as its particular cases.
2019 Vol. 35 (4): 695707 [
Abstract
] (
24
) [
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179KB] (
77
)
708
Derivative Formula and Coupling Property for Linear SDEs Driven by Lévy Processes
Zhao DONG, Yulin SONG, Yingchao XIE
In this paper we investigate an integration by parts formula for Lévy processes by using lower bound conditions of the corresponding Lévy measure. As applications, derivative formula and coupling property are derived for transition semigroups of linear SDEs driven by Lévy processes.
2019 Vol. 35 (4): 708721 [
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21
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1KB] [
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188KB] (
61
)
722
On the Integrability of the Abel and of the Extended Liénard Equations
Man Kwong Mak, Tiberiu HARKO
We present some exact integrability cases of the extended Liénard equation
y
" +
f
(
y
)(
y
')
^{n}
+
k
(
y
)(
y
')
^{m}
+
g
(
y
)
y
' +
h
(
y
)=0, with
n
> 0 and
m
> 0 arbitrary constants, while
f
(
y
),
k
(
y
),
g
(
y
), and
h
(
y
) are arbitrary functions. The solutions are obtained by transforming the equation Liénard equation to an equivalent first kind first order Abel type equation given by
dv
/
dy
=
f
(
y
)
v
^{3n}
+
k
(
y
)
v
^{3m}
+
g
(
y
)
v
^{2}
+
h
(
y
)
v
^{3}
, with
v
=1/
y
'. As a first step in our study we obtain three integrability cases of the extended quadraticcubic Liénard equation, corresponding to
n
=2 and
m
=3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Liénard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Liénard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with
g
(
y
) ≡ 0 and
h
(
y
) ≡ 0, and arbitrary
n
and
m
, thus allowing to obtain the general solution of the corresponding Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.
2019 Vol. 35 (4): 722736 [
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] (
21
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198KB] (
64
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737
Pullback Attractors for a 3D Nonautonomous NavierStokesVoight Equations
Yuming QIN, Xinguang YANG, Xin LIU
This paper is concerned with the existence of pullback attractors for the three dimensional nonautonomous NavierStokesVoight equations for the processes generated by the weak and strong solutions. The main difficulty is how to establish the pullback asymptotic compactness via energy equation approach under suitable assumption on external force.
2019 Vol. 35 (4): 737752 [
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24
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1KB] [
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186KB] (
57
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753
Concentrated Trading and the Survival of Overconfident Traders
Huainian ZHANG, Deqing ZHOU
We build a strategic trading model where the overconfident trader earns more than the rational trader and the mechanism for this result differs from that of Kyle and Wang (1997). In this paper, discretionary and nondiscretionary liquidity traders coexist. The overconfident insider is less worried by the market maker and thus induces a lower liquidity cost. In this way, he attracts all the trading from the discretionary liquidity traders, which enables the survival of the overconfident trader.
2019 Vol. 35 (4): 753760 [
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] (
24
) [
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1KB] [
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147KB] (
49
)
761
An Exact Generalized Test for Homogeneity of Inverse Gaussian Scale Parameters
Xuhua LIU, Xingzhong XU, Weiyan MU
In this paper, we propose a new generalized
p
value for testing homogeneity of scale parameters
λ
_{i}
from
k
independent inverse Gaussian populations. The proposed generalized
p
value is proved to have exact frequentist property, and it is also invariant under the group of scale transformation. Simulation results indicate that the proposed test is better than existing approximate
χ
^{2}
test.
2019 Vol. 35 (4): 761769 [
Abstract
] (
16
) [
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1KB] [
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153KB] (
41
)
770
Infinitely Many Solutions to a Class of
p
Laplace Equations
YiHua DENG
In this paper, we study a class of
p
Laplace equations. Using variational methods, we prove that there are two solutions and one of these solutions is nonnegative. Using recurrence method, we prove that there are infinitely many solutions to this class of equations.
2019 Vol. 35 (4): 770779 [
Abstract
] (
20
) [
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1KB] [
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164KB] (
54
)
780
GMM and Misspecification Correction for Misspecified Models with Diverging Number of Parameters
Qi ZHANG, Kaiping WANG, Deli LI, Lu LIN
Misspecified models have attracted much attention in some fields such as statistics and econometrics. When a global misspecification exists, even the model contains a large number of parameters and predictors, the misspecification cannot disappear and sometimes it instead goes further away from the true one. Then the inference and correction for such a model are of very importance. In this paper we use the generalized method of moments (GMM) to infer the misspecified model with diverging numbers of parameters and predictors, and to investigate its asymptotic behaviors, such as local and global consistency, and asymptotic normality. Furthermore, we suggest a semiparametric correction to reduce the global misspefication and, consequently, to improve the estimation and enhance the modeling. The theoretical results and the numerical comparisons show that the corrected estimation and fitting are better than the existing ones.
2019 Vol. 35 (4): 780797 [
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13
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1KB] [
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228KB] (
52
)
798
Global Regularity of Solutions for a Onedimensional Nuclear Fluid with Nonmonotone Pressure
Baowei FENG
We consider a onedimensional continuous thermal model of nuclear matter, which is described by a compressible NavierStokesPoission system with a nonmonotone equation of state owing to the effective Skyrme nuclear interaction between particles. We prove the global existence of solutions in
H
^{4}
space for a free boundary value problem with a possible destabilizing influence of the pressure which is not always positive, provided a sufficient thermal dissipation is present and first obtain the existence of classical solutions.
2019 Vol. 35 (4): 798811 [
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28
) [
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171KB] (
55
)
812
Inequality Estimates of an Inhomogeneous Semilinear Biharmonic Equation in Entire Space
Fen YANG
In this paper, we consider the inequality estimates of the positive solutions for the inhomogeneous biharmonic equation
△
^{2}
u
+
u
^{p}
+
f
(
x
)=0 in R
^{n}
, (*)
where △
^{2}
is the biharmonic operator,
^{p}
> 1,
^{n}
≥ 5 and 0 ≢
f
∈
C
(R
^{n}
) is a given nonnegative function. We obtain different inequality estimates of Eq.(*), with which the necessary conditions of existence on
f
and
p
are also established.
2019 Vol. 35 (4): 812819 [
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15
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139KB] (
60
)
820
Regularity of Global Attractor for Atmospheric Circulation Equations with Humidity Effect
Jiaojiao PAN, Qian JIANG, Tingwei RUAN, Hong LUO
In this article, regularity of the global attractor for atmospheric circulation equations with humidity effect is considered. It is proved that atmospheric circulation equations with humidity effect possess a global attractor in
H
^{k}
(Ω,
R
^{4}
) for any
k
≥ 0, which attracts any bounded set of
H
^{k}
(Ω,
R
^{4}
) in the
H
^{k}
norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor.
2019 Vol. 35 (4): 820829 [
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] (
21
) [
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1KB] [
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161KB] (
52
)
830
An Inexact Affine Scaling LevenbergMarquardt Method Under Local Error Bound Conditions
Zhujun WANG, Li CAI, Yifan SU, Zhen PENG
We propose an inexact affine scaling LevenbergMarquardt method for solving boundconstrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition. The affine scaling LevenbergMarquardt equation is based on a minimization of the squared Euclidean norm of linearized model adding a quadratic affine scaling matrix to find a solution which belongs to the bounded constraints on variable. The global convergence and the superlinear convergence rate are proved. Numerical results show that the new algorithm is efficient.
2019 Vol. 35 (4): 830844 [
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] (
22
) [
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200KB] (
47
)
845
Feature Screening for Ultrahighdimensional Censored Data with Varying Coefficient Singleindex Model
Yi LIU
In this paper, we study the sure independence screening of ultrahighdimensional censored data with varying coefficient singleindex model. This general model framework covers a large number of commonly used survival models. The property that the proposed method is not derived for a specific model is appealing in ultrahigh dimensional regressions, as it is difficult to specify a correct model for ultrahigh dimensional predictors. Once the assuming data generating process does not meet the actual one, the screening method based on the model will be problematic. We establish the sure screening property and consistency in ranking property of the proposed method. Simulations are conducted to study the finite sample performances, and the results demonstrate that the proposed method is competitive compared with the existing methods. We also illustrate the results via the analysis of data from The National Alzheimers Coordinating Center (NACC).
2019 Vol. 35 (4): 845861 [
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] (
22
) [
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1KB] [
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259KB] (
49
)
862
The Efficient Computation of Aircraft Range Problem
Fang YU, Jinchuan CUI
The goal of efficient computation is to determine reasonable computing cost in polynomial time by using data structure of instance, and analyze the computing cost of satisfactory solution which can meet user's requirements. When faced with NPhard problem, we usually assess computational performance in the worst case. Polynomial algorithm cannot handle with NPhard problem, so we research on NPhard problem from efficient computation point of view. The work is intended to fill the blank of computational complexity theory. We focus on the cluster structure of instance data of aircraft range problem. By studying the partition and complexity measurement of cluster, we establish a connection between the aircraft range problem and Nvehicle exploration problem, and construct the efficient computation mechanism for aircraft range problem. The last examples show that the effect is significant when we use efficient computation mechanism on aircraft range problem. Decision makers can calculate the computing cost before actually computing.
2019 Vol. 35 (4): 862872 [
Abstract
] (
26
) [
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1KB] [
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1766KB] (
53
)
873
Jensen's Inequality Under Nonlinear Expectation Generated by BSDE with Jumps
Na ZHANG, Guangyan JIA
In this paper, we study Jensen's inequality under
f
expectation, which is a nonlinear expectation generated by backward stochastic differential equations (BSDEs) with jumps. We connect
f
convex functions with the viscosity solutions of a kind of integral partial differential equations (IPDEs) with nonlocal terms. And find that under Lipschitz condition, the
f
convex function is still convex in the usual sense, i.e., the jumps shrink the range of ‘convex’ functions.
2019 Vol. 35 (4): 873884 [
Abstract
] (
21
) [
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1KB] [
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185KB] (
56
)
885
A Note on Ranking in the PlackettLuce Model for Multiple Comparisons
Jing LUO, Hong QIN
Ranking and rating individuals is a fundamental problem in multiple comparisons. One of the most wellknown approaches is the PlackettLuce model, in which the ordering is decided by the maximum likelihood estimator. However, the maximum likelihood estimate (MLE) does not exist when some individuals are never ranked lower than others or lose all their races. In this note, we proposed a penalized likelihood method to address this problem. As the penalized parameter goes to zero, the penalized MLE converges to the original MLE. Further, there exists a critical point in which the penalized likelihood ranking is independent of the choice of the penalized parameter. Several numerical examples are provided.
2019 Vol. 35 (4): 885892 [
Abstract
] (
52
) [
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1KB] [
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147KB] (
58
)
893
Estimating Restricted Common Structural Changes for Panel Data
Liwen ZHANG, Zhongyi ZHU
We consider estimating multiple structural changes occurring at unknown common dates in a panel data regression model with restrictions imposed on the coefficients. We establish the consistency and rate of convergence of the structural change estimates and the asymptotic distribution of the parameter estimates. It is shown that the efficiency of the parameter estimators is increased using the restrictions. An efficient dynamic algorithm is proposed to obtain the break date estimates and the parameter estimates. In addition, we propose a testing procedure for the existence of the change points with the restrictions and derive the asymptotic distribution under the null hypothesis of the test statistics. Simulation studies are presented to investigate the performance of the proposed method in finite samples.
2019 Vol. 35 (4): 893908 [
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] (
21
) [
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1KB] [
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273KB] (
64
)
909
Confidence Intervals of the Generalized Pareto Distribution Parameters Based on Upper Record Values
Xu ZHAO, Weihu CHENG, Yang ZHANG, Shaojie WEI, Zhenhai YANG
In this paper, we proposed a new efficient approach to construct confidence intervals for the location and scale parameters of the generalized Pareto distribution (GPD) when the shape parameter is known. The superiority of our method is that the distributions of pivots are exact, but not approximate distributions. The proposed interval estimation provides the shortest interval for the GPD parameter whether or not the confident distribution of the pivot is symmetric. We first estimate the location and scale parameters of the GPD using least squares and then, construct confidence intervals based on the equal probability density principle. The results of various simulation studies illustrate that our interval estimators show the better performance than competing method.
2019 Vol. 35 (4): 909918 [
Abstract
] (
11
) [
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1KB] [
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150KB] (
62
)
919
On a Spanning
K
tree Containing Specified Vertices in a Graph
Feifei SONG, Zhiquan HU
A
k
tree is a tree with maximum degree at most
k
. In this paper, we give a sharp degree sum condition for a graph to have a spanning
k
tree in which specified vertices have degree less than
t
, where 1 ≤
t
≤
k
. We denote by
σ
_{k}
(
G
) the minimum value of the degree sum of
k
independent vertices in a graph
G
. Let
k
≥ 2,
s
≥ 0 and 1 ≤
t
≤
k
be integers, and suppose
G
is an (
s
+ 1)connected graph with
σ
_{k}
(
G
) ≥ 
G
+ (
kt
)
s
1. Then for any
s
specified vertices,
G
contains a spanning
k
tree in which every specified vertex has degree at most
t
. This improves a result obtained by Matsuda and Matsumura.
2019 Vol. 35 (4): 919923 [
Abstract
] (
25
) [
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1KB] [
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130KB] (
51
)
924
Linear Arboricity of NICPlanar Graphs
Bei NIU, Xin ZHANG
A graph is NICplanar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. It is proved that every NICplanar graph with minimum degree at least 2 (resp.3) contains either an edge with degree sum at most 23 (resp. 17) or a 2alternating cycle (resp. 3alternating quadrilateral). By applying those structural theorems, we confirm the Linear Arboricity Conjecture for NICplanar graphs with maximum degree at least 14 and determine the linear arboricity of NICplanar graphs with maximum degree at least 21.
2019 Vol. 35 (4): 924934 [
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] (
18
) [
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1KB] [
PDF
305KB] (
56
)
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