中国科学院数学与系统科学研究院期刊网

15 September 2023, Volume 66 Issue 5
    

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  • Xi Sheng YU, Yu Wei YAO
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 801-814. https://doi.org/10.12386/A20200164
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    Option pricing with discrete dividend payments is still a challenge. This paper proposes a novel model by taking the dividends into consideration, and establishes the option price theorem for obtaining the option price. Theoretical analysis shows that the proposed new model can fully take the impact of dividend payments on option price such as the dividend paying time, amount and number, and hence it can produce an accurate price for option. We also conduct a theoretical comparison of the pricing between the newly-proposed model and classic/benchmark, with which the relation and pricing differences between the new model and these models are deeply detected. The numerical results also show that the proposed model can produce highly accurate prices for options and has strong pricing robustness. Based on this, our model can be an excellent alternative of pricing European options written on the underlying asset paying discrete dividends.
  • Liu Yan LI, Jun Ping LI
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 815-826. https://doi.org/10.12386/b20210647
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    This paper considers a supercritical Galton-Watson process with immigration $\{X_n\}_{n\geq0}$. It is well-known that there is a sequence of constants $\{c_n\}_{n\geq0}$ such that $X_n/c_n\to V$ almost surely as $n\rightarrow\infty$. Using Cramér transforms, we obtain lower deviations for the process $\{X_n\}_{n\geq0}$, which refer to the asymptotic properties of $P(X_n=k)$ for sufficiently large $k$ satisfying $k_n\leq k\leq c_n$ and $k_n\rightarrow \infty$.
  • He Ying WANG, Rui LIU, Qi Yao BAO
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 827-834. https://doi.org/10.12386/A20220008
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    Quantum detection investigates the injectivity of quantum measurements on quantum states. Since every measurement of quantum system can be characterized by a positive operator-valued measure (POVM), and every Parseval frame corresponds to a rank-one POVM. In this paper, we mainly consider the quantum injectivity problem of Gabor frames, and give a sufficient condition for the quantum injectivity of a Gabor frame $\left\{\pi (m,n) \varphi \right\}_{(m,n) \in \Lambda}$, namely it is a full Gabor frame and satisfies $\langle \pi (m,n) \varphi,\varphi \rangle \ne 0$ for $m=0,\ 1 \le n \le \frac{N}{2}$, $1 \le m \le \frac{N-1}{2},\ 0 \le n \le N-1$ and $\frac{N-1}{2} < m \le \frac{N}{2},\ 0 \le n \le \frac{N}{2}$. We also give its stability with a quantitative error estimate and its applications for low dimensional cases.
  • Yan Hui ZHANG, Tao QIAN
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 835-844. https://doi.org/10.12386/b20220026
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    In this paper, using complex analysis methods, we will show that an $h^p$ harmonic function $u$ can be decomposed into the sum of one singular function and one absolutely continuously function on unit ball $B$ of $\mathbb{R}^n$ for $p\geq 1.$ Then we will obtain the corresponding results of functions in $h^p$ space of the upper half space of $\mathbb{R}^n$ by the Kelvin transform.
  • Shao Tao HU, Yuan Heng WANG, Gang CAI
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 845-854. https://doi.org/10.12386/A20220013
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    We introduce a new Tseng’s extragradient algorithm for solving pseudomonotone variational inequality problems in Hilbert spaces. We prove that the sequence generated by our proposed algorithm converges strongly to an element of solution set for variational inequality problems. The results obtained in this paper extend and improve many recent ones in the literature.
  • Ran Ran ZHANG, Zhi Bo HUANG, Chuang Xin CHEN
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 855-866. https://doi.org/10.12386/A20220020
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    We consider the uniqueness of the meromorphic solution $f(z)$ of the second order linear difference equation $ p_2(z)y(z+2)+p_1(z)y(z+1)+p_0(z)y(z)=0, $ where $p_2(z), p_1(z), p_0(z)$ are nonzero polynomials with $p_2(z)+p_1(z)+p_0(z)\not\equiv0$. We give the forms of $f(z)$ if $f(z)$ shares $0, 1, \infty$ CM with any meromorphic function $g(z)$. Furthermore, if $g(z)$ is also a solution of the above equation, we obtain the exact forms of this equation. As a corollary, we see that if a meromorphic function $g(z)$ shares $0, 1, \infty$ CM with the gamma function $\Gamma(z)$, then $g(z)\equiv \Gamma (z)$.
  • Qi YAN, Xian An JIN
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 867-880. https://doi.org/10.12386/A20220018
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    In 2020, Gross, Mansour and Tucker introduced the partial-dual Eulergenus polynomial and gave the interpolating conjecture: The partial-dual Euler-genus polynomial for any non-orientable ribbon graph is interpolating. Recently, we gave two infinite classes of counterexamples to the conjecture, but they contain only one or two non-orientable loops. In this paper, we further give two classes of non-interpolating partial-dual Euler-genus polynomials, in which all loops in the first class of ribbon graphs are non-orientable and the number of orientable loops in the second class of ribbon graphs can be chosen arbitrarily.
  • Yong TANG, Yu Ping WANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 881-888. https://doi.org/10.12386/A20220031
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    The inverse spectral problems for Sturm-Liouville operators with one boundary condition having the spectral parameter are studied from mixed spectral data in this paper. The authors show that if the potential $q(x)$ on $[a_0,1]$ is given a priori, then the potential $q(x)$ on $[0,1]$ can be uniquely determined by parts of one spectrum. In addition, we prove that the potential $q(x)$ on $[0,1]$ can be uniquely determined by one spectrum with two eigenvalues missing.
  • Bing Mao DENG, Cui Ping ZENG, Dan LIU, De Gui YANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 889-898. https://doi.org/10.12386/A20220036
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    We studied the normality concerning repelling periodic poits, and obtained two results as follow: (1) Let $\mathcal{F}$ be a family of holomorphic functions in a domain $D$, and let $k\ge 2$ be a positive integer. If, for each $f\in \mathcal{F}$, all zeros of $f(z)-z$ have multiplicity at least $3$, and its iteration $f^k$ has at most $3k-1$ distinct repelling fixed points in $D$, then $\mathcal{F}$ is normal in $D$. There are examples show that all conditions are necessary in this result; (2) Let $\mathcal{F}$ be a family of meromorphic functions in a domain $D$, and let $k\ge 3$ be a positive integer. If, for each $f\in \mathcal{F}$, all zeros and poles of $f(z)-z$ have multiplicity at least $3$, and its iteration $f^k$ has at most $2k-1$ distinct repelling fixed points in $D$, then $\mathcal{F}$ is normal in $D$.
  • Xiao Fei SUN, Kang Ning WANG, Lu LIN
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 899-916. https://doi.org/10.12386/A20220037
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    Composite quantile regression has good properties in robustness and estimation efficiency. For the longitudinal data single-index models, we propose profile composite quantile regression based estimating equations and smooth-threshold variable selection methods. The new methods can incorporate the intra-subject correlation by using copula functions, and inherit the advantages of composite quantile regression. Under some mild conditions, the asymptotical properties are established. Simulation studies and real data analysis are included to illustrate the finite sample performance.
  • Jun WANG, Li WANG, Qiao Cheng ZHONG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 917-926. https://doi.org/10.12386/b20220150
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    This paper is devoted to the following fractional Schrödinger-Poisson systems: $\left\{ {\begin{array}{*{20}{c}} {{{\left( { - \Delta } \right)}^s}u + V\left( x \right)u + \phi \left( x \right)u = f\left( {x,u} \right),}\\ {{{\left( { - \Delta } \right)}^t}\phi \left( x \right) = {u^2},} \end{array}} \right.\begin{array}{*{20}{c}} {x \in {^3},}\\ {x \in {^3},} \end{array}$ where $(-\Delta)^{s}$ is the fractional Laplacian, $s, t \in (0, 1),$ $V : \mathbb{R}^3 \to \mathbb{R}$ is continuous. In contrast to most studies, the paper considers that the potentials $V$ is indefinite. With the help of Morse theory, the existence of nontrivial solutions for the above problem is obtained.
  • Qiu Yu LI, Nai Huan JING
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 927-938. https://doi.org/10.12386/A20220040
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    Quantum toroidal algebras or double affine quantum algebras are special cases ($N=2$) of the recently defined quantum $N$-toroidal algebras, which generalize the toroidal Lie algebras and $N$-toroidal Lie algebras. In this paper, we will construct a level one representation of the quantum $N$-toroidal algebra for the exceptional type $G_2$, which can be regarded as a generalization of the basic representation of the quantum affine algebra in type $G_2$.
  • Han Yu HU, Yun Hua ZHOU
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 939-950. https://doi.org/10.12386/A20220042
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    We study some properties of mean linear independence for the bundle map induced by a diffeomorphisim of smooth Riemannian manifold. We first define a $\theta$ function by using the distances between vectors in frames and then give an equivalent definition of mean linear independence. At the same time, a variational principle on $\theta$ function is also investigated. Second, we prove that any tangent vector belongs to a mean linear independent $k$-frame, where $k$ is the style number of the system at the base point of the vector.
  • Li Heng SANG, Zhen Long CHEN
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 951-970. https://doi.org/10.12386/B20220216
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    This paper is concerned with Hausdorff measures of the image and the graph for spherical Gaussian random fields $T=\{T(t), t\in \mathbb{S}^2\}$ valued in $\mathbb{R}^d$ on the unit sphere $\mathbb{S}^2$ with $2<\alpha <4$. For each compact set $D$ on $\mathbb{S}^2$, we show that when $4<(\alpha -2)d$, with probability one, $0<\phi-m(T(D))<\infty$, where $\phi-m(\cdot)$ denotes the Hausdorff measure with Hausdorff function $\phi(r)=r^{\frac{4}{\alpha-2}}\log\log\frac{1}{r}$. Meanwhile, we obtain the Hausdorff measures of the graph, that is, with probability one, $0<\phi-m({\rm Gr}(T(D)))<\infty$ for $(\alpha -2)d<4$ where $\phi(r)=r^{2+(1-\frac{\alpha-2}{2})d}(\log\log\frac{1}{r})^{\frac{(\alpha-2)d}{4}}$ and for $(\alpha-2)d>4$ where $\phi(r)=r^{\frac{4}{\alpha-2}}\log\log\frac{1}{r}$. This paper extends the corresponding results in Euclidean space to the sphere case.
  • Jin Lin GUAN, Yan TANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 971-980. https://doi.org/10.12386/A20220045
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    We propose a new iterative algorithm based on three-step implicit midpoint rule of nonexpansive mappings in real uniformly smooth Banach spaces. Under very mild conditions, we prove a strong convergence theorem for finding the common solution to the fixed points of three nonexpansive mappings and a convex optimization problem. As an application, we apply our main result to solve a split feasibility problem. Our result improves and extends the corresponding ones announced by some others in the earlier and recent literature.
  • Shuo Yang LI, Meng GAO, Hong Ya GAO
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 981-988. https://doi.org/10.12386/B20220229
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    This paper deals with regularity properties for vector-valued minimizers $u=(u^1,\ldots, u^{n+1}) :\Omega \subset \mathbb R^n \rightarrow \mathbb R^{n+1}$ of variational integrals with splitting structure of the form $$ {\cal J} (u,\Omega) = \int_{\Omega} \sum_{i=1}^{n+1} \left\{f^{i}(x,Du^{i})+ g^{i}(x,({\rm adj} _{n}Du)^{i}) \right\} dx, $$ where $f^i, g^i:\Omega \times \mathbb R^{n} \rightarrow \mathbb R$, $i=1,\ldots,n+1$, are Carathéodory functions satisfying some structural conditions. Regularity properties are derived under suitable assumptions.
  • Xian Yi LI, Hai Yang ZHU
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 989-1002. https://doi.org/10.12386/B20220320
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    In this paper, we prove that the unique nonnegative equilibrium of a rational difference equation with higher order is globally attractive. As application, our results not only improve many known ones, but also solve several open problems and conjectures.
  • Gui Jun LIU, Xiao Feng WANG, Jian Jun CHEN
    Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 1003-1018. https://doi.org/10.12386/B20220455
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    In this paper, IDA spaces of locally integrable functions whose integral distance to holomorphic function is finite and their geometric theory are studied. We characterize the boundedness and compactness of Hankel operators from the doubling Fock spaces $F^p_\phi$ to the doubling Lebesgue spaces $L^q_\phi$ for all possible $1\leq p,\, q<\infty$ by IDA spaces, where $\phi$ is a nonzero subharmonic function such that $\Delta\phi dA$ is a doubling measure. Moreover, the Schatten-$p$ class of Hankel operators on Hilbert spaces $F^2_\phi$ are all considered.