关于Tsallis熵与随机序的一些结果

本文讨论了具有相同Tsallis熵或相同Tsallis相对熵的两个连续随机变量随机等价的一些条件,以及随机序与Tsallis熵序的一些关系.     

随机利率模型下基于Tsallis熵分布的可转债定价

本文主要研究基于Tsallis熵分布且存在瞬时违约风险的情况下,随机利率服从Vasicek利率模型的可转换债券的定价问题。标的股票价格过程服从Tsallis熵分布的前提下,构建投资组合,利用无套利原理得到可转债价格所满足的偏微分方程,进一步采用有限元法得到可转债价格的数值解。根据长江证券、利欧股份以及吉林敖东股票的市场真实数据,利用Tsallis熵分布模拟收益率序列,并得到基于Tsallis熵分布的股价模型优于几何布朗运动模型下的最优参数,在此基础上,绘制股价基于Tsallis熵分布下三种标的股票所对应可转债的理论价格的三维图及与市场实际价格的对比图。研究结果发现,对应标的股票价格基于Tsallis熵分布下的可转债理论价格与市场真实价格更为接近。     

随机利率下基于Tsallis熵分布的幂式期权定价

考虑到无风险利率的随机性以及股票收益率分布的尖峰厚尾和长期相依性,利用具有长程记忆及统计反馈性质的Tsallis熵分布建立股票价格的运动模型,在无风险利率服从Vasicek模型下,运用保险精算定价法得到了幂式期权的定价公式,推广了经典的Black-Scholes定价公式,扩展了已有文献的结论.     

基于Tsallis分布和更新过程的欧式期权定价

考虑到股价所具有的均值回复性、长记忆性和收益率尖峰后尾的特征,利用指数O-U过程和Tsallis熵分布分别对传统B-S定价模型的漂移项、随机波动项进行改进,并假设跳跃源服从比泊松过程更一般的更新过程,利用无套利思想和广义Ito公式,给出在股票价格服从一类更新跳-扩散过程下满足的偏微分方程,最后运用Feynman-Kac公式及等价鞅方法,计算欧式期权价格.     

非对称Ⅲ型裂纹表面受运动载荷的动态扩展问题

通过复变函数论的方法,对非对称Ⅲ型裂纹表面受运动载荷的动态扩展问题进行了研究．采用自相似函数的方法可以获得解析解的一般表达式．应用该法可以迅速地将所讨论的问题转化为Riemann-Hilbert问题,并求得了非对称扩展裂纹表面分别受到常数运动载荷、阶跃运动载荷作用下的应力、位移和动态应力强度因子解析解．利用这些解并采用叠加原理,就可以求得任意复杂问题的解．     

一种基于Yager熵最大化属性权重组合方法

研究了区间型多属性决策中属性权重组合问题。首先,提出了一种客观权重确定方法——平均差极大化方法;其次,基于区间型理想点和熵极大化原理,提出属性权重的组合模型。然后,利用属性的组合权重和投影模型,对方案进行排序。最后,通过算例说明了该方法的可行性和有效性。     

物理熵、信息熵及其演化方程

在非平衡统计物理熵演化方程和熵产生率简明公式的启示下,发展了动态信息熵理论,建立了信息熵密度在时间和态变量空间变化的非线性演化方程.它的数学形式和物理意义与物理熵演化方程类似：信息熵密度的变化同样是由漂移、扩散和产生三者共同引起的.信息熵产生率简明公式与物理熵的亦类似.进而研究了物理熵和信息熵的同异及这两种统计熵的可能统一,讨论了熵增加原理、平衡态最大熵原理和最大信息熵原理三者相互间及其与熵演化方程之间的关联.     

二维曲线型随机变量的信息度量

主要在二维曲线型随机变量的基础上,引入了信息论中常讨论的信息量和信息熵,尤其是条件熵的建立.     

基于Tsallis理论的中国股市收益分布研究

应用Tsallis提出的非广延统计力学理论以及与之密切相关的非线性Fokker-Planck方程所描述的动力系统,根据我国上证指数和深证指数2004年1月1日～2008年11月13日的高频数据,分析了在三种不同的时间标度下股指收益的概率分布,发现Tsallis分布可以很好地描述两市收益分布的尖峰厚尾有限方差等特征,同时也给出了市场微观动力学层面的解释。揭示出我国上海和深圳股市的价格过程并不符合随机游走,而是反常扩散过程,两市具有十分接近的非线性动力系统特征。所得结论对于研究我国金融市场的资产配置和定价、风险管理和制度建设都具有重要的意义。     

双曲型守恒律方程的熵稳定格式的一些讨论

本文讨论双曲型守恒律方程的熵稳定格式.对于给定的熵对,格式所满足的熵条件中的数值熵通量是不唯一的.Tadmor的充分条件可以唯一地确定标量方程的熵守恒通量,但不能唯一确定方程组的熵守恒通量,却可以给出方程组的空间一阶精度的熵守恒格式.也讨论了在熵守恒通量上添加数值粘性得到的显式熵稳定格式需要满足的条件及常见的时间离散对熵守恒和熵稳定的影响.     

Generalized Informational Entropy and Noncanonical Distribution in Equilibrium Statistical Mechanics

Based on the Jaynes principle of maximum for informational entropy, we find a generalized probability distribution and construct a generalized equilibrium statistical mechanics (ESM) for a wide class of objects to which the usual (canonical) ESM cannot be applied. We consistently consider the case of a continuous, not discrete, random variable characterizing the state of the object. For large values of the argument, the resulting distribution is characterized by a power-law, not exponential, asymptotic behavior, and the corresponding power asymptotic expression agrees with the empirical laws established for these objects. The -deformed Boltzmann–Gibbs–Shannon functional satisfying the requirements of the entropy axiomatics and leading to the canonical ESM for =0 is used as the original entropy functional. We also consider nonlinear transformations of this functional. We show that depending on how the averages of the dynamical characteristics of the object are defined, the different (Tsallis, Renyi, and Hardy–Littlewood–Pólya) versions of the generalized ESM can be used, and we give their comparative analysis. We find conditions under which the Gibbs–Helmholtz thermodynamic relations hold and the Legendre transformation can be applied to the generalized entropy and the Massieu–Planck function. We consider the Tsallis and Renyi ESM versions in detail for the case of a one-dimensional probabilistic object with a single dynamical characteristic whose role is played by a generalized positive energy with a monotonic power growth. We obtain constraints on the Renyi index under which the equilibrium distribution relates to a definite class of stable Gaussian or Levy–Khinchin distributions.     

The unique solution of the Karcher equation and the self-adjointness of the Karcher mean

Lawson and Lim showed that the Karcher equation for positive invertible operators on a Hilbert space has a unique solution using the method of the implicit function theorem of a Banach space. In this paper, in the framework of the operator inequality, we show the equivalence of the unique solution of the Karcher equation and the self-adjointness of the Karcher mean. For this, we reform the notion of the operator power means of negative order by virtue of the Tsallis relative operator entropy of negative order.     

Improvement of an Estimate of the Global Existence Theorem for Solutions of the Bogoliubov Equations

For a one-dimensional system of particles interacting like elastic balls, we improve an estimate of the existence theorem for the global solution of the Cauchy problem for the Bogoliubov equations with initial data from the space of sequences of measurable functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 420–424, December, 2005.     

L2‐estimates for the DG IIPG‐0 scheme

We discuss the optimality in L² of a variant of the Incomplete Discontinuous Galerkin Interior Penalty method (IIPG) for second order linear elliptic problems. We prove optimal estimate, in two and three dimensions, for the lowest order case under suitable regularity assumptions on the data and on the mesh. We also provide numerical evidence, in one dimension, of the necessity of the regularity assumptions. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

A linear system solver based on a modified Krylov subspace method for breakdown recovery

Despite its usefulness in solving eigenvalue problems and linear systems of equations, the nonsymmetric Lanczos method is known to suffer from a potential breakdown problem. Previous and recent approaches for handling the Lanczos exact and near-breakdowns include, for example, the look-ahead schemes by Parlett-Taylor-Liu [23], Freund-Gutknecht-Nachtigal [9], and Brezinski-Redivo Zaglia-Sadok [4]; the combined look-ahead and restart scheme by Joubert [18]; and the low-rank modified Lanczos scheme by Huckle [17]. In this paper, we present yet another scheme based on a modified Krylov subspace approach for the solution of nonsymmetric linear systems. When a breakdown occurs, our approach seeks a modified dual Krylov subspace, which is the sum of the original subspace and a new Krylov subspaceK _m (w _j ,A ^T ) wherew _j is a newstart vector (this approach has been studied by Ye [26] for eigenvalue computations). Based on this strategy, we have developed a practical algorithm for linear systems called the MLAN/QM algorithm, which also incorporates the residual quasi-minimization as proposed in [12]. We present a few convergence bounds for the method as well as numerical results to show its effectiveness.Research supported by Natural Sciences and Engineering Research Council of Canada.     

A composite step bi-conjugate gradient algorithm for nonsymmetric linear systems

The Bi-Conjugate Gradient (BCG) algorithm is the simplest and most natural generalization of the classical conjugate gradient method for solving nonsymmetric linear systems. It is well-known that the method suffers from two kinds of breakdowns. The first is due to the breakdown of the underlying Lanczos process and the second is due to the fact that some iterates are not well-defined by the Galerkin condition on the associated Krylov subspaces. In this paper, we derive a simple modification of the BCG algorithm, the Composite Step BCG (CSBCG) algorithm, which is able to compute all the well-defined BCG iterates stably, assuming that the underlying Lanczos process does not break down. The main idea is to skip over a step for which the BCG iterate is not defined.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.The work of this author was supported by the Office of Naval Research under contracts N00014-90J-1695 and N00014-92J-1890, the Department of Energy under contract DE-FG03-87ER25307, the National Science Foundation under contracts ASC 90-03002 and 92-01266, and the Army Research Office under contract DAAL03-91-G-0150. Part of this work was completed during a visit to the Computer Science Dept., The Chinese University of Hong Kong.