随机分形

本文概括了随机分形的主要结果，综述了随机分形的最新进展和目前的动态，提出了一些末解决的问题，全文共分为三部分：（１）由随机过程和随机场（如Ｌｅｖｙ过程，Ｇａｕｓｓ场，自相似过程等）产生的各种随机分形集（如象集、水平集、Ｋ重点集等）的Ｈａｕｓｄｏｒｆｆ维数、测度和ｐａｃｋｉｎｇ维数、测试；（２）随机Ｃａｎｔｏｒ型集和统计自相似集的维数和测试；（３）分形集（如Ｓｐｉｅｒｐｉｎｓｋｉ ｇａｓｋｅｔ，ｎｅ     

两参数Wiener过程的增量的一个下极限

本文得到一个关于两参数Ｗｉｅｎｅｒ过程的反向Ｃｓ¨ｏｒｇ¨ｏＲéｖéｓｚ概率不等式，并由此改进了林正炎关于两参数Ｗｉｅｎｅｒ过程增量的下极限结果．     

M／M／1／k后馈系统中的随机过程

本文对Ｍ／Ｍ／１／ｋ后馈排队系统中各随机过程的Ｐｏｉｓｓｏｎ性进行了讨论，推广了Ｂｒｅｍａｕｄ（［２］，［３］）的相应结果。所得结论表明Ｍ／Ｍ／１／ｋ后馈系统与Ｍ／Ｍ／１后馈系统情况有所不同，即在某些情况下，除总输出过程外，还有其它的过程也可能是Ｐｏｉｓｓｏｎ过程。顺便又地Ｍ／Ｍ／Ｃ／ｋ前馈后馈排队系统的动态数学模型进行了严格的讨论。     

Necessary conditions for renewal processes approximating to Wiener processes

ＮｅｃｅｓｓａｒｙＣｏｎｄｉｔｉｏｎｓｆｏｒＲｅｎｅｗａｌＰｒｏｃｅｓｓｅｓＡｐｐｒｏｘｉｍａｔｉｎｇｔｏＷｉｅｎｅｒＰｒｏｃｅｓｓｅｓ＊ＺｈａｎｇＬｉｘｉｎａｂｓｔｒａｃｔ．Ｌｅｔ｛Ｘ,Ｘｉ;ｉ≥１｝ｂｅａｓｅｑｕｅｎｃｅｏｆｉ．ｉ．ｄ．ｒ．ｖ＇．...     

Saks空间中（A）型相容映射的叠合点及公共不动点（英）

ＣｏｉｎｃｉｄｅｎｃｅＰｏｉｎｔｓａｎｄＣｏｍｍｏｎＦｉｘｅｄＰｏｉｎｔｓｆｏｒＣｏｍｐａｔｉｂｌｅＭａｐｓｏｆＴｙｐｅ（Ａ）ｏｎＳａｋｓＳｐａｃｅｓＰ．Ｐ．Ｍｕｒｔｈｙ，Ｂ．Ｋ．Ｓｈａｒｔｒｉａ，Ｙ．Ｊ．ＣｈｏＣｏｉｎｃｉｄｅｎｃｅＰｏｉｎｔｓａｎ...     

关于Gauss过程若干强极限定理

本文讨论了具有平稳增量的Ｇａｕｓｓ过程，在（０≤ｒ≤∞）时这类Ｇａｕｓｓ过程的增量有多小的问题．把有关的Ｗｉｅｎｅｒ过程的结果，在一定的条件下，推广到这类Ｇａｕｓｓ过程上去．此外还讨论了另一种形式的不可微模．     

解直角三角形目标检测

一、填空（每空２分，共３０分）（１）在△ＡＢＣ中：∠Ｃ＝９０°，ａ＝１２，ｂ＝９，则ｓｉｎＡ＝，ｃｔｇＡ＝．（２）在△ＡＢＣ中，∠Ｃ＝９０°，ｓｉｎＡ＝４５，ＡＢ＝１０，那么ＢＣ＝，ｃｏｓＢ＝．（３）已知ｃｏｓ５４°３６′＝０．５７９３，查表求得同一行中它的修正值是５，则ｃｏｓ５４°３４′＝．（４）用“＜”号连结下列各数：ｓｉｎ３０°，ｔｇ４５°，ｃｔｇ９０°，ｃｏｓ４５°，ｃｔｇ６０°，ｃｏｓ３０°：．（５）化简：（ｓｉｎ６０°－１）２＋｜１＋ｃｏｓ３０°｜＝．（６）在△ＡＢＣ中，∠Ｂ是锐角，…     

On the Supports of One Class of Measure-valued Process

ＯｎｔｈｅＳｕｐｐｏｒｔｓｏｆＯｎｅＣｌａｓｓｏｆＭｅａｓｕｒｅ－ｖａｌｕｅｄＰｒｏｃｅｓｓＴａｎｇＪｉａｓｈａｎ唐加山（ＤｅｐｔｏｆＭａｔｈ．，ＢｅｉｊｉｎｇＮｏｒｍａｌｕｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊｉｎｇ，１００８７５，Ｐ．Ｒ，Ｃ．）Ｃｏｍｍｕｎ...     

一个命题的再研究

命题　△ＡＢＣ中，∠Ａ、∠Ｂ、∠Ｃ所对边分别是ａ、ｂ、ｃ，求证　　ｓｉｎＡ－ｓｉｎＢｂｃ＋ｓｉｎＢ－ｓｉｎＣｃａ＋ｓｉｎＣ－ｓｉｎＡａｂ　≥０．（１）（《数学通报》１９９７年５月号问题１０７２）文［１］对上述命题给出了一种简捷证法．通过对（１）式证法的研究，笔者得到了以下几个命题．命题１　设△ＡＢＣ中，∠Ａ、∠Ｂ、∠Ｃ所对边分别是ａ、ｂ、ｃ，则有：　　ｓｉｎＡ－ｓｉｎＢｃａ＋ｓｉｎＢ－ｓｉｎＣａｂ＋ｓｉｎＣ－ｓｉｎＡｂｃ　≤０．（２）证明　由正弦定理知，不等式（２）等价于ａ－ｂｃａ＋ｂ－ｃａｂ＋…     

数学问题解答

数学问题解答１９９６年３月号问题解答（解答由问题提供人给出）１００１已知ΔＡＢＣ中，三内角为Ａ，Ｂ，Ｃ．试证：ｃｏｓ２Ａ＋ｃｏｓ２Ｂ＋ｃｏｓ２Ｃ＋２ｃｏｓＡｃｏｓＢｃｏｓＣ＝１．证明容易知道在ΔＡＢＣ中成立：（其中ａ，ｂ，ｃ是ΔＡＢＣ的三边长）ａ＝ｂ...     

Generalized stochastic processes in algebras of generalized functions

Stochastic processes with paths in a generalized function algebra are defined and it is shown that there exists an embedding of generalized functional stochastic processes into such ones. Gaussian stochastic processes with paths in an algebra of generalized functions are characterized by their first and second moments and an application to stochastic differential equations is given.     

Gaussian and sparse processes are limits of generalized Poisson processes

The theory of sparse stochastic processes offers a broad class of statistical models to study signals, far beyond the more classical class of Gaussian processes. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by Lévy white noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Lévy process—from Gaussian to sparse—can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects.     

利率均值回复模型下标的资产期权定价

在利率均值回复金融市场中 ,给出了财富贴现过程的随机微分方程 ;证明了与之联系的倒向随机微分方程解的存在唯一性 .最后 ,从倒向随机微分方程的解出发 ,得到了欧式期权定价的条件期望定价公式 .     

On the construction of non-affine jump-diffusion models

We describe a method for construction of jump analogues of certain one-dimensional diffusion processes satisfying solvable stochastic differential equations. The method is based on the reduction of the original stochastic differential equations to the ones with linear diffusion coefficients, which are reducible to the associated ordinary differential equations, by using the appropriate integrating factor processes. The analogues are constructed by means of adding the jump components linearly into the reduced stochastic differential equations. We illustrate the method by constructing jump analogues of several diffusion processes and expand the notion of market price of risk to the resulting non-affine jump-diffusion models.     

On weak convergence of random fields

We suggest simple and easily verifiable, yet general, conditions under which multi-parameter stochastic processes converge weakly to a continuous stochastic process. Connections to, and extensions of, R. Dudley’s results play an important role in our considerations, and we therefore discuss them in detail. As an illustration of general results, we consider multi-parameter stochastic processes that can be decomposed into differences of two coordinate-wise non-decreasing processes, in which case the aforementioned conditions become even simpler. To illustrate how the herein developed general approach can be used in specific situations, we present a detailed analysis of a two-parameter sequential empirical process.     

Sampling and reconstruction for shift-invariant stochastic processes

In this paper, combining stochastic processes with shift-invariant spaces, we introduce shift-invariant stochastic processes. It is a general case of the classical band-limited stochastic processes and a kind of non-band-limited stochastic processes. Two sampling theorems are obtained for the shift-invariant stochastic processes. The results for band-limited stochastic processes and shift-invariant spaces are generalized by our new results.     

批量马尔可夫到达过程概述

批量马尔可夫到达过程(Batch Markovian Arrival Processes,BMAP)对平稳点过程类具有稠密性,能够描述许多到达过程,在计算机、可靠性、通信和库存等领域的随机建模中获得广泛应用,是一类非常重要的随机点过程.通过对BMAP理论主要文献的分析,系统介绍了BMAP的概念和主要性质,回顾了BMAP应用成果和拟合工作的发展,展望了BMAP理论的发展前景.     

非时齐马氏过程的随机单调性

本文研究了非时齐马氏过程的随机单调性问题.利用时齐的马氏过程随机单调性的相关证明方法,加以改进,获得了非时齐马氏过程随机单调性的显式判定方法,并进一步将这一充分性条件推广为等价条件.     

带随机过程的随机规划问题最优解过程的平稳性与马氏性

证明了带随机过程的随机规划问题其最优争集中至少存在一列最优解均为可测的随机过程；且如果问题中的随机过程具有平稳性与马氏性，则此时间问题的最优解过程亦具有相应的特性。     

Discrete time modeling of mean-reverting stochastic processes for real option valuation

In this paper the recombining binomial lattice approach for modeling real options and valuing managerial flexibility is generalized to address a common issue in many practical applications, underlying stochastic processes that are mean-reverting. Binomial lattices were first introduced to approximate stochastic processes for valuation of financial options, and they provide a convenient framework for numerical analysis. Unfortunately, the standard approach to constructing binomial lattices can result in invalid probabilities of up and down moves in the lattice when a mean-reverting stochastic process is to be approximated. There have been several alternative methods introduced for modeling mean-reverting processes, including simulation-based approaches and trinomial trees, however they unfortunately complicate the numerical analysis of valuation problems. The approach developed in this paper utilizes a more general binomial approximation methodology from the existing literature to model simple homoskedastic mean-reverting stochastic processes as recombining lattices. This approach is then extended to model dual correlated one-factor mean-reverting processes. These models facilitate the evaluation of options with early-exercise characteristics, as well as multiple concurrent options.