Maximum Entropy Estimates for Risk-Neutral Probability Measures with Non-Strictly-Convex Data

This article investigates use of the Principle of Maximum Entropy for approximation of the risk-neutral probability density on the price of a financial asset as inferred from market prices on associated options. The usual strict convexity assumption on the market-price to strike-price function is relaxed, provided one is willing to accept a partially supported risk-neutral density. This provides a natural and useful extension of the standard theory. We present a rigorous analysis of the related optimization problem via convex duality and constraint qualification on both bounded and unbounded price domains. The relevance of this work for applications is in explaining precisely the consequences of any gap between convexity and strict convexity in the price function. The computational feasibility of the method and analytic consequences arising from non-strictly-convex price functions are illustrated with a numerical example.     

广义线性模型极大似然估计的强相合性与渐近正态性

本文研究了若干重要类型的离散响应变量广义线性模型,在sum from i=1 to n ZiZi'的最小特征根大于cnα(对某个c>0,α>0)等条件下证明了回归参数向量的极大似然估计的强相合性与渐近正态性,其中设计阵序列{||Zn||}可以为无界序列.     

Asymptotic Estimates of Stirling Numbers

New asymptotic estimates are given of the Stirling numbers and , of first and second kind, respectively, as n tends to infinity. The approximations are uniformly valid with respect to the second parameter m.     

Asymptotic Estimates of Elementary Probability Distributions

Several new asymptotic estimates (with precise error bounds) are derived for Poisson and binomial distributions as the parameters tend to infinity. The analytic methods used are also applicable to other discrete distribution functions.     

期权定价的最大熵方法

以欧式期权为例,用标的资产(如股票风险资产)和无风险资产复制期权,并用自融资无套利原理分析金融市场的资产价值变化情况.在此基础上,通过最大熵原理来求得资产组合中每个资产所占的比重,进而得出期权定价模型,由于最大熵原理所求得概率分布是目前所知求概率分布方法中最客观、无偏的,所以求得的新模型不受金融市场类型和标的资产价格分布的限制,具有较强的客观、无偏、可预测性.通过对期权的常用算例计算,发现新模型比B-S模型以及一些其它熵期权定价模型有更准确的标的资产价格分布、更低的回溯测试误差.     

Maximum Entropy Condition in Queueing Theory

The main results in queueing theory are obtained when the queueing system is in a steady-state condition and if the requirements of a birth-and-death stochastic process are satisfied. The aim of this paper is to obtain a probabilistic model when the queueing system is in a maximum entropy condition. For applying the entropic approach, the only information required is represented by mean values (mean arrival rates, mean service rates, the mean number of customers in the system). For some one-server queueing systems, when the expected number of customers is given, the maximum entropy condition gives the same probability distribution of the possible states of the system as the birth-and-death process applied to an M/M/1 system in a steady-state condition. For other queueing systems, as M/G/1 for instance, the entropic approach gives a simple probability distribution of possible states, while no close expression for such a probability distribution is known in the general framework of a birth-and-death process.     

Calculating Maximum Entropy Flows in Networks

This paper describes methods for calculating the most likely values of link flows in networks with incomplete data. The object is to present a thorough and rigorous treatment of maximum entropy flow estimation methods and to develop a methodological framework capable of handling different types of network problems. A multiple probability space conditional entropy approach is described for the general network problem. Results are presented and discussed for an example network intended for water supply.     

Restoration of Degraded Images with Maximum Entropy

We present a Maximum Entropy based approach to the restoration ofdegraded images as an alternative to restorationtechniques using inverse Wiener filtering.The method we discuss applies in particularto images corrupted by a relativelyhigh system noise. A variety of experimental results supportingour imaging model are included.     

Maximum Entropy Analysis of Multiple-server Queueing Systems

This paper deals with a general queueing system with many servers in steady state. Applying the method of entropy maximization with several constraints and the asymptotic property of the GI/G/c queueing systems, we provide approximate formulae for the distributions of the number of customers and the waiting time. In order to examine the accuracy of the approximate formulae, we present several numerical examples for illustration.     

Nondeterminism of Linear Operators and Lower Entropy Estimates

Let u be a (bounded, linear) operator from a Hilbert space ℋ into the Banach space C(T), the space of continuous functions on the compact metric space T. We introduce and investigate numbers τ _n (u), n≥1, measuring the degree of determinism of the operator u. The slower τ _n (u) decreases, the less determined are functions in the range of u by their values on a certain set of points. It is shown that n ^−1/2 τ _n (u)≤2e _n (u), where e _n (u) are the (dyadic) entropy numbers of u. Furthermore, we transform the notion of strong local nondeterminism from the language of stochastic processes into that of linear operators. This property, together with a lower entropy estimate for the compact space T, leads to a lower estimate for τ _n (u), hence also for e _n (u). These results are used to prove sharp lower entropy estimates for some integral operators, among them, Riemann–Liouville operators with values in C(T) for some fractal set T. Some multi-dimensional extensions are treated as well.      

Relative Entropy, Asymptotic Pairs and Chaos

In this paper, we prove that positive conditional entropy impliesthe existence of asymptotic pairs and scrambled sets on fibers.Moreover, we introduce the notion of conditional topologicalentropy for a subset using Bowen's definition of separated andspanning sets, and prove that the conditional topological entropyof a system relative to a factor is the supremum of conditionaltopological entropy of its scrambled sets on fibers.     

混合样本时密度估计的渐近正态性

对多元密度函数的最近邻估计和核估计,在样本为不混合的场合下,本文获得了它们的渐近正态性。     

Asymptotic Estimates for Gagliardo–Nirenberg Embedding Constants

Sharp asymptotic information is determined for the Gagliardo–Nirenberg embedding constants in high dimension. This analysis is motivated by the earlier observation that the logarithmic Sobolev inequality controls the Nash inequality. Moreover, one sees here that Hardy's inequality can be interpreted as the asymptotic limit of the logarithmic Sobolev inequality.     

估计死亡率分布的一个最大熵模型

本文提出了一种估计死亡率分布的新模型一最大熵模型。该模型直接从样本信息出发，不需要对待估分布的概率密度函数或先验分布作任何假定，从而克服了极大似然估计和贝叶斯估计的不足。而且通过两个例子的计算结果，表明该方法与样本数据的拟合效果要好于其它两种方法。     

The Entropy in Learning Theory. Error Estimates

We continue the investigation of some problems in learning theory in the setting formulated by F. Cucker and S. Smale. The goal is to find an estimator on the base of given data that approximates well the regression function of an unknown Borel probability measure defined on We assume that belongs to a function class It is known from previous works that the behavior of the entropy numbers of in the uniform norm plays an important role in the above problem. The standard way of measuring the error between a target function and an estimator is to use the norm ( is the marginal probability measure on X generated by ). This method has been used in previous papers. We continue to use this method in this paper. The use of the norm in measuring the error has motivated us to study the case when we make an assumption on the entropy numbers of in the norm. This is the main new ingredient of thispaper. We construct good estimators in different settings: (1) we know both and ; (2) we know but we do not know and (3) we only know that is from a known collection of classes but we do not know An estimator from the third setting is called a universal estimator.     

Asymptotic Normality of Cross-correlogram Estimates of the Response Function

The problem of estimation of an unknown response function of a time-invariant continuous linear system is considered. Discrete-time sample input–output cross-correlograms are taken as estimates of the response function. The inputs are supposed to be zero-mean stationary Gaussian processes close, in some sense, to a white noise. Both asymptotic normality of finite-dimensional distributions of the estimates and their asymptotic normality in spaces of continuous functions are studied. Our basic tool is a new integral representation for cumulants of the estimate as a finite sum of integrals involving cyclic products of kernels. Some inequalities for these integrals are obtained and their asymptotic behaviour is studied.