Martingale Approach to Stochastic Control with Discretionary Stopping

We develop a martingale approach for continuous-time stochastic control with discretionary stopping. The relevant Dynamic Programming Equation and Maximum Principle are presented. Necessary and sufficient conditions are provided for the optimality of a control strategy; these are analogues of the "equalization" and "thriftiness" conditions introduced by Dubins and Savage (1976) in a related, discrete-time context. The existence of a thrifty control strategy is established.     

Partial Sum Process for Records

Suppose the upper records from a sequence of i.i.d. random variables is in the domain of attraction of a normal distribution. Consider the D(0,1]-valued process {Z_n(·)} constructed by usual interpolation of the partial sums of the records. We prove that under some mild conditions, {Z_n} converges to a limiting Gaussian process in D(0,1]. As a consequence, the partial sums of records is asymptotically normal. AMS 2000 Subject Classification Primary—60F17 Secondary—60G70     

Records with confirmation

In the present paper, the concept of records with confirmation is proposed. This new concept implies that such records cannot be contaminated with sample outliers. On determining records with confirmation, we discuss their distributional and limit properties. A test based on records with confirmation is provided in our work. We also present simulation techniques for generating record values with confirmation.     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.     

A Characterization of Counterexamples to the Kodaira-Ramanujam Vanishing Theorem on Surfaces in Positive Characteristic

The author gives a characterization of counterexamples to the Kodaira-Ramanujam vanishing theorem on smooth projective surfaces in positive characteristic. More precisely, it is reproved that if there is a counterexample to the Kodaira-Ramanujam vanishing theorem on a smooth projective surface X in positive characteristic, then X is either a quasi-elliptic surface of Kodaira dimension 1 or a surface of general type. Furthermore, it is proved that up to blow-ups, X admits a fibration to a smooth projective curve, such that each fiber is a singular curve.     

Martingale property of empirical processes

It is shown that for a large collection of independent martingales, the martingale property is preserved on the empirical processes. Under the assumptions of independence and identical finite-dimensional distributions, it is proved that a large collection of stochastic processes are martingales essentially if and only if the empirical processes are also martingales. These two results have implications on the testability of the martingale property in scientific modeling. Extensions to submartingales and supermartingales are given.

     

On a Constant Related to the Prime Counting Function

Let \({\pi(x)}\) be the number of primes not exceeding x. We produce new explicit bounds for \({\pi(x)}\) and we use them to obtain a fine frame for the remainder term in the asymptotic formula of the sum \({\sum_{2\leq n\leq x}1/\pi(n)}\).     

Some Martingales Related to the Integral of Brownian Motion. Applications to the Passage Times and Transience

Let (B _t) _t0 be the standard linear Brownian motion started at y and set (X _t, B _t). In this paper we introduce some martingales related to the Markov process (U _t) _t0, which allow us to calculate explicitly the probability laws of several passage times associated to U in a probabilistic way. With the aid of an appropriate supermartingale, we also establish the transience of the process (U _t) _t0.     

分布乘法问题评述

论述了研讨分布乘法的两种基本途径，还分析了八十年代关于分布乘积研究的两个特点．文章结尾指出了九十年代分布乘法理论可能的发展趋势．     

On characterizing the bivariate exponential and geometric distributions

In this note, a characterization of the Gumbel's bivariate exponential distribution based on the properities of the conditional moments is discussed. The result forms a sort of bivariate analogue of the characterization of the univariate exponential distribution given by Sahobov and Geshev (1974) (cited in Lau and Rao ((1982), Sankhy Ser. A, 44, 87)). A discrete version of the property provides a similar conclusion relating to a bivariate geometric distribution.     

On a characterization theorem for locally compact abelian groups

The well-known Skitovich-Darmois theorem asserts that a Gaussian distribution is characterized by the independence of two linear forms of independent random variables. The similar result was proved by Heyde, where instead of the independence, the symmetry of the conditional distribution of one linear form given another was considered. In this article we prove that the Heyde theorem on a locally compact Abelian group X remains true if and only if X contains no elements of order two. We describe also all distributions on the two-dimensional torus which are characterized by the symmetry of the conditional distribution of one linear form given another. In so doing we assume that the coefficients of the forms are topological automorphisms of X and the characteristic functions of the considering random variables do not vanish.     

Some Extensions of M-Fractions Related to Strong Stieltjes Distributions

The purpose of this paper is to show the symmetric relations that appear between the coefficients of some even and odd extensions of the M-fractions related to a certain kind of symmetric strong Stieltjes distribution.     

触销式双障碍买权价值过程鞅性

主要证明了在不存在交易成本的完全市场条件下连续时间欧式触销式双障碍买权贴现到0时刻的价值过程为鞅,并且给出了对应单障碍买权价值过程的鞅性质.同时还讨论了执行价格为随机的触销式双障碍买权的鞅性质,给出了任意时刻 t(0≤ t≤T)其内在价值的表达式.     

Counting Function of Characteristic Values and Magnetic Resonances

We consider the meromorphic operator-valued function I ? K(z) = I ? A(z)/z where A is holomorphic on the domain 𝒟 ? ?, and has values in the class of compact operators acting in a given Hilbert space. Under the assumption that A(0) is a selfadjoint operator which can be of infinite rank, we study the distribution near the origin of the characteristic values of I ? K, i.e. the complex numbers w ≠ 0 for which the operator I ? K(w) is not invertible, and we show that generically the characteristic values of I ? K converge to 0 with the same rate as the eigenvalues of A(0).

We apply our abstract results to the investigation of the resonances of the operator H = H ₀ + V where H ₀ is the shifted 3D Schrödinger operator with constant magnetic field of scalar intensity b > 0, and V: ?³ → ? is the electric potential which admits a suitable decay at infinity. It is well known that the spectrum σ(H ₀) of H ₀ is purely absolutely continuous, coincides with [0, + ∞[, and the so-called Landau levels 2bq with integer q ≥ 0, play the role of thresholds in σ(H ₀). We study the asymptotic distribution of the resonances near any given Landau level, and under generic assumptions obtain the main asymptotic term of the corresponding resonance counting function, written explicitly in the terms of appropriate Toeplitz operators.     

Characterization of probability distributions by conditional expectation of function of record statistics

In this paper, two general classes of distributions have been characterized through conditional expectation of power of difference of two record statistics. Further, some particular cases and examples are also discussed.     

The Asymptotic Distributions of Sums of Records

LetX ₁,X ₂... be a sequence of i.i.d. random variables and let be the associated (upper) record sequence. Resnick (1973) identified the class of all possible limit distributions for . Here we focus on sums of records, . We describe three cases in which T _n can be normalized to have a non-trivial limiting distribution. The problem of identifying all possible limit laws for normalized sums of records remains open.     

Characteristic vector fields of generic distributions of corank 2

We study generic distributions D⊂TM$D\subset TM$ of corank 2 on manifolds M   of dimension n?5$n?5$. We describe singular curves of such distributions, also called abnormal curves. For n   even the singular directions (tangent to singular curves) are discrete lines in D(x)$D(x)$, while for n   odd they form a Veronese curve in a projectivized subspace of D(x)$D(x)$, at generic x∈M$x\in M$. We show that singular curves of a generic distribution determine the distribution on the subset of M where they generate at least two different directions. In particular, this happens on the whole of M if n is odd. The distribution is determined by characteristic vector fields and their Lie brackets of appropriate order. We characterize pairs of vector fields which can appear as characteristic vector fields of a generic corank 2 distribution, when n is even.     

Random translation,dilation and contraction of order statistics

Distributional properties of the order statistics are utilized to characterize distributions of interest. Further, the distributions of one sided random dilation and contraction are obtained and important deductions are discussed.     

An Upper Bound on the Binomial Process Approximation to the Exceedance Process

In this paper we deal with an approximation to the point process of exceedances over a higher threshold. We compute sharp bounds on the remainder terms when actual distributions of exceedances are replaced by appropriate generalized Pareto distributions (GPD). The bound will be formulated in terms of the von Mises function. The ultimate as well as the penultimate approximations are considered; in the latter case the shape parameter of the approximating GPD depends on the threshold.     

On the Characteristic Properties of Exponential Distribution

New characterizations for the exponential distribution are given in terms of record values and the probabilities of finite sums of independent and identically distributed nonnegative random variables provided that the underlying distribution is either new better than used or new worse than used.