Pasting of two processes with independent increments

A terminating stochastically continuous strictly Markov process is obtained as a result of pasting two nonterminating homogeneous stochastically continuous Markov processes with independent increments, one of which is semicontinuous. It is shown that this process can be extended to a complete homogeneous stochastically continuous strictly Markov Feller process. Previously, this problem has been solved by the author under stronger restrictions-both pasting processes were semicontinuous.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 487–491, April, 1993.     

Equivalence of measures induced by infinitely divisible processes

We find sufficient conditions for the equivalence of two measures on function space induced by infinitely divisible processes. The processes are not assumed to be stochastically continuous or to have independent increments. The theorem proved here is equivalent to known results in the special case of stochastically continuous processes with independent increments.     

Extremal Processes with One Jump

Convergence of a sequence of deterministic functions in the Skorohod topology implies convergence of the jumps. For processes with independent additive increments the fixed discontinuities converge. In this paper it will be shown that this is not true for processes with independent max-increments. The limit in of a sequence of stochastically continuous extremal processes may have fixed discontinuities. Our construction makes use of stochastically continuous extremal processes whose sample functions have only one jump.     

Homomorphisms for distributive operations in partial algebras. Applications to linear operators and measure theory

Summary In this paper the concept of distributivity introduced earlier [1] is used to show that a homomorphism with respect to two distributive operations which is extended as a homomorphism with respect to operation 1 remains necessarily also a homomorphism with respect to operation 2 on the 1-closure of the original domain of definition. The result is illustrated by applications to continuous extensions of homomorphisms between -complete vector lattices, association of families of stochastically independent systems of sets and integration of products of independent functions.     

Multivariate reliability modelling based on dependent dynamic shock models

In this paper, we stochastically model positively dependent multivariate reliability distributions based on stochastically dependent dynamic shock models. In the first part, we consider a shock model with delayed failures. This shock model will be used to construct a class of absolutely continuous multivariate reliability distributions. Explicit parametric forms for the multivariate reliability functions are suggested. Multivariate ageing properties and dependence structures of the class are discussed as well. In the second part, we obtain two types of absolutely continuous multivariate exponential distributions based on further generalized shock models.     

Stochastically larger component of a random vector

Summary Definitions of different strengths are given to the notion of ‘a stochastically larger component of a two-dimensional random vector.’ Some of them reduce to the known definitions of stochastic order relationship when the components are stochastically independent. The definitions and the approach are related to nonparametric problems.     

Optimal Bayesian fault prediction scheme for a partially observable system subject to random failure

A new method for predicting failures of a partially observable system is presented. System deterioration is modeled as a hidden, 3-state continuous time homogeneous Markov process. States 0 and 1, which are not observable, represent good and warning conditions, respectively. Only the failure state 2 is assumed to be observable. The system is subject to condition monitoring at equidistant, discrete time epochs. The vector observation process is stochastically related to the system state. The objective is to develop a method for optimally predicting impending system failures. Model parameters are estimated using EM algorithm and a cost-optimal Bayesian fault prediction scheme is proposed. The method is illustrated using real data obtained from spectrometric analysis of oil samples collected at regular time epochs from transmission units of heavy hauler trucks used in mining industry. A comparison with other methods is given, which illustrates effectiveness of our approach.     

Random Linear Functionals Arising in Stochastic Integration

The space of signed measures on the Borel σ-algebra of a Polish space is incomplete with respect to the bounded Lipschitz norm. Elements of its completion are called hypermeasures. They can be regarded as linear functionals on the space of bounded Lipschitz functions. It is shown that, under mild assumptions, every stochastically continuous random linear functional on this space is a modification of a random hypermeasure.      

On the Levy-Baxter theorems for shot-noise fields. I

We consider shot-noise fields generated by countably additive stochastically continuous homogeneous random measures with independent values on disjoint sets. We establish necessary and sufficient conditions under which the shot-noise fields possess the Levy-Baxter property on fixed and increasing parametric sets. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1463–1476, November, 1998.     

Stochastic processes with independent increments,taking values in a Hilbert space

Let $\text{H}$ be a real separable Hilbert space; let X(t), t?[0, 1], be a separable, stochastically continuous, $\text{H}$-valued stochastic process with independent increments. Then a decomposition of X(t) into a uniformly convergent sum of independent processes is found. In this decomposition one of the processes is Gaussian with continuous sample functions, and the remainder of the processes have sample functions whose discontinuities correspond to those of certain real-valued Poisson processes. The decomposition of X(t) leads to a Levy-Khintchine representation of the characteristic functional of X(t). In addition, the case when X(t) has finite variance is explored, and, as a consequence of the above decomposition, a Kolmogorov-type representation of the characteristic functional of X(t) is derived.     

Intersections and polar functions of fractional brownian sheets

Let B^H={B^H（t）,t∈R^N＋}be a real-valued（N,d）fractional Brownian sheet with Hurst index H=（H1,…,HN）.The characteristics of the polar functions for B^H are discussed.The relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar-functions of B^H is obtained.The Hausdorff dimension about the fixed points and the inequality about the Kolmogorov’s entropy index for B^H are presented.Furthermore,it is proved that any two independent fractional Brownian sheets are nonintersecting in some conditions.A problem proposed by LeGall about the existence of no-polar continuous functions satisfying the Holder condition is also solved.     

Transformations preserving the Hausdorff-Besicovitch dimension

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R ¹ resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution function from the above class is a DP-function. Relations between the entropy of probability distributions, their Hausdorff-Besicovitch dimension and their DP-properties are discussed. Examples are given of singular distribution functions preserving the fractal dimension and of strictly increasing absolutely continuous functions which do not belong to the DP-class.      

Life expectancy under random censorship

This paper deals with estimation of life expectancy used in survival analysis and competing risk study under the condition that the data are randomly censored by K independent censoring variables. The estimator constructed is based on a theorem due to Berman [2], and it involves an empirical distribution function which is related to the Kaplan-Meier estimate used in biometry. It is shown that the estimator, considered as a function of age, converges weakly to a Gaussian process. It is found that for the estimator to have finite limiting variance requires the assumption that the censoring variables be stochastically larger than the “survival” random variable under investigation.     

寿命分布NBUE性质的一个新刻画及应用

本文介绍绝对连续型NBUE寿命分布的一个新的刻画：一个寿命分布具有NBUE性当且仅当它随机地大于具有相应的平衡分布的随机变量。讨论了此刻画在可靠性与寿命检验中的应用，并建立了严格NBUE性的一个非参数检验方法。     

B值随机元阵列的矩完全收敛性

在随机元阵列随机有界于某非负随机变量的条件下,得到了B值行独立的随机元阵列的矩完全收敛性的一些充分条件.同时研究了p型Banach空间中行独立的随机元阵列的矩完全收敛性.     

Cones,semicontinuous functions,and continuous lattices

The purpose of this paper is to introduce a notion which is a generalization of convex sets and to use this notion to construct continuous lattices which are shown to be related to lattices of lower-semicontinuous functions. The end results of this development is a characterization of lattices of lower-semicontinuous functions in terms of a class of continuous lattices introduced in this paper (see Theorem 8). Then material is introduced which leads to a complementary result in Theorem 11 which characterizes the continuous lattices that can be lattices of lower-semicontinuous functions.     

Monotone matrices and monotone Markov processes

A stochastic matrix is “monotone” [4] if its row-vectors are stochastically increasing. Closure properties, characterizations and the availability of a second maximal eigenvalue are developed. Such monotonicity is present in a variety of processes in discrete and continous time. In particular, birth-death processes are monotone. Conditions for the sequential monotonicity of a process are given and related inequalities presented.     

TIME INCONSISTENCY AND REPUTATION IN MONETARY POLICY: A STRATEGIC MODELLING IN CONTINUOUS TIME

This article develops a model to examine the equilibrium behavior of the time inconsistency problem in a continuous time economy with stochastic and endogenized dis-tortion. First, the authors introduce the notion of sequentially rational equilibrium, and show that the time inconsistency problem may be solved with trigger reputation strategies for stochastic setting. The conditions for the existence of sequentially rational equilibrium are provided. Then, the concept of sequentially rational stochastically stable equilibrium is introduced. The authors compare the relative stability between the cooperative behavior and uncooperative behavior, and show that the cooperative equilibrium in this monetary policy game is a sequentially rational stochastically stable equilibrium and the uncooper-ative equilibrium is sequentially rational stochastically unstable equilibrium. In the long run, the zero inflation monetary policies are inherently more stable than the discretion rules, and once established, they tend to persist for longer periods of the time.     

Affine processes are regular

We show that stochastically continuous, time-homogeneous affine processes on the canonical state space ${\mathbb{R}_{\geq 0}^m \times \mathbb{R}^n}$ are always regular. In the paper of Duffie et?al. (Ann Appl Probab 13(3):984?C1053, 2003) regularity was used as a crucial basic assumption. It was left open whether this regularity condition is automatically satisfied for stochastically continuous affine processes. We now show that the regularity assumption is indeed superfluous, since regularity follows from stochastic continuity and the exponentially affine form of the characteristic function. For the proof we combine classic results on the differentiability of transformation semigroups with the method of the moving frame which has been recently found to be useful in the theory of SPDEs.