Importance and Sensitivity Analysis in Dynamic Reliability

In dynamic reliability, the evolution of a system is governed by a piecewise deterministic Markov process, which is characterized by different input data. Assuming such data to depend on some parameter p ∈ P, our aim is to compute the first-order derivative with respect to each p ∈ P of some functionals of the process, which may help to rank input data according to their relative importance, in view of sensitivity analysis. The functionals of interest are expected values of some function of the process, cumulated on some finite time interval [0,t], and their asymptotic values per unit time. Typical quantities of interest hence are cumulated (production) availability, or mean number of failures on some finite time interval and similar asymptotic quantities. The computation of the first-order derivative with respect to p ∈ P is made through a probabilistic counterpart of the adjoint state method, from the numerical analysis field. Examples are provided, showing the good efficiency of this method, especially in case of a large P.     

SINGLE BIRTH PROCESSES

1.IntroductionLetusstartfromthetime-continuouscase.Then,theQ-matrixofasinglebirthprocessisasfollows:Throughoutthenote,weconsideronlytotallystableandconservativeQ-matrix:forallDuetothefactthattheboundaryofsuchQ-matrixisatmostsingleexit,thesinglebirthp...     

Malliavin calculus for Markov chains using perturbations of time

In this article, we develop a Malliavin calculus associated to a time-continuous Markov chain with finite state space. We apply it to get a criterion of density for solutions of stochastic differential equation involving the Markov chain and also to compute greeks.     

Probabilities for Queueing Systems with Embedded Markov Chains

Abstract

Transition probabilities of embedded Markov chain for single-server queues are considered when the distribution of the inter-arrival time or that of the service time is specified. A comprehensive collection of formulas is derived for the transition probabilities, covering some seventeen flexible families. The corresponding estimation procedures are also derived by the method of moments. It is expected that this work could serve as a useful reference for the modeling of queuing systems with embedded Markov chains.     

Markov process functionals in finance and insurance

The Markov property of Markov process functionals which are frequently used in economy, finance, engineering and statistic analysis is studied. The conditions to judge Markov property of some important Markov process functionals are presented, the following conclusions are obtained： the multidimensional process with independent increments is a multidimensional Markov process; the functional in the form of path integral of process with independent increments is a Markov process; the surplus process with the doubly stochastic Poisson process is a vector Markov process. The conditions for linear transformation of vector Markov process being still a Markov process are given.     

Optimization of Weighted Monte Carlo Methods with Respect to Auxiliary Variables

We consider the problems whose mathematical model is determined by some Markov chain terminating with probability one; moreover, we have to estimate linear functionals of a solution to an integral equation of the second kind with the corresponding substochastic kernel and free term [1]. To construct weighted modifications of numerical statistical models, we supplement the coordinates of the phase space with auxiliary variables whose random values functionally define the transitions in the initial chain. Having implemented each auxiliary random variable, we multiply the weight by the ratio of the corresponding densities of the initial and numerically modeled distributions. We solve the minimization problem for the variances of estimators of linear functionals by choosing the modeled distribution of the first auxiliary random variable.     

On stochastic processes in a multilevel control bulk queueing system

This article analyzes some stochastic processes that arise in a bulk single server queue with continuously operating server, state dependent compound Poisson input flow and general state dependent service process. The authors treat the queueing process as a semi–regenerative process and obtain the invariant probability measure and the transient distribution for the embedded Markov chain. The stationary probability measure for the queueing process with continuous time parameter is found by using semi-regenerative techniques. The authors also study the input and output processes and establish ergodic theorems for some functionals of these processes. The results are obtained in terms of the invariant probability measure for the embedded process and the stationary measure for the queueing process with continuous time parameter     

Birthing Markov processes at random rates

Summary This paper studies processes constructed by birthing the trajectories of a given Markov process along time according to random probabilities. Getoor has considered the case where the random probabilities are determined by comultiplicative functionals and proved for right processes that the post-birth process has the Markov property. Here randomizations of comultiplicative functionals are described which give rise to conditionally Markov processes. The main argument is developed for general Markov processes and the transition probabilities of the new process, including those from the pre-birth state, are explicited.     

Generalized inverses and their application to applied probability problems

The main aim of this paper is to examine the applicability of generalized inverses to a wide variety of problems in applied probability where a Markov chain is present either directly or indirectly through some form of imbedding. By characterizing all generalized inverses of I—P, where P is the transition matrix of a finite irreducible discrete time Markov chain, we are able to obtain general procedures for finding stationary distributions, moments of the first passage time distributions, and asymptotic forms for the moments of the occupation-time random variables. It is shown that all known explicit methods for examining these problems can be expressed in this generalized inverse framework. More generally, in the context of a Markov renewal process setting the aforementioned problems are also examined using generalized inverses of I—P. As a special case, Markov chains in continuous time are considered, and we show that the generalized inverse technique can be applied directly to the infinitesimal generator of the process, instead of to I—P, where P is the transition matrix of the discrete time jump Markov chain.     

常利率复合二项双险种风险模型的研究

提出了含利率因素的复合二项双险种风险模型,并在有关假设的基础上,给出了此模型下保险公司稳定经营的必要条件;证明了索赔时刻的盈余过程是一马氏过程和调节系数的存在性,并采用递归方法得到了模型的破产概率的上界估计.     

On distribution of functionals of Brownian motion with linear drift

The paper deals with methods of calculation of the distributions of functionals of Brownian motion with linear drift. Various stopping time moments are considered. The cases where the moments take values equal to infinity are of special interest. Bibliography:5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 205–217. Translated by A. Sudakov.     

Tensor approach to mixed high-order moments of absorbing Markov chains

The moments of the number of the visits in an absorbing Markov chain are considered. The first moments and the non-mixed second moments of the number of the visits are calculated in classical textbooks such as the book of J. Kemeny and J. Snell “Finite Markov Chains”. The first moments and the non-mixed second moments can be easily expressed in a matrix form using the fundamental matrix of the absorbing Markov chain. Since the representation of the mixed moments of higher orders in a matrix form is not straightforward, if ever possible, they were not calculated. The gap is filled now. A tensor approach to the mixed high-order moments is proposed and compact closed-form expressions for the moments are discover.     

Discrete chaotic calculus and covariance identities

We show that for the binomial process (or Bernoulli random walk) the orthogonal functionals constructed in Kroeker, J.P. (1980) "Wiener analysis of functionals of a Markov chain: application to neural transformations of random signals", Biol. Cybernetics 36 , 243-248, [14] for Markov chains can be expressed using the Krawtchouk polynomials, and by iterated stochastic integrals. This allows to construct a chaotic calculus based on gradient and divergence operators and structure equations, and to establish a Clark representation formula. As an application we obtain simple infinite dimensional proofs of covariance identities on the discrete cube.     

Stability and monotonicity for interactive Markov chains

In a Markov chain model of a social process, interest often centers on the distribution of the population by state. One question, the stability question, is whether this distribution converges to an equilibrium value. For an ordinary Markov chain (a chain with constant transition probabilities), complete answers are available. For an interactive Markov chain (a chain which allows the transition probabilities governing each individual to depend on the locations by state of the rest of the population), few stability results are available. This paper presents new results. Roughly, the main result is that an interactive Markov chain with unique equilibrium will be stable if the chain satisfies a certain monotonicity property. The property is a generalization to interactive Markov chains of the standard definition of monotonicity for ordinary Markov chains.     

Interest Guarantees in Banking

Interest guarantees on loans and savings contracts are viewed as financial claims and priced by the no arbitrage principle in continuous time Markov interest models of diffusion type and of Markov chain type. Various forms of loan contracts and guarantees are considered, an important distinction being made between loans with fixed repayments and loans with fixed amortizations. Differential equations are obtained for the values of the guarantees, and some closed form expressions are obtained for standard contracts in certain well structured models.     

Central Limit Theorem Started at a Point for?Stationary Processes and Additive Functionals of?Reversible Markov Chains

In this paper we study the almost sure central limit theorem started at a point for additive functionals of a stationary and ergodic Markov chain via a martingale approximation in the almost sure sense. Some of the results provide sufficient conditions for general stationary sequences. We use these results to study the quenched CLT for additive functionals of reversible Markov chains.     

Distributions of functionals of the Brownian motion stopped at the moment inverse to the sojourn time

The paper deals with methods of computing the distributions of functionals of the Brownian motion stopped at some random moments. The moments are obtained by means of the minimum and maximum operations from the moments inverse to some additive functionals. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 39–56.     

Bivariate Markov chain embeddable variables of polynomial type

The primary aim of the present article is to provide a general framework for investigating the joint distribution of run length accumulating/enumerating variables by the aid of a Markov chain embedding technique. To achieve that we introduce first a class of bivariate discrete random variables whose joint distribution can be described by the aid of a Markov chain and develop formulae for their joint probability mass function, generating functions and moments. The results are then exploited for the derivation of the distribution of a bivariate run-related statistic. Finally, some interesting uses of our results in reliability theory and educational psychology are highlighted. Research supported by General Secretary of Research and Technology of Greece under grand PENED 2001.     

An approach to the study of multistate insurance contracts

We derive a matrix representation for formulas of moments of cash value of future payment streams arising from multistate insurance contract, where the evolution of the insured risk and the interest rate are random. As an application, we derive formulas for net single and period premiums. The general theory is illustrated with a case where the evolution of the insured risk is modeled by a Markov chain. Copyright © 2012 John Wiley & Sons, Ltd.     

Filtering of a Multi-Dimension Stochastic Volatility Model

In this article, we study a stochastic volatility model for a class of risky assets. We assume that the volatilities of the assets are driven by a common state of economy, which is unobservable and represented by a hidden Markov chain. Under this hidden Markov model (HMM), we develop recursively computable filtering equations for certain functionals of the chain. Expectation maximization (EM) parameter estimation is then used. Applications to an optimal asset allocation problem with mean-variance utility are given.