Chapman-Kolmogorov Equation Research Articles

On double-boundary non-crossing probability for a class of compound processes with applications

We develop an efficient method for computing the probability that a non-decreasing, pure jump (compound) stochastic process stays between arbitrary upper and lower boundaries (i.e., deterministic functions, possibly discontinuous) within a finite time period. The compound process is composed of a process modelling the arrivals of certain events (e.g., demands for a product in inventory systems, customers in queuing, or claims/capital gains in insurance/dual risk models), and a sequence of independent and identically distributed random variables modelling the sizes of the events. The events arrival process is assumed to belong to the wide class of point processes with conditional stationary independent increments which includes (non-)homogeneous Poisson, binomial, negative binomial, mixed Poisson and doubly stochastic Poisson (i.e., Cox) processes as special cases. The proposed method is based on expressing the non-exit probability through Chapman–Kolmogorov equations, re-expressing them in terms of a circular convolution of two vectors which is then computed applying fast Fourier transform (FFT). We further demonstrate that our FFT-based method is computationally efficient and can be successfully applied in the context of inventory management (to determine an optimal replenishment policy), ruin theory (to evaluate ruin probabilities and related quantities) and double-barrier option pricing or simply computing non-exit probabilities for Brownian motion with general boundaries.

REFLECTED BROWNIAN MOTION ON SIMPLE NESTED FRACTALS

For a large class of planar simple nested fractals, we prove the existence of the reflected diffusion on a complex of an arbitrary size. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions for the fractal under which this projection can be well defined, and illustrate them by numerous examples. We then construct a proper version of the transition probability densities for the reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman–Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property.

A novel method based on augmented Markov vector process for the time-variant extreme value distribution of stochastic dynamical systems enforced by Poisson white noise

The probability density function (PDF) of the time-variant extreme value process for structural responses is of great importance. Poisson white noise excitation occurs widely in practical engineering problems. The extreme value distribution of the response of systems excited by Poisson white noise processes is still not yet readily available. For this purpose, in the present paper, a novel method based on the augmented Markov vector process for the PDF of the time-variant extreme value process for a Poisson white noise driven dynamical system is proposed. Specifically, the augmented Markov vector (AMV) process is constructed by combining the extreme value process and its underlying response process. Then the joint probability density of the AMV can be evaluated by solving the Chapman-Kolmogorov Equation, e.g., via the path integral solution (PIS). Further, the PDF of the time-variant extreme value process is obtained, and can be used, say, to estimate the dynamic reliability of a stochastic system. For the purpose of illustration and verification, several numerical examples are studied and compared with Monte Carlo solution. Problems to be further studied are also discussed.

Heat Kernel Estimates of Fractional Schr\xf6dinger Operators with Negative Hardy Potential

We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potential Δα/2 − λ|x|−αΔα/2−λ|x|−α\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {\\Delta}^{\\alpha/2} -\\lambda |x|^{-\\alpha} $$\\end{document} on , where α ∈ (0, d ∧ 2) and λ > 0. The proof is purely analytical and elementary. In particular, for upper bounds of heat kernel we use the Chapman-Kolmogorov equation and adopt self-improving argument.

Cost optimization and ANFIS computing for admission control of M/M/1/K queue with general retrial times and discouragement

The present study deals with the admission control policy for the single server finite capacity queueing model with discouraged customers and general distributed retrial times. The arriving customer is forced to join the retrial orbit on finding the server busy. From the orbit, the customer can retry for the service again after sometime. Due to the long queue in front of the server, the customers may be discouraged and leave the system without receiving the service. The noble feature of proposed study is to control the arrivals in the system based on F-policy according to which as soon as the system capacity becomes full, no more customers are permitted to join the system until the system capacity again drops to threshold level ‘F’. The steady state queue size distribution of the system size is evaluated by introducing the supplementary variables corresponding to remaining retrial times and framing Chapman–Kolmogorov equations. The recursive and soft computing based artificial neuro fuzzy inference system (ANFIS) approaches are applied to solve governing equations and to establish various performance indices. Furthermore, a cost function is also constructed to evaluate the optimal service rate and the corresponding expected cost of the system. Genetic algorithm (GA) and quasi-Newton method (QNM) are used to minimize the expected cost of the system to decide the optimal decision parameter corresponding to service rate.

STOCHASTIC EXPERT ESTIMATION OF TECHNICAL CONDITION OF CONSTRUCTION IN AUTOMATED C-STEAM OF CONTROL OF BRIDGES

Purpose. The purpose of the work is to develop a model for assessing the technical condition of a structure as a function of the operation time. The theoretical basis for the development of a stochastic rating model for the technical condition of a structure during operation is the Markov theory of random processes. Phenomenological models of cumulative accumulation of damaged due to the natural degradation of elements during the life cycle of operation are considered. The wear of the structure element is described by a Markov discrete process with continuous time. The Markov process, whose evolution over time depends only on a fixed modern state, has found wide application in the system of operation of road bridges. Degradation of elements during operation is considered as a stream of failures that are physically a manifestation of damage to the elements of a structure under the influence of loads and the environment. The degradation of bridge elements is interpreted as a stationary simplest flow of Poisson type. A mathematical model of a random process with continuous time and discrete states is described by the well-known Kolmogorov-Chapman equations. Methodology. Theoretical study of the processes of degradation of elements of bridges made in the framework of probability theory and mathematical statistics. Findings. The resulting model of expert rating assessment of the technical condition of the facility is based on a time dependent transition matrix. It is proved that the transition matrix developed according to the operating system is the most realistic basis for the prediction of degradation processes. The analysis of foreign studies of works on the method of obtaining a transition matrix was performed. A complete algorithm for calculating the rating assessment of the technical condition of the structure is given. The model algorithm is illustrated by a practical example. Scientific novelty. The study is pioneering. For the first time in the system of operation of road bridges, a stochastic expert assessment of the technical condition of the structure is proposed. Practical value. The resulting model is a practical tool for managing the reliability and resource of road bridges.

Path Integral approach via Laplace’s method of integration for nonstationary response of nonlinear systems

In this paper the nonstationary response of a class of nonlinear systems subject to broad-band stochastic excitations is examined. A version of the Path Integral (PI) approach is developed for determining the evolution of the response probability density function (PDF). Specifically, the PI approach, utilized for evaluating the response PDF in short time steps based on the Chapman–Kolmogorov equation, is here employed in conjunction with the Laplace’s method of integration. In this manner, an approximate analytical solution of the integral involved in this equation is obtained, thus circumventing the repetitive integrations generally required in the conventional numerical implementation of the procedure. Further, the method is extended to nonlinear oscillators, approximately modeling the amplitude of the system response as a one-dimensional Markovian process. Various nonlinear systems are considered in the numerical applications, including Duffing and Van der Pol oscillators. Appropriate comparisons with Monte Carlo simulation data are presented, demonstrating the efficiency and accuracy of the proposed approach.

Stochastic modelling and availability analysis of a critical engineering system

Purpose The purpose of this paper is to present the technique for evaluating the performance of a condensate system of a coal-based thermal power plant situated in the northern part of India. The data which used for system availability evaluation are not precise and are uncertain and, further, collected from concerned power plant history sheets and from discussion through plant personnel. Design/methodology/approach In the proposed model, traditional Markov birth-death process using a probabilistic approach is used to analyze the performance of a complex repairable condensate system of power plant up to a desired degree of accuracy. This approach has been demonstrated by breaking the condensate system into six subsystems arranged in series with two feasible states, namely, working and failed, labeled in a transition diagram and modeled as a Markov process, using Chapman–Kolmogorov equations, which are used for development of a probabilistic stochastic model for availability analysis in a more effecting manner, considering some suitable assumptions. Findings This study of analysis of reliability and availability can help in increasing the plant production and performance. The analysis is done with the help of availability matrices, which are developed using different combinations of failures and repair rates of all subsystems. To achieve the goal of maximum power generation, it is required to run the various subsystem of the concerned system of plant, failure free for a long duration. Therefore, the present approach may be a more powerful analysis tool to access the performance of all subsystems of a condensate system in terms of availability level achieved in availability matrices. The results of present study are found to be highly beneficial to the plant management for making maintenance decisions. Originality/value The present paper suggests a suitable technique for stochastic modeling and availability evaluation of an industrial system using Markovian approach and drawing a transition diagram to represent the operational behavior of the system. The present methodology includes the advantage of the ability to model and develop a more complex industrial system and helps in improving the performance and handling the uncertainties and possibilities of an industrial system.

A 3‐D Multiuser HAP‐MIMO Channel Model Based on Dynamic Evolution of LOS Components

A theoretical Three dimensional (3D) multiuser High altitude platform (HAP) Multipleinput multiple-output (MIMO) channel model based on dynamic evolution of the Line-of-sight (LOS) components is proposed. In this paper, we consider that the LOS components could re-appear after they have vanished, and a two-state Continuous-time Markov chain (CTMC) is used to model the dynamic evolution of the LOS components. The survival probabilities of the LOS components are derived by using Chapman-Kolmogorov (C-K) equations, and the closed-form expressions are derived in this paper. Using the survival probabilities, the spatial correlation functions are derived. Measurements show that a two-state CTMC is indispensable to investigate the dynamic properties of the LOS components.

Markov processes of cubic stochastic matrices: Quadratic stochastic processes

We consider Markov processes of cubic stochastic (in a fixed sense) matrices which are also called quadratic stochastic process (QSPs). A QSP is a particular case of a continuous-time dynamical system whose states are stochastic cubic matrices satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are several kinds of multiplications between cubic matrices we have to fix first a multiplication and then consider the KCE with respect to the fixed multiplication. Moreover, the notion of stochastic cubic matrix also varies depending on the real models of application. The existence of a stochastic (at each time) solution to the KCE provides the existence of a QSP. In this paper, our aim is to construct QSPs for two specially chosen notions of stochastic cubic matrices and two multiplications of such matrices (known as Maksimov's multiplications). We construct a wide class of QSPs and give some time-dependent behavior of such processes. We give an example with applications to the Biology, constructing a QSP which describes the time behavior (dynamics) of a population with the possibility of twin births.

Performance modeling and assessment of maintenance priorities for steam generation unit of a sugar plant

Purpose The purpose of this paper is to deal with the performance modeling and assessment of maintenance priorities for steam generation unit of a sugar plant. Design/methodology/approach The unit comprises of four subsystems, i.e., Bagasse elevator, Bagasse carrier, boiler and feed pump. The Chapman–Kolmogorov equations are generated on the basis of transition diagram and further solved recursively to obtain the performance modeling with the help of normalizing condition using the Markov approach. Findings Decision matrices are formed with the help of different combinations of failure and repair rates of all subsystems. The performance of steam generation unit is evaluated in terms of availability levels depicted in decision matrices and plots of failure rates and repair rates of various subsystems. The maintenance priorities of various subsystems of steam generation unit are decided on the basis of effect of failure and repair rates of subsystems on the availability of steam generation unit. The key finding is that the boiler subsystem is the most critical subsystem and hence should be kept on top maintenance priority for performance enhancement of the steam generation unit. Originality/value The acceptance of both performance modeling and maintenance priorities decision by the management of sugar plant will result in the enhancement of unit availability and reduction of maintenance cost.

Choice Models Adjusted to Non-Available Items and Network Effects

Discrete choice modeling is one of the main tools of estimation utilities and preference probabilities among multiple alternatives in economics, psychology, social sciences, and marketing research. One of popular DCM tools is the Best-Worst Scaling, also known as Maximum Difference. Data for such modeling is given by respondents presented with several items, and each respondent chooses the best alternative. Estimation of utilities is usually performed in a multinomial-logit modeling which produces utilities and choice probabilities. This article describes how to obtain probability estimation adjusted to possible absence of items in actual purchasing. We apply Markov chain modeling in the form of Chapman-Kolmogorov equations and its steady-state solution for stochastic matrix can be obtained analytically. An adjustment to choice probability with network effects is also considered. Numerical example by marketing research data is used.

Chapman-Kolmogorov Equations for a Complete Set of Distinct Reliability States of an Object

The Chapman-Kolmogorov equations indicated in the title are a pretext to demonstrate a mathematically unrecognised truth about the effect of the reliability states of elements (which are generally understood as “subjects”) on the reliability states of a complete set of the same elements, which is called an object. Of importance here are not just the reliability characteristics of individual elements, but the independencies, dependencies and interdependencies between the elements. The relations were described in the language of graph theory. The availability matrix of the language of graph theory was translated to determine the size and probabilities of distinct reliability states of the object, the derivatives of their similarities, and the transition rates adequate to those derivatives. This article continues the research work which identifies the relationship of the properties of a complete set of distinct reliability states of an object with a widely understood theory of systems. The previous papers referred, among others, to: risk, safety, structure entropy, the reliability of the results of checks, and – most of all – technical diagnostics, both in the area of its algorithms and of its optimisation. The object’s serial reliability structure was not assumed in any of those papers, recognising that it would be a serious abuse. The research results were referred to all possible structures of a three-element object. It is believed that by virtue of the block diagrams appropriate to those structures, the readers hereof are provided with a realistic opportunity to practically (and inexpensively) verify the ideas presented here.

A New State Estimation Method with Radar Measurement Missing

This paper describes a method that addresses the transient loss of observations in sea surface target state estimations. A six degrees of freedom swing platform fixed with a MiniRadaScan is used to simulate the loss of observations. The state transition model based on the historical observation data fit prediction is designed because the existing state estimation method can only use the system model prediction while the observation is missing. An observation data sliding window width adaptive adjustment strategy is proposed that can improve the fitting accuracy of the state transition model. To solve the problem where the weight value of the Gaussian components of the Gaussian mixture filter is not changed in the time update stage while the observation is missing, an adaptive adjustment strategy for the weight is proposed based on the Chapman-Kolmogorov equation, which can improve the estimation precision under the conditions of the missing observation. The simulation test demonstrates the proposed accuracy and real-time performance of the proposed algorithm.

Stochastic activation in a genetic switch model

We study a biological autoregulation process, involving a protein that enhances its own transcription, in a parameter region where bistability would be present in the absence of fluctuations. We calculate the rate of fluctuation-induced rare transitions between locally-stable states using a path integral formulation and Master and Chapman-Kolmogorov equations. As in simpler models for rare transitions, the rate has the form of the exponential of a quantity $S_0$ (a "barrier") multiplied by a prefactor $\eta$. We calculate $S_0$ and $\eta$ first in the bursting limit (where the ratio $\gamma$ of the protein and mRNA lifetimes is very large). In this limit, the calculation can be done almost entirely analytically, and the results are in good agreement with simulations. For finite $\gamma$ numerical calculations are generally required. However, $S_0$ can be calculated analytically to first order in $1/\gamma$, and the result agrees well with the full numerical calculation for all $\gamma > 1$. Employing a method used previously on other problems, we find we can account qualitatively for the way the prefactor $\eta$ varies with $\gamma$, but its value is 15-20% higher than that inferred from simulations.

Continuous time random walk with local particle-particle interaction.

The continuous time random walk (CTRW) is often applied to the study of particle motion in disordered media. Yet most such applications do not allow for particle-particle (walker-walker) interaction. In this paper, we consider a CTRW with particle-particle interaction; however, for simplicity, we restrain the interaction to be local. The generalized Chapman-Kolmogorov equation is modified by introducing a perturbation function that fluctuates around 1, which models the effect of interaction. Subsequently, a time-fractional nonlinear advection-diffusion equation is derived from this walking system. Under the initial condition of condensed particles at the origin and the free-boundary condition, we numerically solve this equation with both attractive and repulsive particle-particle interactions. Moreover, a Monte Carlo simulation is devised to verify the results of the above numerical work. The equation and the simulation unanimously predict that this walking system converges to the conventional one in the long-time limit. However, for systems where the free-boundary condition and long-time limit are not simultaneously satisfied, this convergence does not hold.

Performance modeling and maintenance priorities decision for the water flow system of a coal-based thermal power plant

PurposeThe purpose of this paper is to deal with the formation of performance modeling and maintenance priorities for the water flow system (WFS) of a coal-based thermal power plant.Design/methodology/approachThe system consists of five subsystems, i.e. condenser, condensate extraction pump, Low Pressure Heater, deaerator and boiler feed pump. The Chapman-Kolmogorov equations are generated on the basis of transition diagram and further solved recursively to obtain the performance modeling with the help of normalizing condition using Markov approach.FindingsAvailability matrices are formed with the help of different combinations of failures and repair rates of all subsystems. The performance of all subsystems is evaluated in terms of availability level achieved in availability matrices and plots of failure rates and repair rates of various subsystems. The maintenance priorities of various subsystems of WFS are decided on the basis of repair rate.Originality/valueThe adoption of both performance modeling and maintenance priorities decision by the management of thermal power plant will result in the enhancement of system availability and reduction in maintenance cost.

Note on Surjective Polynomial Operators

A linear Markov chain is a discrete time stochastic process whose transitions depend only on the current state of the process. A nonlinear Markov chain is a discrete time stochastic process whose transitions may depend on both the current state and the current distribution of the process. These processes arise naturally in the study of the limit behavior of a large number of weakly interacting Markov processes. The nonlinear Markov processes were introduced by McKean and have been extensively studied in the context of nonlinear Chapman-Kolmogorov equations as well as nonlinear Fokker-Planck equations. The nonlinear Markov chain over a finite state space can be identified by a continuous mapping (a nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex) of the finite state space and by a family of transition matrices depending on occupation probability distributions of states. Particularly, a linear Markov operator is a linear operator associated with a square stochastic matrix. It is well-known that a linear Markov operator is a surjection of the simplex if and only if it is a bijection. The similar problem was open for a nonlinear Markov operator associated with a stochastic hyper-matrix. We solve it in this paper. Namely, we show that a nonlinear Markov operator associated with a stochastic hyper-matrix is a surjection of the simplex if and only if it is a permutation of the Lotka-Volterra operator.

A Semigroup Approach to Fractional Poisson Processes

It is well-known that fractional Poisson processes (FPP) constitute an important example of a non-Markovian structure. That is, the FPP has no Markov semigroup associated via the customary Chapman–Kolmogorov equation. This is physically interpreted as the existence of a memory effect. Here, solving a difference-differential equation, we construct a family of contraction semigroups \((T_\alpha )_{\alpha \in ]0,1]}\), \(T_\alpha =(T_\alpha (t))_{t\ge 0}\). If \(C([0,\infty [,B(X))\) denotes the Banach space of continuous maps from \([0,\infty [\) into the Banach space of endomorphisms of a Banach space X, it holds that \(T_\alpha \in C([0,\infty [,B(X))\) and \(\alpha \mapsto T_\alpha \) is a continuous map from ]0, 1] into \(C([0,\infty [,B(X))\). Moreover, \(T_1\) becomes the Markov semigroup of a Poisson process.

The Markov process admits a consistent steady-state thermodynamic formalism

The search for a unified formulation for describing various non-equilibrium processes is a central task of modern non-equilibrium thermodynamics. In this paper, a novel steady-state thermodynamic formalism was established for general Markov processes described by the Chapman-Kolmogorov equation. Furthermore, corresponding formalisms of steady-state thermodynamics for the master equation and Fokker-Planck equation could be rigorously derived in mathematics. To be concrete, we proved that (1) in the limit of continuous time, the steady-state thermodynamic formalism for the Chapman-Kolmogorov equation fully agrees with that for the master equation; (2) a similar one-to-one correspondence could be established rigorously between the master equation and Fokker-Planck equation in the limit of large system size; (3) when a Markov process is restrained to one-step jump, the steady-state thermodynamic formalism for the Fokker-Planck equation with discrete state variables also goes to that for master equations, as the discretization step gets smaller and smaller. Our analysis indicated that general Markov processes admit a unified and self-consistent non-equilibrium steady-state thermodynamic formalism, regardless of underlying detailed models.