Dupire It\^{o}’s formula for the exponential synchronization of stochastic semi-Markov jump systems with mixed delay under impulsive control

This paper emphasizes the exponential synchronization for a class of stochastic semi-Markov jump systems with mixed delay via stochastic hybrid impulsive control. The impulsive sequence includes synchronous and asynchronous impulses with the impulsive gains being a sequence of stochastic variables. Inspired by the idea of average, a concept of ``average stochastic impulsive gain" is used to qualify the impulse intensity. Our approach expands Dupire functional It\^{o}$"s formula to the stochastic semi-Markov jump systems with mixed delay for the first time. Moreover, in view of the established Lyapunov functional, graph theory, and stochastic analysis theory, some exponential synchronization criteria for the systems are derived. The theoretical results are applied to a class of Chua"s circuit systems with semi-Markov jump and mixed delay. Some synchronization criteria for the circuit systems are provided. The simulation results verify the effectiveness of the theoretical results.     

Sufficient Stochastic Maximum Principle for the Optimal Control of Semi-Markov Modulated Jump-Diffusion with Application to Financial Optimization

The finite state semi-Markov process is a generalization over the Markov chain in which the sojourn time distribution is any general distribution. In this article, we provide a sufficient stochastic maximum principle for the optimal control of a semi-Markov modulated jump-diffusion process in which the drift, diffusion, and the jump kernel of the jump-diffusion process is modulated by a semi-Markov process. We also connect the sufficient stochastic maximum principle with the dynamic programming equation. We apply our results to finite horizon risk-sensitive control portfolio optimization problem and to a quadratic loss minimization problem.     

Stochastic Processes with Age-Dependent Transition Rates

We study stochastic processes with age-dependent transition rates. A typical example of such a process is a semi-Markov process which is completely determined by the holding time distributions in each state and the transition probabilities of the embedded Markov chain. The process we construct generalizes semi-Markov processes. One important feature of this process is that unlike semi-Markov processes the transition probabilities of this process are age-dependent. Under certain condition we establish the Feller property of the process. Finally, we compute the limiting distribution of the process.     

A Non-Homogeneous Continuous Time Semi-Markov Model for the Study of Accumulated Claim Process

The accumulated claim process is the summed total of all claims starting from time t. The semi-Markov environment, at authors’ opinion, is able to follow the evolution of this process. In the paper a continuous time non-homogeneous semi-Markov model with a denumerable set of states will be used to follow the stochastic evolution of the accumulated claim process.     

The semi-Markov beta-Stacy process: a Bayesian non-parametric prior for semi-Markov processes.

The literature on Bayesian methods for the analysis of discrete-time semi-Markov processes is sparse. In this paper, we introduce the semi-Markov beta-Stacy process, a stochastic process useful for the Bayesian non-parametric analysis of semi-Markov processes. The semi-Markov beta-Stacy process is conjugate with respect to data generated by a semi-Markov process, a property which makes it easy to obtain probabilistic forecasts. Its predictive distributions are characterized by a reinforced random walk on a system of urns.

     

Stochastic integral with respect to a semi-Markov process of diffusion type

We consider a multidimensional semi-Markov process of diffusion type. A stochastic integral with respect to the semi-Markov process is defined in terms of asymptotics related to the first exit time from a small neighborhood of the starting point of the process, and, in particular, in terms of its characteristic operator. This integral is equal to the sum of two other integrals: the first one is a curvilinear integral with respect to an additive functional defined in terms of the expected first exit time from a small neighborhood, and the second one is a stochastic integral with respect to a martingale of special kind. To prove the existence and to derive the properties of the integral, both the method of deducing sequences and that of inscribed ellipsoids are used. For Markov processes of diffusion type, the new definition of the stochastic integral is reduced to the standard one. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 251–276.     

Stochastic integral in the case of infinite expected first exit time

We develop a method of analysis of a multidimensional semi-Markov process of diffusion type in the case of infinite expectation of the first exit time from a small neighborhood of the initial point. A generalization of Dynkin’s formula for this case is obtained. Itô’s formula for a stochastic integral over a multidimensional semi-Markov process of diffusion type is derived. Bibliography: 4 titles.     

Stability of stochastic jump-parameter semi-Markov linear systems of differential equations

The asymptotic stability of stochastic Itô-type jump-parameter semi-Markov systems of linear differential equations is examined. A system of integral matrix equations is derived which has the property that the existence of a positive definite solution of the system implies the asymptotic stability of the stochastic semi-Markov system. Finally, an illustrative example is presented.     

A semi-Markov model of disease recurrence in insured dogs

We use a semi-Markov model to analyse the stochastic dynamics of disease occurrence of dogs insured in Canada from 1990 to 1999, and the probability pattern of death from illness. After statistically justifying the use of a stochastic model, we demonstrate that a stationary first-order semi-Markov process is appropriate for analysing the available data set. The probability transition function is estimated and its stationarity is tested statistically. Homogeneity of the semi-Markov model with respect to important covariates (such as geographic location, insurance plan, breed and age) is also statistically examined. We conclude with discussions and implications of our results in veterinary contents. Copyright © 2007 John Wiley & Sons, Ltd.     

Simplicial fixed point algorithms: Volume 129 in: Mathematical Centre Tracts,Mathematisch Centrum,Amsterdam, 1980, iv + 172 pages,Dfl.21.00

This work provides a reliability analysis for power units which are deliberately shutdown when undemanded. A gas turbine is a common example of such a unit. Two appropriate interrelated reliability criteria are mathematically formulated and the relation to their ordinary counterparts is investigated. Using semi-Markov and renewal theory methods these criteria are computed under different assumptions with regard to the stochastic behavior of the demand on the unit and the time to failure and repair time distributions.     

Dual approximation of random evolutions in an averaging scheme

We consider a double approximation of semi-Markov random evolutions, namely, the averaging and diffusion approximation, when the balance condition is not fulfilled. Double approximation algorithms are applicable for reserve and transport processes and other stochastic systems in a semi-Markov random medium.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 400–408, March, 1992.     

Some Reward Paths in Semi-Markov Models with Stochastic Selection of the Transition Probabilities

In the present paper, the reward paths in non homogeneous semi-Markov systems in discrete time are examined with stochastic selection of the transition probabilities. The mean entrance probabilities and the mean rewards in the course of time are evaluated. Then the rate of the total reward for the homogeneous case is examined and the mean total reward is evaluated by means of p.g.f’s.      

Semi-Markov failure rates processes

Usually, a reliability function is defined by a failure rate which is a real function taking the non-negative real values. In this paper the failure rate is assumed to be a stochastic process with non-negative and right continuous trajectories. The reliability function is defined as an expectation of a function of that random process. Particularly, the failure rate defined by the semi-Markov processes is considered here. The theorems dealing with the renewal equations for the conditional reliability functions with a semi-Markov process as a failure rate are presented in this paper. A system of that kind of equations for the discrete state space semi-Markov process is applied for calculating the reliability function for the 3-states semi-Markov random walk. Using the introduced system of renewal equations for the countable state space, the reliability function for the Furry-Yule failure rate process is obtained.     

Full backward non-homogeneous semi-Markov processes for disability insurance models: A Catalunya real data application

In this paper a stochastic model for disability insurance contracts is presented. The model is based on a discrete time non-homogeneous semi-Markov process to which the backward recurrence time process is joined. This permits us to study in a more complete way the disability evolution and to face the duration problem in a more effective way. The model is applied to a sample of contracts drawn at random from a mutual insurance company.     

State estimation for partially observed Markov chains

Recursive equations are derived for the conditional distribution of the state of a Markov chain, given observations of a function of the state. Mainly continuous time chains are considered. The equations for the conditional distribution are given in matrix form and in differential equation form. The conditional distribution itself forms a Markov process. Special cases considered are doubly stochastic Poisson processes with a Markovian intensity, Markov chains with a random time, and Markovian approximations of semi-Markov processes. Further the results are used to compute the Radon-Nikodym derivative for two probability measures for a Markov chain, when a function of the state is observed.     

Mean time to failure and availability of semi-Markov missions with maximal repair

We analyze mean time to failure and availability of semi-Markov missions that consist of phases with random sequence and durations. It is assumed that the system is a complex one with nonidentical components whose failure properties depend on the mission process. The stochastic structure of the mission is described by a Markov renewal process. We characterize mean time to failure and system availability under the maximal repair policy where the whole system is replaced by a brand new after successfully completing a phase before the next phase starts. Special cases involving Markovian missions are also considered to obtain explicit formulas.     

Stochastic Systems with Averaging in the Scheme of Diffusion Approximation

We propose a system approach to the asymptotic analysis of stochastic systems in the scheme of series with averaging and diffusion approximation. Stochastic systems are defined by Markov processes with locally independent increments in a Euclidean space with random switchings that are described by jump Markov and semi-Markov processes. We use the asymptotic analysis of Markov and semi-Markov random evolutions. We construct the diffusion approximation using the asymptotic decomposition of generating operators and solutions of problems of singular perturbation for reducibly inverse operators. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 9, pp. 1235–1252, September, 2005.     

半马氏过程的一维分布及构造

本文求出了半马氏过程跳跃链的转移概率,给出了半马氏过程的逗留时间分布和一维分布,构造了半马氏过程$X(t,\omega)$,最后证明了半马氏过程的两种定义是等价的.     

Optimal control over evolution stochastic systems and its application to stochastic models of financial mathematics

We consider problems of optimal stabilization of controlled evolution stochastic systems in semi-Markov media and their application to financial stochastic models. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 687–698, May, 1998.