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Abstract
This paper provides the definitions and basic properties related to a discrete state space semi-Markov process. The semi-Markov process is constructed by the so called Markov renewal process that is a special case the two-dimensional Markov sequence. The Markov renewal process is defined by the transition probabilities matrix, called the renewal kernel and an initial distribution or by another characteristics which are equivalent to the renewal kernel. The counting process corresponding to the semi-Markov process allows to determine concept of the process regularity. In the paper are also shown the other methods of determining the semi-Markov process. The presented concepts are illustrated a simple example.

Monounireducible nonhomogeneous semi-
Markov processes are defined and investigated. The mono-
unireducible topological structure is a sufficient condition
that guarantees the absorption of the semi-Markov process
in a state of the process. This situation is of fundamental
importance in the modelling of credit rating migrations because permits the derivation of the distribution function of
the time of default. An application in credit rating modelling is given in order to illustrate the results.

ABSTRACTRecently, artificial neural networks (ANN) have gained attention as a promising modeling tool for building intelligent systems. A number of applications have been reported in areas varying from pattern recognition to bankruptcy prediction. In this paper, we present a creative methodology that integrates computer simulation, semi‐Markov optimization, and ANN techniques for automated knowledge acquisition in real‐time scheduling. The integrated approach focuses on the synergy between operations research and ANN in eliciting human knowledge, filtering inconsistent data, and building competent models capable of performing at the expert level. The new approach includes three main components. First, computer simulation is used to collect expert decisions. This step allows expert knowledge to be obtained in a non‐intrusive way and minimizes the difficulties involved in interviewing experts, constructing repertory grids, or using other similar structures required for manual knowledge acquisition. The data collected from computer simulation are then optimized using a semi‐Markov decision model to remove data redundancies, inconsistencies, and errors. Finally, the optimized data are used to build ANN‐based expert systems. The integrated approach is evaluated by comparing it with the human expert and using ANN alone in the domain of real‐time scheduling. The results indicate that ANN‐based systems perform worse than human experts from whom the data were collected, but the integrated approach outperforms human experts and ANN models alone.

Credit risk is one of the most important contemporary problems for banks and insurance companies. Indeed, for banks, more than forty percent of the equities are necessary to cover this risk. Though this problem is studied by large rating agencies with substantial economic, social and financial tools, building stochastic models is nevertheless necessary to complete this descriptive orientation. This book presents a complete presentation of such a category of models using homogeneous and non-homogeneous semi-Markov processes developed by the authors in several recent papers.

Abstract
A semi-Markov stochastic process is used for solving in a reliability problem in the paper. The problem concerns of two different component cold standby system and a switch. To obtain the reliability characteristic and parameters of the system we construct so called an embedded semi-Markov process in the process describing operation process of the system. In the model the conditional time to failure of the system is represented by a random variable denoting the first passage time from the given state to the specified subset of states. We apply theorems of the Semi-Markov processes theory concerning the conditional reliability functions to calculate the reliability function and mean time to failure of the system. Often an exact reliability function of the system by using Laplace transform is difficult to calculate, frequently impossible. The semi-Markov processes perturbation theory, allows to obtain an approximate reliability function of the system in that case.

In considering life insurance contracts, durational effects may appear at two levels. The first is concerned with the underlying biometrical risk, meaning that dependencies of transition probabilities on the previous duration in a certain state can be observed. Secondly, there is a need for duration-depending actuarial payments. The model presented here, based on semi-Markov processes, allows one to directly model dependencies on the previous duration. Relying on real data, numerical examples dealing with disability insurance as well as German private health insurance outline the impact of using duration-depending transition rates.

In this paper, we present a discrete time regime switching binomial-like model of the term structure where the regime switches are governed by a discrete time semi-Markov process. We model the evolution of the prices of zero-coupon when given an initial term structure as in the model by Ho and Lee that we aim to extend. We discuss and derive conditions for the model to be arbitrage free and relate this to the notion of martingale measure. We explicitly show that due to the extra source of uncertainty coming from the underlying semi-Markov process, there are an infinite number of equivalent martingale measures. The notion of path independence is also studied in some detail, especially in the presence of regime switches. We deal with the market incompleteness by giving an explicit characterization of the minimal entropy martingale measure. We give an application to the pricing of a European bond option both in a Markov and semi-Markov framework. Finally, we conclude.

Adaptive algorithms for state estimation are currently of tremendous interest. Such estimation techniques have particular military usefulness in automatic gunfire control systems. The conventional Kalman filter, developed by Kalman and Bucy, optimally solves the state estimation problem concerning linear systems with Gaussian disturbance and error processes. The maneuvering target tracking problem generally involves nonlinear system properties as well as non-Gaussian disturbance processes. The study presented here explores several solutions. to this problem. An adaptive state estimator centered about the familiar Kalman filter has been developed for applications in three-dimensional maneuvering target tracking. Target maneuvers are modeled in a general manner by a semi-Markov process. The semi-Markov modeling is based on very intuitively appealing assumptions. Specifically, target maneuvers are randomly selected from a range (possibly infinite) of maneuver commands. The selected command is sustained for a random holding time before another command is selected. Dynamics of the selection and holding process may be stationary or time varying. By incorporating the semi-Markov modeling into a Baysian estimation scheme, an adaptive state estimator can be designed to identify the particular maneuver command influencing the target. The algorithm has the distinct advantages of requiring only one Kalman filter and non-growing computer storage requirements. Several techniques of implementing the adaptive algorithm have been developed. The merits of rectangular and spherical modeling have been explored. Most importantly, the planar discrete level semi-Markov algorithm, originally developed for sonar applications, has been extended to a continuum of levels, as well as extended to three-dimensional tracking. The developed algorithms have been fully evaluated by computer simulations. Emphasis has been placed on computational burden as well as overall tracking performance. Results are presented that show.that the developed estimators largely eliminate severe tracking errors that occur when more simplistic target models are incorporated. ; Ph. D.

AbstractFor semi‐Markov decision processes with discounted rewards we derive the well known results regarding the structure of optimal strategies (nonrandomized, stationary Markov strategies) and the standard algorithms (linear programming, policy iteration). Our analysis is completely based on a primal linear programming formulation of the problem.

AbstractThis paper addresses the problem of computing the expected discounted return in finite Markov and semi‐Markov chains. The objective is to reveal insights into two questions. First, which iterative methods hold the most promise? Second, when are interative methods preferred to Gaussian elimination? A set of twenty‐seven randomly generated problems is used to compare the performance of the methods considered. The observations that apply to the problems generated here are as follows: Gauss‐Seidel is not preferred to Pre‐Jacobi in general. However, if the matrix is reordered in a certain way and the author's row sum extrapolation is used, then Gauss‐Seidel is preferred. Transforming a semi‐Markov problem into a Markov one using a transformation that comes from Schweitzer does not yield improved performance. A method analogous to symmetric successive overrelaxation (SSOR) in numerical analysis yields improved performance, especially when the row‐sum extrapolation is used only sparingly. This method is then compared to Gaussian elimination and is found to be superior for most of the problems generated.

Zeitstetige Mehrzustandsmodelle haben seit geraumer Zeit Anwendung in diversen Disziplinen gefunden. Die Markovannahme ist attraktiv, aber häufig nicht angemessen. Semi-Markov-Modelle bilden eine allgemeinere Alternative. Es wird erläutert, wie ein allgemeiner nichtparametrischer Schätzer für Intensitätsfunktionen in den Kontext von Semi-Markov-Modellen eingebettet und dementsprechend genutzt werden kann. Darüber hinaus werden auf diesem Schätzer aufbauend zwei vielseitige Anpassungstests für Intensitätsmodelle entwickelt. Die Methoden werden an einem epidemiologischen Datensatz illustriert.