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Author: Yi Lu
Publisher:
ISBN:
Category : Markov processes
Languages : en
Pages : 0
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Book Description
Periodic non-homogeneous Poisson processes and Poisson models under Markovian environments are studied in this thesis. By accounting for periodic seasonal variations and random fluctuations in the underlying risk, these models generalize the classical homogeneous Poison risk model. Non-homogenous Poisson processes with periodic claim intensity rates are proposed for the claim counting process of risk theory. We introduce a doubly periodic Poisson model with short and long-term trends. Beta-type intensity functions are presented as illustrations. Doubly periodic Poisson models are appropriate when the seasonality does not repeat the exact same short-term pattern every year, but has a peak intensity that varies over a longer period. This reflects periodic environments like those forming hurricanes, in alternating El Niño/La Niña years. The properties of the model and the statistical inference of the model parameters are discussed. An application of the model to the dataset of Atlantic Hurricanes Affecting the United States (1899-2000) is discussed in detail. Further we introduce a periodic regime-switching Cox risk model by considering both, seasonal variations and stochastic fluctuations in the claims intensity. The intensity process, governed by a periodic function with a random peak level, is proposed. The periodic intensity function follows a deterministic pattern in each short-term period, and is illustrated by a beta-type function. A finite-state Markov chain defines the level process, explaining the random effect due to different underlying risk years. The properties of this regime-switching claim counting process are discussed in detail. By properly defining the Lundberg coefficient; Lundberg-type bounds for finite time ruin probabilities in the two-state risk model case are derived. A detailed derivation of the likelihood function and the maximum likelihood estimates of the model parameters is also given. Statistical applications of the model to the Atlantic hurricanes affecting the United States dataset are discussed under two different level classifications schemes. The Markov-modulated risk model is considered to reflect a risk process or insurance business alternating between a finite number of Poisson models. Here we assume that the claim inter-arrivals, claim severities and premiums of the model are influenced by an external Markovian environment. The effect of this external environment may be characterized, at any time, by a state variable, representing for example, certain types of epidemics, a variety of weather conditions or of different states of the economy.
Author: Yi Lu
Publisher:
ISBN:
Category : Markov processes
Languages : en
Pages : 0
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Book Description
Periodic non-homogeneous Poisson processes and Poisson models under Markovian environments are studied in this thesis. By accounting for periodic seasonal variations and random fluctuations in the underlying risk, these models generalize the classical homogeneous Poison risk model. Non-homogenous Poisson processes with periodic claim intensity rates are proposed for the claim counting process of risk theory. We introduce a doubly periodic Poisson model with short and long-term trends. Beta-type intensity functions are presented as illustrations. Doubly periodic Poisson models are appropriate when the seasonality does not repeat the exact same short-term pattern every year, but has a peak intensity that varies over a longer period. This reflects periodic environments like those forming hurricanes, in alternating El Niño/La Niña years. The properties of the model and the statistical inference of the model parameters are discussed. An application of the model to the dataset of Atlantic Hurricanes Affecting the United States (1899-2000) is discussed in detail. Further we introduce a periodic regime-switching Cox risk model by considering both, seasonal variations and stochastic fluctuations in the claims intensity. The intensity process, governed by a periodic function with a random peak level, is proposed. The periodic intensity function follows a deterministic pattern in each short-term period, and is illustrated by a beta-type function. A finite-state Markov chain defines the level process, explaining the random effect due to different underlying risk years. The properties of this regime-switching claim counting process are discussed in detail. By properly defining the Lundberg coefficient; Lundberg-type bounds for finite time ruin probabilities in the two-state risk model case are derived. A detailed derivation of the likelihood function and the maximum likelihood estimates of the model parameters is also given. Statistical applications of the model to the Atlantic hurricanes affecting the United States dataset are discussed under two different level classifications schemes. The Markov-modulated risk model is considered to reflect a risk process or insurance business alternating between a finite number of Poisson models. Here we assume that the claim inter-arrivals, claim severities and premiums of the model are influenced by an external Markovian environment. The effect of this external environment may be characterized, at any time, by a state variable, representing for example, certain types of epidemics, a variety of weather conditions or of different states of the economy.
Author: Kiyosi Itô
Publisher: Springer
ISBN: 981100272X
Category : Mathematics
Languages : en
Pages : 54
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Book Description
An extension problem (often called a boundary problem) of Markov processes has been studied, particularly in the case of one-dimensional diffusion processes, by W. Feller, K. Itô, and H. P. McKean, among others. In this book, Itô discussed a case of a general Markov process with state space S and a specified point a ∈ S called a boundary. The problem is to obtain all possible recurrent extensions of a given minimal process (i.e., the process on S \ {a} which is absorbed on reaching the boundary a). The study in this lecture is restricted to a simpler case of the boundary a being a discontinuous entrance point, leaving a more general case of a continuous entrance point to future works. He established a one-to-one correspondence between a recurrent extension and a pair of a positive measure k(db) on S \ {a} (called the jumping-in measure and a non-negative number m
Author: Yi Lu
Publisher:
ISBN:
Category : Poisson processes
Languages : en
Pages : 0
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Book Description
Compound non-homogenous Poisson processes with periodic claim intensity rates are studied in this work. A risk process related to a short term periodic environment and the periodicity for its compound claim counting process are discussed. The ruin probabilities of compound non-homogenous Poisson processes with periodic intensity function are also discussed, in which the embedded discrete risk model and the average arrival rate risk model are presented and bounds for the ruin probability of the continuous-time risk model are derived. We introduce a more general Poisson process model with double periodicity. Here the periodic environment does not repeat the exact same pattern every year but varies the short term peak over a relatively long period, with different levels in each year. Illustrations of periodicity for short and long term Poisson models and numerical examples for ruin probabilities are also given.
Author: J. F. C. Kingman
Publisher: Clarendon Press
ISBN: 0191591246
Category : Mathematics
Languages : en
Pages : 118
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Book Description
In the theory of random processes there are two that are fundamental, and occur over and over again, often in surprising ways. There is a real sense in which the deepest results are concerned with their interplay. One, the Bachelier Wiener model of Brownian motion, has been the subject of many books. The other, the Poisson process, seems at first sight humbler and less worthy of study in its own right. Nearly every book mentions it, but most hurry past to more general point processes or Markov chains. This comparative neglect is ill judged, and stems from a lack of perception of the real importance of the Poisson process. This distortion partly comes about from a restriction to one dimension, while the theory becomes more natural in more general context. This book attempts to redress the balance. It records Kingman's fascination with the beauty and wide applicability of Poisson processes in one or more dimensions. The mathematical theory is powerful, and a few key results often produce surprising consequences.
Author: Yury A. Kutoyants
Publisher: Springer Nature
ISBN: 3031370546
Category : Mathematics
Languages : en
Pages : 683
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Book Description
This book covers an extensive class of models involving inhomogeneous Poisson processes and deals with their identification, i.e. the solution of certain estimation or hypothesis testing problems based on the given dataset. These processes are mathematically easy-to-handle and appear in numerous disciplines, including astronomy, biology, ecology, geology, seismology, medicine, physics, statistical mechanics, economics, image processing, forestry, telecommunications, insurance and finance, reliability, queuing theory, wireless networks, and localisation of sources. Beginning with the definitions and properties of some fundamental notions (stochastic integral, likelihood ratio, limit theorems, etc.), the book goes on to analyse a wide class of estimators for regular and singular statistical models. Special attention is paid to problems of change-point type, and in particular cusp-type change-point models, then the focus turns to the asymptotically efficient nonparametric estimation of the mean function, the intensity function, and of some functionals. Traditional hypothesis testing, including some goodness-of-fit tests, is also discussed. The theory is then applied to three classes of problems: misspecification in regularity (MiR),corresponding to situations where the chosen change-point model and that of the real data have different regularity; optical communication with phase and frequency modulation of periodic intensity functions; and localization of a radioactive (Poisson) source on the plane using K detectors. Each chapter concludes with a series of problems, and state-of-the-art references are provided, making the book invaluable to researchers and students working in areas which actively use inhomogeneous Poisson processes.
Author: A. T. Bharucha-Reid
Publisher: Courier Corporation
ISBN: 0486150356
Category : Mathematics
Languages : en
Pages : 485
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Book Description
This graduate-level text and reference in probability, with numerous applications to several fields of science, presents nonmeasure-theoretic introduction to theory of Markov processes. The work also covers mathematical models based on the theory, employed in various applied fields. Prerequisites are a knowledge of elementary probability theory, mathematical statistics, and analysis. Appendixes. Bibliographies. 1960 edition.
Author:
Publisher:
ISBN:
Category : Dissertations, Academic
Languages : en
Pages : 918
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Author: Mogens Bladt
Publisher: Springer
ISBN: 1493970496
Category : Mathematics
Languages : en
Pages : 749
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Book Description
This book contains an in-depth treatment of matrix-exponential (ME) distributions and their sub-class of phase-type (PH) distributions. Loosely speaking, an ME distribution is obtained through replacing the intensity parameter in an exponential distribution by a matrix. The ME distributions can also be identified as the class of non-negative distributions with rational Laplace transforms. If the matrix has the structure of a sub-intensity matrix for a Markov jump process we obtain a PH distribution which allows for nice probabilistic interpretations facilitating the derivation of exact solutions and closed form formulas. The full potential of ME and PH unfolds in their use in stochastic modelling. Several chapters on generic applications, like renewal theory, random walks and regenerative processes, are included together with some specific examples from queueing theory and insurance risk. We emphasize our intention towards applications by including an extensive treatment on statistical methods for PH distributions and related processes that will allow practitioners to calibrate models to real data. Aimed as a textbook for graduate students in applied probability and statistics, the book provides all the necessary background on Poisson processes, Markov chains, jump processes, martingales and re-generative methods. It is our hope that the provided background may encourage researchers and practitioners from other fields, like biology, genetics and medicine, who wish to become acquainted with the matrix-exponential method and its applications.
Author: James R. Kirkwood
Publisher: CRC Press
ISBN: 1482240742
Category : Business & Economics
Languages : en
Pages : 336
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Book Description
Clear, rigorous, and intuitive, Markov Processes provides a bridge from an undergraduate probability course to a course in stochastic processes and also as a reference for those that want to see detailed proofs of the theorems of Markov processes. It contains copious computational examples that motivate and illustrate the theorems. The text is desi
Author: Tomasz Rolski
Publisher:
ISBN:
Category :
Languages : en
Pages : 58
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Book Description
