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"Markov processes are processes that have limited memory. In particular, their dependence on the past is only through the previous state. They are used to model the behavior of many systems including communications systems, transportation networks, image segmentation and analysis, biological systems and DNA sequence analysis, random atomic motion and diffusion in physics, social mobility, population studies, epidemiology, animal and insect migration, queueing systems, resource management, dams, financial engineering, actuarial science, and decision systems.Covering a wide range of areas of application of Markov processes, this second edition is revised to highlight the most important aspects as well as the most recent trends and applications of Markov processes. The author spent over 16 years in the industry before returning to academia, and he has applied many of the principles covered in this book in multiple research projects."

"ABSTRAKModel epidemik SIS (Susceptible Infected Susceptible) diaplikasikan dalampembuatan model matematis penyebaran penyakit influenza. Model penyebaranpenyakit flu dibuat dengan pendekatan stokastik. Model stokastik yang digunakandalam skripsi ini adalah model Continuous Time Markov Chain (CTMC). Padamodel CTMC, dikonstruksi probabilitas transisi, ekspektasi, dan limit distribusidari banyaknya individu yang terinfeksi penyakit flu dengan asumsi banyaknyaindividu terinfeksi hanya dapat bertambah satu, berkurang satu atau tetap dalaminterval waktu yang sangat pendek (
t 􀀀 0). Ekspektasi dari banyaknya individuyang terinfeksi flu tidak dapat diselesaikan secara langsung, tetapi dapat diketahuibahwa rata- rata pada model stokastik lebih kecil dibandingkan dengan solusideterministik. Dari kajian tentang limit distribusi, didapatkan bahwa probabilitastidak ada individu terinfeksi adalah satu saat t 􀀀 ª. Simulasi numerik padapenyebaran penyakit flu diberikan sebagai pendukung untuk interpretasi model

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ABSTRACTMathematical model for the spread of influenza using SIS (Susceptible InfectedSusceptible) Epidemic Model for constant total human population size is discussedin this undergraduate thesis. These influenza model was made with stochasticapproach. Stochastic model that used in this thesis is Continuous Time MarkovChain (CTMC). Transition probability, expectation, and limiting distribution forthe number of infected people were constructed in CTMC with assumption that thenumber of infected people might change by increasing one, decreasing one, or stillin the time interval that tends to zero (
t 􀀀 0). The expectation for the number ofinfected people cannot be solved directly, but we will know that the mean of thestochastic SIS epidemic model is less than the deterministic solution. Fromlimiting distribution analyses, probability that there are no infected people att 􀀀 ª is one. Some numerical simulation for the spread of influenza is given togive a better interpretation and a better understanding about the modelinterpretation"

"This self-contained, practical, entry-level text integrates the basic principles of applied mathematics, applied probability, and computational science for a clear presentation of stochastic processes and control for jump diffusions in continuous time. The author covers the important problem of controlling these systems and, through the use of a jump calculus construction, discusses the strong role of discontinuous and nonsmooth properties versus random properties in stochastic systems. The book emphasizes modeling and problem solving and presents sample applications in financial engineering and biomedical modeling. Computational and analytic exercises and examples are included throughout. While classical applied mathematics is used in most of the chapters to set up systematic derivations and essential proofs, the final chapter bridges the gap between the applied and the abstract worlds to give readers an understanding of the more abstract literature on jump diffusions."

"While there are several texts on how to solve and analyze stochastic programs, this is the first text to address basic questions about how to model uncertainty, and how to reformulate a deterministic model so that it can be analyzed in a stochastic setting. This text would be suitable as a stand-alone or supplement for a second course in OR/MS or in optimization-oriented engineering disciplines where the instructor wants to explain where models come from and what the fundamental issues are. "

"[This book is an introduction to the theory of continuous-time stochastic processes. A balance of theory and applications, the work features concrete examples of modeling real-world problems from biology, medicine, finance, and insurance using stochastic methods. This textbook, offers a rigorous and self-contained introduction to the theory of continuous-time stochastic processes, stochastic integrals, and stochastic differential equations. Expertly balancing theory and applications, the work features concrete examples of modeling real-world problems from biology, medicine, industrial applications, finance, and insurance using stochastic methods. Key topics include: Markov processes Stochastic differential equations Arbitrage-free markets and financial derivatives Insurance risk Population dynamics, and epidemics Agent-based models., This book is an introduction to the theory of continuous-time stochastic processes. A balance of theory and applications, the work features concrete examples of modeling real-world problems from biology, medicine, finance, and insurance using stochastic methods. This textbook, offers a rigorous and self-contained introduction to the theory of continuous-time stochastic processes, stochastic integrals, and stochastic differential equations. Expertly balancing theory and applications, the work features concrete examples of modeling real-world problems from biology, medicine, industrial applications, finance, and insurance using stochastic methods. Key topics include: Markov processes Stochastic differential equations Arbitrage-free markets and financial derivatives Insurance risk Population dynamics, and epidemics Agent-based models.]"

"ABSTRAK

Tingkat mortalitas digunakan dalam menghitung besar premi, anuitas pensiun, cadangan asuransi hidup, dan berbagai produk asuransi jiwa lainnya. Untuk itu perlu dilakukan peramalan tingkat mortalitas untuk masa yang akan datang. Kenaikan tingkat mortalitas dipandang sebagai akibat dari proses penuaan manusia yang didasarkan pada suatu indeks kesehatan, yaitu usia fisiologis. Rantai Markov Waktu Kontinu dengan satu absorbing state digunakan untuk memodelkan proses penuaan. Waktu yang dihabiskan sebelum masuk ke dalam absorbing state didefinisikan sebagai waktu bertahan hidup hingga terjadi kematian dan mengikuti Coxian phase type distribution. Fungsi survival dari distribusi yang digunakan dalam peramalan tingkat mortalitas dapat ditentukan. Penaksiran parameter model diperoleh dengan meminimumkan jumlah kuadrat errors dari fungsi survival. Kemudian dilakukan fitting model untuk melihat hasil peramalan tingkat mortalitas untuk data laki-laki dan perempuan. Hasil simulasi menyatakan bahwa model menunjukkan fit yang memuaskan dan dapat digunakan dalam meramalkan tingkat mortalitas usia tua pada data laki-laki dan semua usia pada data perempuan.

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ABSTRACT

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Mortality rates are used in calculating premiums, pension annuities, life insurance reserves, and other life insurance products. Therefore, it is necessary to forecast the mortality rate for the future time. Increasing in mortality rates are seen as a result of the aging process based on a health index called physiological age. Continuous Time Markov Chain with one absorbing state is used to model the aging process. The time spent before entering the absorbing state is defined as the survival time until death occurs and under the Coxian phase type distribution. The survival function can be determined from this distribution and used in forecasting mortality rates. The parameters estimation is obtained by minimizing sum squares of errors from the survival function. Then model fitting are performed to see the result of forecasting mortality rates for man and woman data. Simulation results indicate that the model show satisfactory fit and can be used in forecasting old age mortality for man and all age for woman.

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"This introductory textbook explains how and why probability models are applied to scientific fields such as medicine, biology, physics, oceanography, economics, and psychology to solve problems about stochastic processes. It does not just show how a problem is solved but explains why by formulating questions and first steps in the solutions. Stochastic Processes is ideal for a course aiming to give examples of the wide variety of empirical phenomena for which stochastic processes provide mathematical models. It introduces the methods of probability model building and provides the reader with mathematically sound techniques as well as the ability to further study the theory of stochastic processes. Originally published in 1962, this was the first comprehensive survey of stochastic processes requiring only a minimal background in introductory probability theory and mathematical analysis. Stochastic Processes continues to be unique, with many topics and examples still not discussed in other textbooks. As new fields of applications (such as finance and DNA analysis) become important, researchers will continue to find the fundamental and accessible topics explained in this book essential background for their research."