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"Uncertainty and risk are integral to engineering because real systems have inherent ambiguities that arise naturally or due to our inability to model complex physics. The authors discuss probability theory, stochastic processes, estimation, and stochastic control strategies and show how probability can be used to model uncertainty in control and estimation problems. The material is practical and rich in research opportunities.The authors provide a comprehensive treatment of stochastic systems from the foundations of probability to stochastic optimal control. The book covers discrete- and continuous-time stochastic dynamic systems leading to the derivation of the Kalman filter, its properties, and its relation to the frequency domain Wiener filter as well as the dynamic programming derivation of the linear quadratic Gaussian (LQG) and the linear exponential Gaussian (LEG) controllers and their relation to H2 and H controllers and system robustness.Stochastic Processes, Estimation, and Control is divided into three related sections. First, the authors present the concepts of probability theory, random variables, and stochastic processes, which lead to the topics of expectation, conditional expectation, and discrete-time estimation and the Kalman filter. After establishing this foundation, stochastic calculus and continuous-time estimation are introduced. Finally, dynamic programming for both discrete-time and continuous-time systems leads to the solution of optimal stochastic control problems, resulting in controllers with significant practical application."

"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."

"Teori katastrofe menjelaskan bahwa perubahan kecil (smooth change) pada suatu parameter akan menyebabkan rusaknya kestabilan dan menimbulkan perubahan perilaku sistem yang drastis secara tiba-tiba. Dengan menggunakan kalkulus stokastik, fungsi delta Dirac, transformasi Fourier terhadap fungsi karakteristik serta persamaan Fokker-Planck, dapat dijelaskan hubungan antara model katastrofe cusp stokastik dengan suatu fungsi densitas probabilitas (FDP) stasioner. Pada tesis ini ditunjukkan bahwa model katastrofe cusp stokastik dapat digunakan untuk menjelaskan peristiwa krisis pasar saham, yaitu krisis Black Monday pada 19 Oktober 1987 di pasar saham Amerika. Estimasi parameter dengan metode momen menunjukkan bahwa terdapat perubahan nilai diskriminan Cardan dari positif ke negatif, sehingga menunjukkan adanya kasus perubahan FDP dari unimodal ke bimodal. Peristiwa katastrofe pada data Black Monday menunjukkan bahwa krisis ini dipengaruhi oleh faktor internal.

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Catastrophe theory explains that a smooth change of parameters can perturb the system stability to a sudden discontinuous state. Using stochastic calculus, Dirac delta function, Fourier transform of characteristic function, and Fokker-Planck equation we show the connection between stochastic cusp catastrophe model to a stationer probability density function (PDF).

This thesis shows that stochastic cusp catastrophe model can explains U.S stock market crash in October 19, 1987 called Black Monday. Parameter estimations using momen method shows change of Cardan discriminant from positive to negative which explain the stationer PDF in unimodal case to bimodal case. Catastrophe in Black Monday data explains that the crisis influenced by internal factor."