The result is a good reference book that is completely recommendable to anyone who want to enter Stochastic Calculus with the idea of really understanding the foundations. This probably justifies the title of the work. The point of view is clear and rigurous, but avoiding unnecessary machinery. A good bibliography is provided at the end for those who want to go deeper. The final chapter provides solutions and hints to all these exercices. Every chapter includes many examples and exercices that are very useful to put in practise, step by astep, the ideas, notions and properties that are introduced. The book is organized in eleven chapters: starting from the fundamentals of Probability and random variables, going to the definition and main properties of Stochastic Processes, providing the definition and properties of Ito integral and Stochastic Differential Equations, and giving, at the end, a rich collection of applications. From this point of view, senior undergraduate students or graduate students can also benefit from this book. But the spirit seems to be close to people already possessing a mathematical maturity on which one can build with rigour the basic ideas on Stochastic Calculus and its Applications. This does not mean that the book cannot be used as a course manual. The book under review can be an excellent source for these people. As a consequence, many scientists, engineerings and economists, even with a good mathematical background, oftem find themselves in the situation where they have to learn the foundations of this discipline from the very beginning, sometimes struggling with a not very helpful literature. For mathematicians, this book can be used as a first text on stochastic calculus or as a companion to more rigorous texts by a way of examples and exercises.The spectacular development of Stochastic Calculus in the last decades has not been accompanied by an increasing presence in Science courses. The spectacular development of Stochastic. By Marco Castrillon Lopez 18 / Aug / 2017. The book covers models in mathematical finance, biology and engineering. An Informal Introduction to stochastic Calculus with Applications. Using such structure, the text will provide a mathematically literate reader with rapid introduction to the subject and its advanced applications. It contains many solved examples and exercises making it suitable for self study.In the book many of the concepts are introduced through worked-out examples, eventually leading to a complete, rigorous statement of the general result, and either a complete proof, a partial proof or a reference. It is also suitable for researchers to gain working knowledge of the subject. It may be used as a textbook by graduate and advanced undergraduate students in stochastic processes, financial mathematics and engineering. Not everything is proved, but enough proofs are given to make it a mathematically rigorous exposition.This book aims to present the theory of stochastic calculus and its applications to an audience which possesses only a basic knowledge of calculus and probability. In biology, it is applied to populations' models, and in engineering it is applied to filter signal from noise. In finance, the stochastic calculus is applied to pricing options by no arbitrage. These results suce for a rigorous treatment of important applications, such as ltering theory. It also gives its main applications in finance, biology and engineering. stochastic calculus, including its chain rule, the fundamental theorems on the represen-tation of martingales as stochastic integrals and on the equivalent change of probability measure, as well as elements of stochastic dierential equations. This book presents a concise and rigorous treatment of stochastic calculus.
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