Measure and integration wereonceconsidered,especially by many ofthe more practically inclined, to be an esoteric area ofabstract m...
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Measure and integration wereonceconsidered,especially by many ofthe more practically inclined, to be an esoteric area ofabstract mathematics best left to pure mathematicians. However,it has become increasingly obvious in recent years that this area is now an indispensable, even unavoidable, language and provides a fundamental methodology for modern probability theory, stochas tic analysis and their applications, especially in financial mathematics. Our aim in writing this book is to provide a smooth and fast introduction to the language and basic results ofmodern probability theory and stochastic differential equations with help ofthe computer manipulator software package MAPLE. It is intended for advanced undergraduate students or graduates, not necessarily in mathematics, to provide an overviewand intuitive background for more advanced studies as wellas somepractical skillsin the use of MAPLE software in the context of probability and its applications. This book is not a conventional mathematics book. Like such books it provides precise definitions and mathematical statements, particularly those based on measure and integration theory, but instead ofmathematical proofs it uses numerous MAPLE experiments and examples to help the reader un derstand intuitively the ideas under discussion. The pace increases from ex tensive and detailed explanations in the first chapters to a more advanced presentation in the latter part of the book. The MAPLE is handled in a sim ilar way, at first with simple commands, then some simple procedures are gardually developed and, finally, the stochastic package is introduced.
An elementary introduction to stochastic calculus with the additional benefit of using the symbolic computation programme MAPLE Includes supplementary material: sn.pub/extras Klappentext The authors provide a fast introduction to probabilistic and statistical concepts necessary to understand the basic ideas and methods of stochastic differential equations. The book is based on measure theory which is introduced as smoothly as possible. It is intended for advanced undergraduate students or graduates, not necessarily in mathematics, providing an overview and intuitive background for more advanced studies as well as some practical skills in the use of MAPLE in the context of probability and its applications. Although this book contains definitions and theorems, it differs from conventional mathematics books in its use of MAPLE worksheets instead of formal proofs to enable the reader to gain an intuitive understanding of the ideas under consideration. As prerequisites the authors assume a familiarity with basic calculus and linear algebra, as well as with elementary ordinary differential equations and, in the final chapter, simple numerical methods for such ODEs. Although statistics is not systematically treated, they introduce statistical concepts such as sampling, estimators, hypothesis testing, confidence intervals, significance levels and p-values and use them in a large number of examples, problems and simulations. Inhalt 1 Probability Basics.- 2 Measure and Integral.- 3 Random Variables and Distributions.- 4 Parameters of Probability Distributions.- 5 A Tour of Important Distributions.- 6 Numerical Simulations and Statistical Inference.- 7 Stochastic Processes.- 8 Stochastic Calculus.- 9 Stochastic Differential Equations.- 10 Numerical Methods for SDEs.- Bibliographical Notes.- References.
From Elementary Probability to Stochastic Differential Equations with MAPLE®