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The following link is an abstract of the book by A D Aleksandrov on his view on Mathematics. A D Aleksandrov’s view of Mathematics Which introduce the methods, contents and meaning of mathematics. 


This Analysis book by Steven is a great introduction book covers most concrete analysis areas. By concrete, I mean the contents (Analysis) is not abstract, but fundamental for going deeper to higher analysis. It is also an well organized book with lots of example which clarifies the underlying definition or ideas which build up the analysis. It is a good introduction book for peers who are just get in touch with Analysis or even for graduate who are not confident enough with their prior knowledge on Analysis, it even helpful to refresh what is Analysis really about and what is use of it. I personally like the book very much. 

This book brings reader the fully-fledged type of thinking used by professional mathematicians. It develops the more formal approach as a natural outgrowth of the pattern of underlying ideas, building on a school-mathematics background to develop the viewpoint of an advanced practicing mathematician. It covers the nature of mathematical thinking; a review of the intuitive development of familiar number systems; sets, relations, functions; an introduction to logic as used by practicing mathematicians, methods of proof (including how mathematical proof is written); development of axiomatic number systems from natural numbers and proof by induction to the construction of the real and complex numbers; the real number as a complete ordered field; cardinal numbers; foundations in retrospect.It is indeed an interesting and helpful book for readers in transition from ‘school mathematics’ to be a professional mathematician. 

This book is designed for postgraduate course in stochastic analysis. It require readers familiar with measure-theoretic probability and discrete-time processes, it continues to explore stochastic process in a continuous time set-up. The stochastic process chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. 

The theory of stochastic integration and stochastic calculus is developed in this book.  The power of Stochastic calculus is illustrated by results concerning the martingales representation and change of measure on Wiener space, and these development furthered application in financial economics, such as option pricing and consumption optimization. This book also contains a detailed discussion on weak and strong solutions of SdE and a study of local time for semimartingale, with special emphasis on the theory of Brownian local time. 

May 2018
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Stochastic analysis