Knowledge rather than Hope

A Book for Retail Investors and Mathematical Finance Students

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Teaser: Contents, Preface and Chapters 1 & 2     Cover in pdf     R Code     Errata

ALTERSARMUT VERHINDERN: Eine Praktische Anleitung

Dem Deutschen Volke vom Bekehrten Schwaben

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Initially, there was an ambitious plan to write a comprehensive book on quantitative finance with readable and thorough introduction to necessary advanced mathematics. It turned out to be a little bit too much ambitions.
So I decided to limit myself to the [one of the] most important part: to write a user-friendly introduction to the Lebesgue measure theory. Currently, there are many good books on financial mathematics but I do not know any book on measure theory, which I would recommend for a self-study. So I wrote my own :)

I especially appreciate the following comment, which I received: "very clear introduction to the measure theory, much better than Shreve". This comparison with maestro Shreve is particularly flattering for me, since I learnt by Shreve's textbook and it is really wonderful.

The book does not cover Lebesgue integral, nor even measurable functions (except the Chapter 4, where the main idea of function measurability is illustrated). But the fundamental ideas of the Lebesgue measure are discussed comprehensively. So after you read my notes (with pencil in hand), you will be able to read any book on Real Analysis and will easily understand Lebesgue integral and other advanced topics.

The 1st chapter is a teaser for financiers, here I quickly show that the math is valuable for financial modeling. Those, who are interested in Real Analysis only may skip it but I do recommend to have a look on it. The 2nd chapter is the most important, here I slowly and thoroughly introduce the reader to the measure theory. I start with the measure on (0,1) and then on \mathbb{R}^1 (as Lebesgue himself did). You will see, how semirings and sigma-algebras naturally arise and what advantages the Lebesgue measure has. In the 3rd chapter we discuss the abstract measure theory (now, after chapter 2, your mind is mature for this). As a bonus (and this is my own "pedagogical invention") I demonstrate that a semiring as such, without a sigma-additive measure on it, does not mean much. Additionally, I try to show an intuition behind the interplay of the measure theory and the theory of stochastic processes. Chapter 4 is incomplete but it accentuates the important but frequently omitted idea that measurable functions can do without measures...

The only nice-to-have feature which is, so far, missing is a discussion of singular measures. Singular measures are important for Levy processes. However, if you carefully read about Cantor set, singular measures will not be a problem for you.

So good luck and if you like what already done and I once come back to academia then the book is to be continued.

QuantLib notes