Glenn Shafer wrote a very thought provoking book on the mathematical theory of evidence many years ago. Vladimir Vovk was a student of Kolmogorov. At first I thought this would be a book that tries to replace the measure theoretic approach to probability with a game theory (gambling) approach and I was skeptical about it being successful. But these authors are serious and first rate probabilists who understand the theory, foundations and history of probability from all the angles.
The authors sell the book as two books. The first half presents probability theory and the classic probability results in the framework of a game between two opponents that they call Skeptic and the World. If we replaced Skeptic by Investor and World by Market we see it naturally as an investment problem in finance. They see their methodology as natural for a course in the probabilist approach to financial problems. The special topics in finance and important models such as Black-Scholes is then the subject of the second part of the book.
On the other hand if you replace Skeptic with Gambler and World with the House we see this as the natural gambling problem that really goes back to the origin of probability and the days of Fermat and Pascal. It is this third aspect of the book that I enjoy the most. In this first part, their approach is mixed in with the history of probability. The famous probabilists and their theories come to life in photographs and sketches. We get the introduction of Martingales as developed by Jean Ville in his 1939 book. The subjective approach of de Finetti, von Mises approach through collectives, Kolmogorov's establishment of mathematical rigor through the measure theoretic approach, Kolmogorov's later development of complexity theory and the various debates on probability by the great 20th century probabilists including J. L. Doob. We are even introduced to lesser knowns such as the UCLA mathematician Tony Martin (picture in the appendix with his theorem)who in 1990 provided an important advanced mathematical result in game theory that is important in these authors' approach. Curiously, we also see a picture of D. R. Cox a famous statistician (but not known for fundamental contributions to probability).
This book is long on historical perspective, philosophy and clear writing. But don't think that it is a probability for dummies book. The probability theory is deep and the mathematics is not necessarily simple. I think the authors find their approach more appealing than the usual measure theoretic approach that requires the formality of the holy triple (sample space, sigma algebra and probability measure).
This book is good for learning the history of probability from the Bernoulli's and DeMoivre and Laplace through to the 21st century. It is also good for learning the fundamentals of stochastic finance. A great book for probablists and statisticians to have.
Bayes is the only major historical character I can think of who is missing from the book and it is a mystery to me that they would leave him out especially since they do spend time covering deFinetti and subjective probability. They also say that their approach borrows from both the subjective and the frequentist schools. Bayesian methods are mentioned only briefly on page 59 where the "neosubjectivist" movement is discussed and we see a picture of Bruno de Finetti (the first one that I have seen).
If you want to see a different approach to finance that relies heavily on the Ito Calculus, Mike Steele's book published by Princeton University Press offers it to you. That book is one of the few references on the subject that is not in the authors' bibliography. The bibliography is nice and fairly extensive.
This book could be used for a graduate level course in probability from a non-traditional framework. It is good also for business school students with a serious interest in probability and the second half of the book could be used for a course in stochastic finance.
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