This book has exerted a continuing appeal since its original publication in 1970. It develops the theory of probability from axioms on the expectation functional rather than on probability measure, demonstrates that the standard theory unrolls more naturally and economically this way, and that applications of real interest can be addressed almost immediately. A secondary aim of the original text was to introduce fresh examples and convincing applications, and that aim is continued in this edition, a general revision plus the addition of chapters giving an economical introduction to dynamic programming, that is then applied to the allocation problems represented by portfolio selection and the multi-armed bandit. The investment theme is continued with a critical investigation of the concept of risk-free'trading and the associated Black-Sholes formula, while another new chapter develops the basic ideas of large deviations. The book may be seen as an introduction to probability for students with a basic mathematical facility, covering the standard material, but different in that it is unified by its theme and covers an unusual range of modern applications.

Zusammenfassung

"This surprising and beautiful introduction to concepts of probability ... chapters have been added which deal with areas of big actual interest ... ." (Peter Imkeller, zbMATH 0980.60004, 2022)

From the reviews of the fourth edition:

"... a clear success in its unorthodoxy, Probability via Expectation has become a treasured classic."
P.A.L. Emrechts in "Short Book Reviews", Vol. 21/1, April, 2001

"Apart from presenting a case for the development of probability theory by using the expectation operator rather than probability measure as the primitive notion, a second distinctive feature of this book is the very large range of modern applications that it covers. Many of these are addressed by more than 350 exercises interspersed throughout the text. In summary, this well written book is a ... introduction to probability theory and its applications." (Norbert Henze, Metrika, November, 2002)

"Originally published in 1970, this book has stood the test of time. ... the text demonstrates a modern alternative approach to a now classical field. ... The fourth edition contains a number of modifications and corrections. New material on dynamic programming, optimal allocation, options pricing and large deviations is included." (Martin T. Wells, Journal of the American Statistical Association, September 2001)

Inhalt
Uncertainty, Intuition and Expectation.- Expectation.- Probability.- Some Basic Models.- Conditioning.- Applications of the Independence Concept.- The Two Basic Limit Theorems.- Continuous Random Variables and Their Transformations.- Markov Processes in Discrete Time.- Markov Processes in Continuous Time.- Action Optimisation: Dynamic Programming.- Optimal Resource Allocation.- Finance: Option Pricing and the Implied Martingale.- Second-Order Theory.- Consistency and Extension: The Finite-Dimensional Case.- Stochastic Convergence.- Martingales.- Extension: Examples of the Infinite-Dimensional Case.- Large-Deviation Theory.- Quantum Mechanics.