Whittle P. Probability via Expectation

4th Edition. — Springer, 2000. — 369 p. — (Springer Texts in Statistics). — ISBN: 0387989552, 1461267951.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.Uncertainty, Intuition, and Expectation
Axioms for the Expectation Operator
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: 'Risk-Free' Trading and Option Pricing
Second-Order Theory
Consistency and Extension: The Finite-Dimensional Case
Stochastic Convergence
Martingales
Large-Deviation Theory
Extension: Examples ofthe Infinite-Dimensional Case
Quantum Mechanics