Ebook: Probability via Expectation

The third edition of 1992 constituted a major reworking of the original text, and the preface to that edition still represents my position on the issues that stimulated me first to write. The present edition contains a number of minor modifications and corrections, but its principal innovation is the addition of material on dynamic programming, optimal allocation, option pricing and large deviations. These are substantial topics, but ones into which one can gain an insight with less labour than is generally thought. They all involve the expectation concept in an essential fashion, even the treatment of option pricing, which seems initially to forswear expectation in favour of an arbitrage criterion. I am grateful to readers and to Springer-Verlag for their continuing interest in the approach taken in this work. Peter Whittle Preface to the Third Edition This book is a complete revision of the earlier work Probability which appeared in 1970. While revised so radically and incorporating so much new material as to amount to a new text, it preserves both the aim and the approach of the original. That aim was stated as the provision of a 'first text in probability, demanding a reasonable but not extensive knowledge of mathematics, and taking the reader to what one might describe as a good intermediate level' . In doing so it attempted to break away from stereotyped applications, and consider applications of a more novel and significant character.

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This book has exerted a continuing appeal since publication of its original edition 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 demonstrates that applications of real interest can be addressed almost immediately. Early analysts of games of chance found the question "What is the fair price for entering this game?" quite as natural as "What is the probability of winning it?" Modern probability virtually adopts the former view; present-day treatments of conditioning, weak convergence, generalised processes and, notably, quantum mechanics start explicitly from an expectation characterisation. 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 addition of Chapters 11, 12, 13, and 18. Chapter 11 gives an economical introduction to dynamic programming, applied in Chapter 12 to the allocation problems represented by portfolio selection and the multi-armed bandit. The investment theme is continued in Chapter 13 with a critical investigation of the concept of 'risk-free' trading and the associated Black-Sholes formula. Chapter 18 develops the basic ideas of large deviations, now a standard and invaluable component of theory and tool in applications. The book is 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. For these latter reasons it is of interest to a wide class of readers; probabilists will find the alternative approach of interest, physicists ad engineers will find it

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This book has exerted a continuing appeal since publication of its original edition 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 demonstrates that applications of real interest can be addressed almost immediately. Early analysts of games of chance found the question "What is the fair price for entering this game?" quite as natural as "What is the probability of winning it?" Modern probability virtually adopts the former view; present-day treatments of conditioning, weak convergence, generalised processes and, notably, quantum mechanics start explicitly from an expectation characterisation. 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 addition of Chapters 11, 12, 13, and 18. Chapter 11 gives an economical introduction to dynamic programming, applied in Chapter 12 to the allocation problems represented by portfolio selection and the multi-armed bandit. The investment theme is continued in Chapter 13 with a critical investigation of the concept of 'risk-free' trading and the associated Black-Sholes formula. Chapter 18 develops the basic ideas of large deviations, now a standard and invaluable component of theory and tool in applications. The book is 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. For these latter reasons it is of interest to a wide class of readers; probabilists will find the alternative approach of interest, physicists ad engineers will find it
Content:
Front Matter....Pages i-xxi
Uncertainty, Intuition, and Expectation....Pages 1-12
Expectation....Pages 13-38
Probability....Pages 39-50
Some Basic Models....Pages 51-79
Conditioning....Pages 80-101
Applications of the Independence Concept....Pages 102-120
The Two Basic Limit Theorems....Pages 121-140
Continuous Random Variables and Their Transformations....Pages 141-149
Markov Processes in Discrete Time....Pages 150-181
Markov Processes in Continuous Time....Pages 182-214
Action Optimisation; Dynamic Programming....Pages 215-228
Optimal Resource Allocation....Pages 229-240
Finance: ‘Risk-Free’ Trading and Option Pricing....Pages 241-252
Second-Order Theory....Pages 253-267
Consistency and Extension: The Finite-Dimensional Case....Pages 268-281
Stochastic Convergence....Pages 282-289
Martingales....Pages 290-305
Large-Deviation Theory....Pages 306-316
Extension: Examples of the Infinite-Dimensional Case....Pages 317-328
Quantum Mechanics....Pages 329-339
Back Matter....Pages 341-353

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This book has exerted a continuing appeal since publication of its original edition 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 demonstrates that applications of real interest can be addressed almost immediately. Early analysts of games of chance found the question "What is the fair price for entering this game?" quite as natural as "What is the probability of winning it?" Modern probability virtually adopts the former view; present-day treatments of conditioning, weak convergence, generalised processes and, notably, quantum mechanics start explicitly from an expectation characterisation. 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 addition of Chapters 11, 12, 13, and 18. Chapter 11 gives an economical introduction to dynamic programming, applied in Chapter 12 to the allocation problems represented by portfolio selection and the multi-armed bandit. The investment theme is continued in Chapter 13 with a critical investigation of the concept of 'risk-free' trading and the associated Black-Sholes formula. Chapter 18 develops the basic ideas of large deviations, now a standard and invaluable component of theory and tool in applications. The book is 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. For these latter reasons it is of interest to a wide class of readers; probabilists will find the alternative approach of interest, physicists ad engineers will find it
Content:
Front Matter....Pages i-xxi
Uncertainty, Intuition, and Expectation....Pages 1-12
Expectation....Pages 13-38
Probability....Pages 39-50
Some Basic Models....Pages 51-79
Conditioning....Pages 80-101
Applications of the Independence Concept....Pages 102-120
The Two Basic Limit Theorems....Pages 121-140
Continuous Random Variables and Their Transformations....Pages 141-149
Markov Processes in Discrete Time....Pages 150-181
Markov Processes in Continuous Time....Pages 182-214
Action Optimisation; Dynamic Programming....Pages 215-228
Optimal Resource Allocation....Pages 229-240
Finance: ‘Risk-Free’ Trading and Option Pricing....Pages 241-252
Second-Order Theory....Pages 253-267
Consistency and Extension: The Finite-Dimensional Case....Pages 268-281
Stochastic Convergence....Pages 282-289
Martingales....Pages 290-305
Large-Deviation Theory....Pages 306-316
Extension: Examples of the Infinite-Dimensional Case....Pages 317-328
Quantum Mechanics....Pages 329-339
Back Matter....Pages 341-353
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