A comprehensive and accessible presentation of probability
and stochastic processes with emphasis on key theoretical concepts
and real-world applications
With a sophisticated approach, Probability and Stochastic
Processes successfully balances theory and applications in a
pedagogical and accessible format. The book's primary focus
is on key theoretical notions in probability to provide a
foundation for understanding concepts and examples related to
stochastic processes.
Organized into two main sections, the book begins by developing
probability theory with topical coverage on probability measure;
random variables; integration theory; product spaces, conditional
distribution, and conditional expectations; and limit theorems. The
second part explores stochastic processes and related concepts
including the Poisson process, renewal processes, Markov chains,
semi-Markov processes, martingales, and Brownian motion. Featuring
a logical combination of traditional and complex theories as well
as practices, Probability and Stochastic Processes also
includes:
* Multiple examples from disciplines such as business,
mathematical finance, and engineering
* Chapter-by-chapter exercises and examples to allow readers to
test their comprehension of the presented material
* A rigorous treatment of all probability and stochastic
processes concepts
An appropriate textbook for probability and stochastic processes
courses at the upper-undergraduate and graduate level in
mathematics, business, and electrical engineering, Probability
and Stochastic Processes is also an ideal reference for
researchers and practitioners in the fields of mathematics,
engineering, and finance.
Autorentext
Ionut Florescu, PhD, is Research Associate Professor of Financial Engineering and Director of the Hanlon Financial Systems Lab at Stevens Institute of Technology. His areas of research interest include stochastic volatility, stochastic partial differential equations, Monte Carlo methods, and numerical methods for stochastic processes. He is also the coauthor of Handbook of Probability and coeditor of Handbook of Modeling High-Frequency Data in Finance, both published by Wiley.
Klappentext
A comprehensive and accessible presentation of probability and stochastic processes with emphasis on key theoretical concepts and real-world applications
With a sophisticated approach, Probability and Stochastic Processes successfully balances theory and applications in a pedagogical and accessible format. The book's primary focus is on key theoretical notions in probability to provide a foundation for understanding concepts and examples related to stochastic processes.
Organized into two main sections, the book begins by developing probability theory with topical coverage on probability measure; random variables; integration theory; product spaces, conditional distribution, and conditional expectations; and limit theorems. The second part explores stochastic processes and related concepts including the Poisson process, renewal processes, Markov chains, semi-Markov processes, martingales, and Brownian motion. Featuring a logical combination of traditional and complex theories as well as practices, Probability and Stochastic Processes also includes:
- Multiple examples from disciplines such as business, mathematical finance, and engineering
- Chapter-by-chapter exercises and examples to allow readers to test their comprehension of the presented material
- A rigorous treatment of all probability and stochastic processes concepts
An appropriate textbook for probability and stochastic processes courses at the upper-undergraduate and graduate level in mathematics, business, and electrical engineering, Probability and Stochastic Processes is also an ideal reference for researchers and practitioners in the fields of mathematics, engineering, and finance.
Ionut Florescu, PhD, is Research Associate Professor of Financial Engineering and Director of the Hanlon Financial Systems Lab at Stevens Institute of Technology. His areas of research interest include stochastic volatility, stochastic partial differential equations, Monte Carlo methods, and numerical methods for stochastic processes. He is also the coauthor of Handbook of Probability and coeditor of Handbook of Modeling High-Frequency Data in Finance, both published by Wiley.
Inhalt
List of Figures xvii
List of Tables xx
Preface xxi
Acknowledgments xxiii
Introduction 1
Part I Probability
1 Elements of Probability Measure 9
1.1 Probability Spaces 10
1.1.1 Null element of . Almost sure (a.s.) statements. Indicator of a set 21
1.2 Conditional Probability 22
1.3 Independence 29
1.4 Monotone Convergence Properties of Probability 31
1.5 Lebesgue Measure on the Unit Interval (0,1] 37
Problems 40
2 Random Variables 45
2.1 Discrete and Continuous Random Variables 48
2.2 Examples of Commonly Encountered Random Variables 52
2.3 Existence of Random Variables with Prescribed Distribution 65
2.4 Independence 68
2.5 Functions of Random Variables. Calculating Distributions 72
Problems 82
3 Applied Chapter: Generating Random Variables 87
3.1 Generating One-Dimensional Random Variables by Inverting the cdf 88
3.2 Generating One-Dimensional Normal Random Variables 91
3.3 Generating Random Variables. Rejection Sampling Method 94
3.4 Generating Random Variables. Importance Sampling 109
Problems 119
4 Integration Theory 123
4.1 Integral of Measurable Functions 124
4.2 Expectations 130
4.3 Moments of a Random Variable. Variance and the Correlation Coefficient 143
4.4 Functions of Random Variables. The Transport Formula 145
4.5 Applications. Exercises in Probability Reasoning 148
4.6 A Basic Central Limit Theorem: The DeMoivreLaplaceTheorem: 150
Problems 152
5 Conditional Distribution and Conditional Expectation 157
5.1 Product Spaces 158
5.2 Conditional Distribution and Expectation. Calculation in Simple Cases 162
5.3 Conditional Expectation. General Definition 165
5.4 Random Vectors. Moments and Distributions 168
Problems 177
6 Moment Generating Function. Characteristic Function 181
6.1 Sums of Random Variables. Convolutions 181
6.2 Generating Functions and Applications 182
6.3 Moment Generating Function 188
6.4 Characteristic Function 192
6.5 Inversion and Continuity Theorems 199
6.6 Stable Distributions. Lvy Distribution 204
6.6.1 Truncated Lévy flight distribution 206
Problems 208
7 Limit Theorems 213
7.1 Types of Convergence 213
7.1.1 Traditional deterministic convergence types 214
7.1.2 Convergence in Lp 215
7.1.3 Almost sure (a.s.) convergence 216
7.1.4 Convergence in probability. Convergence in distribution 217
7.2 Relationships between Types of Convergence 221
7.2.1 A.S. and Lp 221
7.2.2 Probability, a.s., Lp convergence 223
7.2.3 Uniform Integrability 226
7.2.4 Weak convergence and all the others 228
7.3 Continuous Mapping Theorem. Joint Convergence. Slutsky's Theorem 230
7.4 The Two Big Limit Theorems: LLN and CLT 232
7.4.1 A note on statistics 232
7.4.2 The order statistics 234
7.4.3 Limit theorems for the mean statistics 238
7.5 Extension…