This volume develops probability theory on measure-theoretic foundations and studies stochastic processes.
This volume develops probability theory on measure-theoretic foundations and studies stochastic processes. It emphasizes rigor, structure, and applications.
Part I. Foundations of Probability
Chapter 1. Probability Spaces
1.1 Sigma-algebras and events 1.2 Probability measures 1.3 Examples 1.4 Basic properties 1.5 Interpretations
Chapter 2. Random Variables
2.1 Definitions 2.2 Distributions 2.3 Expectation 2.4 Variance 2.5 Examples
Chapter 3. Independence
3.1 Independent events 3.2 Independent random variables 3.3 Conditional probability 3.4 Conditional expectation 3.5 Examples
Part II. Distributions
Chapter 4. Discrete Distributions
4.1 Bernoulli and binomial 4.2 Poisson distribution 4.3 Geometric distribution 4.4 Applications 4.5 Examples
Chapter 5. Continuous Distributions
5.1 Uniform distribution 5.2 Normal distribution 5.3 Exponential distribution 5.4 Applications 5.5 Examples
Chapter 6. Transform Methods
6.1 Moment generating functions 6.2 Characteristic functions 6.3 Convergence of distributions 6.4 Applications 6.5 Examples
Part III. Limit Theorems
Chapter 7. Law of Large Numbers
7.1 Weak law 7.2 Strong law 7.3 Applications 7.4 Examples 7.5 Extensions
Chapter 8. Central Limit Theorem
8.1 Statement 8.2 Proof ideas 8.3 Applications 8.4 Examples 8.5 Extensions
Chapter 9. Large Deviations (Overview)
9.1 Rare events 9.2 Rate functions 9.3 Applications 9.4 Examples 9.5 Connections
Part IV. Stochastic Processes
Chapter 10. Definitions
10.1 Index sets 10.2 Sample paths 10.3 Finite-dimensional distributions 10.4 Examples 10.5 Applications
Chapter 11. Markov Processes
11.1 Markov property 11.2 Transition probabilities 11.3 Classification of states 11.4 Applications 11.5 Examples
Chapter 12. Martingales
12.1 Definitions 12.2 Martingale convergence 12.3 Optional stopping 12.4 Applications 12.5 Examples
Part V. Continuous-Time Processes
Chapter 13. Brownian Motion
13.1 Construction 13.2 Properties 13.3 Scaling 13.4 Applications 13.5 Examples
Chapter 14. Stochastic Calculus (Overview)
14.1 Itô integral 14.2 Itô formula 14.3 Stochastic differential equations 14.4 Applications 14.5 Examples
Chapter 15. Diffusion Processes
15.1 Generators 15.2 Transition densities 15.3 Applications 15.4 Examples 15.5 Connections
Part VI. Advanced Topics
Chapter 16. Ergodic Theory
16.1 Invariant measures 16.2 Ergodicity 16.3 Applications 16.4 Examples 16.5 Connections
Chapter 17. Random Fields
17.1 Multivariate processes 17.2 Spatial models 17.3 Applications 17.4 Examples 17.5 Connections
Chapter 18. Stochastic Analysis
18.1 Advanced stochastic calculus 18.2 Malliavin calculus (overview) 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Applications
Chapter 19. Statistics
19.1 Estimation 19.2 Hypothesis testing 19.3 Applications 19.4 Examples 19.5 Connections
Chapter 20. Finance
20.1 Option pricing 20.2 Risk models 20.3 Stochastic models 20.4 Applications 20.5 Examples
Chapter 21. Engineering and Science
21.1 Signal processing 21.2 Queueing theory 21.3 Reliability 21.4 Applications 21.5 Examples
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Random matrices 22.2 Interacting particle systems 22.3 Stochastic PDEs 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Dependence structures 23.2 High-dimensional probability 23.3 Computational challenges 23.4 Analytical limits 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of probability theory 24.2 Key contributors 24.3 Evolution of stochastic processes 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Distribution tables B. Convergence theorem summary C. Proof techniques checklist D. Simulation methods E. Cross-reference to other MSC branches