Here’s a comprehensive list of 100 chapter titles for Stochastic Processes from beginner to advanced, covering fundamental concepts, key techniques, and applications in mathematics:
- Introduction to Stochastic Processes: An Overview
- What Makes a Process Stochastic? Key Concepts and Definitions
- Deterministic vs. Stochastic Processes
- Types of Stochastic Processes: A First Glance
- Random Variables: Building Blocks of Stochastic Processes
- Discrete vs. Continuous Stochastic Processes
- Sample Paths and State Spaces of Stochastic Processes
- The Concept of a State in a Stochastic Process
- Stationarity and Ergodicity: The Core Properties of Stochastic Processes
- Markov Chains and Their Importance in Stochastic Modeling
- The Concept of Memoryless Systems: Markov Property
- Introduction to Transition Probabilities and Stochastic Matrices
- The Kolmogorov Forward and Backward Equations
- Basic Operations on Stochastic Processes
- The Notion of Stationary Processes: Definitions and Examples
- Introduction to Discrete-Time Stochastic Processes
- Simple Examples of Discrete-Time Processes
- Markov Chains: Introduction to Discrete States and Transitions
- The Transition Matrix: Properties and Interpretation
- Recurrence and Transience in Markov Chains
- Absorbing Markov Chains and Their Applications
- The Stationary Distribution of a Markov Chain
- Long-Term Behavior and Convergence of Markov Chains
- Classification of States: Recurrent, Transient, and Periodic
- First-Passage Times and Their Importance in Stochastic Modeling
- Absorbing Probabilities and Fundamental Matrices
- Birth-Death Processes in Discrete Time
- Random Walks: A Special Case of Markov Chains
- Gambler’s Ruin Problem and Applications in Random Walks
- Queuing Models and Discrete-Time Markov Processes
- Introduction to Continuous-Time Stochastic Processes
- Poisson Processes: Fundamentals and Applications
- The Exponential Distribution and Its Role in Poisson Processes
- Waiting Times and Inter-Arrival Times in Poisson Processes
- Markov Processes in Continuous Time
- Birth-Death Processes: Continuous-Time Modeling
- Continuous-Time Random Walks and Their Properties
- Continuous Markov Chains: Definition and Analysis
- The Kolmogorov Forward Equation in Continuous Time
- The Poisson Process as a Limit of Discrete-Time Models
- Queueing Theory: Introduction to Continuous-Time Models
- The M/M/1 Queue: Properties and Performance Analysis
- The M/G/1 Queue: General Service Times in Continuous-Time
- Renewal Theory: Continuous-Time Analog of Random Walks
- The Concept of Regenerative Processes in Continuous Time
¶ Part 4: Brownian Motion and Diffusion Processes
- Introduction to Brownian Motion: Historical Development
- Mathematical Definition of Brownian Motion (Wiener Process)
- Properties of Brownian Motion: Continuity and Independence
- The Gaussian Distribution and Its Relation to Brownian Motion
- The Central Limit Theorem and Its Connection to Brownian Motion
- The Stochastic Differential Equation (SDE) for Brownian Motion
- Geometric Brownian Motion: Modeling Stock Prices
- The Itô Calculus: Introduction to Stochastic Integration
- Itô’s Lemma: A Fundamental Result in Stochastic Processes
- The Drift and Volatility in Brownian Motion Models
- The Concept of a Random Walk in Continuous Time
- Applications of Brownian Motion in Physics and Finance
- Stochastic Processes in Biological Modeling: Brownian Motion in Action
- Continuous-Time Random Walks and Their Connection to Diffusion
- Fractional Brownian Motion and Its Applications
¶ Part 5: Stochastic Calculus and Itô’s Lemma
- Introduction to Stochastic Calculus: The Basics
- Itô’s Lemma: Deriving Stochastic Differential Equations
- Stochastic Differential Equations: Solving Basic SDEs
- The Fokker-Planck Equation: Deriving the Probability Distribution
- Stochastic Integration: Concepts and Methods
- Itô vs. Stratonovich Integrals: Differences and Applications
- The Girsanov Theorem: Change of Measure in Stochastic Processes
- Applications of Stochastic Calculus in Finance and Physics
- Solving SDEs Using Numerical Methods: Euler-Maruyama Method
- Applications of Stochastic Calculus in Population Dynamics
- Stochastic Control Theory: Introduction and Overview
- The Hamilton-Jacobi-Bellman Equation in Stochastic Control
- Optimal Stopping Theory: Theory and Applications
- Stochastic Volatility Models in Finance
- The Black-Scholes Model: Derivation Using Stochastic Calculus
- Lévy Processes: Introduction and Properties
- Stable Distributions and Their Role in Stochastic Processes
- Non-Markovian Processes and Their Generalization
- Multidimensional Stochastic Processes: Theory and Applications
- Stochastic Differential Equations with Jumps (Levy Processes)
- Ergodic Theory for Stochastic Processes
- Random Matrices and Stochastic Processes in High Dimensions
- Brownian Motion with Drift: Detailed Analysis
- Stochastic Models of Epidemics and Infectious Diseases
- Random Fields: Applications in Physics and Engineering
- Stochastic Games: An Introduction to Dynamic Decision Making
- Large Deviations Theory: Principles and Applications
- The Perron-Frobenius Theorem and Its Applications in Markov Chains
- Stochastic Process Modelling in Genetics and Population Biology
- Stochastic Networks: Modeling and Applications in Communication Systems
- Applications of Markov Chains in Artificial Intelligence and Machine Learning
- Stochastic Processes in Finance: Modeling Asset Prices
- Queueing Theory in Telecommunications and Network Modeling
- Stochastic Processes in Reliability Engineering: Failure Models
- Stochastic Models in Actuarial Science and Insurance
- The Role of Stochastic Processes in Weather Forecasting
- Stochastic Modeling of Infectious Disease Spread
- Environmental Modeling Using Stochastic Processes
- Stochastic Processes in Control Systems and Robotics
- Emerging Applications of Stochastic Processes in Deep Learning and AI
This list covers the core concepts of stochastic processes from the basics of random variables and Markov chains to advanced techniques like stochastic calculus and applications in various fields such as finance, biology, and engineering. The chapters are designed to gradually build the reader’s knowledge, taking them from introductory material to sophisticated methods and real-world applications.