UsefulLinks
Statistics
Stochastic Processes
1. Foundations of Probability Theory
2. Introduction to Stochastic Processes
3. Discrete-Time Markov Chains
4. Poisson Processes
5. Continuous-Time Markov Chains
6. Renewal Theory
7. Martingales
8. Brownian Motion
9. Stochastic Calculus
10. Stationary Processes
11. Applications in Queueing Theory
12. Applications in Finance
13. Applications in Biology and Population Dynamics
14. Applications in Physics and Engineering
7.
Martingales
7.1.
Discrete-Time Martingales
7.1.1.
Filtrations
7.1.1.1.
Definition and Properties
7.1.1.2.
Natural Filtration
7.1.1.3.
Adapted Processes
7.1.2.
Martingale Definition
7.1.2.1.
Conditional Expectation Property
7.1.2.2.
Integrability Conditions
7.1.3.
Submartingales and Supermartingales
7.1.3.1.
Definitions
7.1.3.2.
Relationships
7.1.4.
Examples
7.1.4.1.
Simple Random Walk
7.1.4.2.
Partial Sums
7.1.4.3.
Branching Process Martingales
7.2.
Martingale Transforms
7.2.1.
Predictable Processes
7.2.2.
Discrete Stochastic Integration
7.2.3.
Preservation of Martingale Property
7.3.
Stopping Times
7.3.1.
Definition and Properties
7.3.2.
Examples of Stopping Times
7.3.3.
Stopped Processes
7.3.4.
Optional Sampling
7.4.
Optional Stopping Theorem
7.4.1.
Statement and Conditions
7.4.2.
Bounded Stopping Times
7.4.3.
Applications
7.4.3.1.
Gambler's Ruin
7.4.3.2.
Random Walks
7.5.
Martingale Convergence
7.5.1.
Upcrossing Inequality
7.5.2.
Martingale Convergence Theorem
7.5.3.
L² Convergence
7.5.4.
Uniform Integrability
7.6.
Applications
7.6.1.
Likelihood Ratios
7.6.2.
Branching Processes
7.6.3.
Urn Models
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8. Brownian Motion