Stochastic Processes

  1. Martingales
    1. Discrete-Time Martingales
      1. Filtrations
        1. Definition and Properties
          1. Natural Filtration
            1. Adapted Processes
            2. Martingale Definition
              1. Conditional Expectation Property
                1. Integrability Conditions
                2. Submartingales and Supermartingales
                  1. Definitions
                    1. Relationships
                    2. Examples
                      1. Simple Random Walk
                        1. Partial Sums
                          1. Branching Process Martingales
                        2. Martingale Transforms
                          1. Predictable Processes
                            1. Discrete Stochastic Integration
                              1. Preservation of Martingale Property
                              2. Stopping Times
                                1. Definition and Properties
                                  1. Examples of Stopping Times
                                    1. Stopped Processes
                                      1. Optional Sampling
                                      2. Optional Stopping Theorem
                                        1. Statement and Conditions
                                          1. Bounded Stopping Times
                                            1. Applications
                                              1. Gambler's Ruin
                                                1. Random Walks
                                              2. Martingale Convergence
                                                1. Upcrossing Inequality
                                                  1. Martingale Convergence Theorem
                                                    1. L² Convergence
                                                      1. Uniform Integrability
                                                      2. Applications
                                                        1. Likelihood Ratios
                                                          1. Branching Processes
                                                            1. Urn Models

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                                                          8. Brownian Motion