Mathematics / advanced
Martingales and conditional structure
Use conditional expectation to study adapted processes, stopping, maximal inequalities, convergence, and change of measure.
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Module 01 advanced
Martingales and conditional structure
Use conditional expectation to study adapted processes, stopping, maximal inequalities, convergence, and change of measure.
- 01
Conditional expectation as projection
Define conditioning by its integral identities, then recover a genuine orthogonal projection when second moments exist.
32 min - 02
Filtrations, adaptedness, and predictability
Track when information becomes available and distinguish a valid pre-increment choice from a rule that sees the next outcome.
30 min - 03
Martingale differences and Doob decomposition
Separate an integrable adapted process into its predictable conditional drift and an unpredictable martingale component.
32 min - 04
Optional sampling and uniform integrability
Identify when stopping preserves conditional means and why an almost-surely finite stopping time alone is insufficient.
35 min - 05
Doob maximal and Lp inequalities
Control the largest observed value using terminal moments, with a careful distinction between weak L1 tails and strong Lp estimates.
34 min - 06
Martingale convergence theorems
Distinguish pathwise convergence, preservation of expectations, uniform integrability, and stronger Lp limits.
34 min - 07
Quadratic variation and martingale brackets
Separate realized squared increments from their predictable conditional means and derive both square-compensation identities.
34 min - 08
Discrete stochastic integrals and isometries
Construct predictable martingale transforms, derive their energy identity, and expose the failure of anticipative weights.
35 min - 09
Martingale central limit theorems
Replace independence by conditional centering while retaining variance stabilization and a conditional small-jump hypothesis.
36 min - 10
Change of measure and density martingales
Construct a new probability law from an integrable density and distinguish finite-time consistency from a terminal absolutely continuous change.
36 min