Stochastic Analysis
A teaching journal through stochastic analysis — conditional expectation to Itô calculus to a first pass at mathematical finance. Written as lectures to a peer, not as compressed personal notes.
Sections
§01Conditional expectation
Conditioning as projection; the defining orthogonality property.
Conditional expectation is the L²-projection of a random variable onto a subspace of information — a σ-algebra. The defining property is orthogonality: the residual is uncorrelated with every random variable measurable with respect to that information. The rest of the topic speaks this language; martingales, Itô integrals, and pricing are all “the right projection” in a larger filtration.