· 5 min
The exact condition for convergence in the mean. Uniform integrability, its characterisation by uniform absolute continuity, and the Vitali theorem that convergence in measure upgrades to L-one convergence exactly when the sequence is uniformly integrable.
· 13 min
The Karhunen-Loeve expansion writes a stochastic process in the eigenbasis of its covariance operator, coordinates that are uncorrelated and mean-square optimal. We prove the expansion and its optimality, then derive it for Brownian motion and the Brownian bridge.
· 31 min
The models built on the order book mechanism. The efficient price is a martingale, the bounce and adverse selection set spreads, queue position sets fills, imbalance sets the microprice, Kyle's lambda turns information into impact, and Almgren-Chriss solves execution.
· 4 min
The density process of an equivalent measure, the Bayes rule for conditional expectation, and the Girsanov theorem that removes a drift. We prove the density is a martingale and the conditional Bayes formula, and state Girsanov with its proof outline.
· 6 min
The Ito integral built from simple predictable integrands by the isometry, extended to the full L^2 class, and shown to be a martingale. We prove the isometry and the extension.
· 7 min
Building the canonical random path. The Levy-Ciesielski construction of Brownian motion as a random series in the Schauder basis, the almost-sure uniform convergence giving continuous paths, and the verification of the defining covariance through Parseval's identity.
· 4 min
The defining fair-game property, optional stopping, and Doob's inequalities. We prove the discrete optional stopping theorem, the maximal and L^p inequalities, and the convergence theorem by upcrossings.
· 5 min
Filtrations, stopping times, and the predictable sigma-algebra that encodes non-anticipation. We prove the basic properties of the stopping-time sigma-algebra and identify predictable processes with the measurable closure of the simple integrands.
· 7 min
Random functions as curves in a Hilbert space. Second-order processes through the geometry of L-squared, mean-square continuity equivalent to a continuous covariance, the mean-square integral, and the covariance operator that the Karhunen-Loeve expansion diagonalises.
· 5 min
The modes of convergence for random variables and the two theorems that govern sample averages. We prove the Markov and Chebyshev inequalities, Borel-Cantelli, a strong law under a fourth moment, and the central limit theorem by characteristic functions.
· 6 min
The distribution stable under linear maps. The Gaussian characteristic function, the Gaussian vector defined by mean and covariance, the equivalence of uncorrelated and independent in the Gaussian case, and the Gaussian process specified by a mean and covariance function.
· 7 min
The Fourier transform of a law. The characteristic function, factorisation over independent sums, the moment expansion, the inversion formula that recovers the law, and the Levy continuity theorem behind the central limit theorem.
· 5 min
Conditional expectation defined by its averaging property, shown to exist via Radon-Nikodym, and identified with orthogonal projection in L^2. We prove the tower property, the pull-out rule, and conditional Jensen.
· 8 min
The factorisation that makes randomness combine. Independence of events, sigma-algebras, and random variables, the equivalence with a product law, the factorisation of expectations through Fubini, the Borel-Cantelli lemmas, and the Kolmogorov zero-one law for tail events.
· 4 min
Absolute continuity, equivalence, and the density that connects two measures.
· 6 min
Probability as measure theory with total mass one. The probability space, random variables and their laws as pushforward measures, expectation as the integral, the change of variables that computes it from the law, and the Markov, Chebyshev, and Jensen inequalities.
· 7 min
A complete construction of statistical arbitrage. We prove arbitrage is measure-class invariant, prove the finite fundamental theorem by separation, then prove an i.i.d. sufficient condition and locate statistical arbitrage in the gap the fundamental theorem leaves open.
· 7 min
When a double integral equals an iterated one. The product sigma-algebra and the measurability of sections, the construction of the product measure, and the Tonelli and Fubini theorems that exchange the order of integration for nonnegative and for integrable functions.
· 8 min
How size is assigned to sets. Sigma-algebras and measures, continuity along monotone limits, closure of measurable functions under pointwise limits, the Caratheodory extension theorem that turns an outer measure into a measure, and the construction of Lebesgue measure.