· 7 min
Equations driven by noise, and the theorem that they have a unique solution. The strong solution, Gronwall's inequality, uniqueness via the Lipschitz condition and the Ito isometry, and existence by Picard iteration with the factorial decay that makes the iterates converge.
· 10 min
The chain rule of stochastic calculus. The second-order Taylor expansion that the quadratic variation of Brownian motion forces, Ito's formula for a function of a Brownian motion and of an Ito process, the integration-by-parts rule, and the solution of geometric Brownian motion.
· 6 min
Why Brownian motion needs its own calculus. The quadratic variation of a path, the theorem that Brownian motion accumulates quadratic variation equal to elapsed time, the resulting infinite total variation, and the covariation that gives the chain rule its extra term.
· 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.
· 6 min
The canonical mean-reverting diffusion. We solve the Ornstein-Uhlenbeck SDE in closed form, derive its Gaussian transition and stationary laws, read off the half-life, and reduce it to an exact AR(1) recursion for estimation.
· 5 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.
· 5 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.
· 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.