· 8 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.
· 6 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.