Theory of stochastic calculus

Topics: The derivative of Brownian motion: white noise. Distributions (generalized functions) and their derivatives, generalized derivative of Brownian motion and its relation to stochastic integrals, the Fourier-sine series of Brownian motion is the Paley-Wiener representation, the Fourier-sine series of the derivative of Brownian motion: white noise.

Some C([0, 1], R)-valued processes. The Brownian bridge, the Ornstein-Uhlenbeck process with a random initial condition, the invariant law of the O-U process, construction of a process u(t, x) that is the solution to the stochastic heat equation on [0, 1] with vanishing Dirichlet boundary conditions.

Reference. R.C. Dalang and M. Sanz-Solé, Stochastic Partial Differential Equations, Space-Time White Noise and Random Fields. Springer Monographs in Mathemacis (2026, to appear). ArXiv: 2402.02119v5

The material presented in this lecture is not part of the exam.