Stochastic Calculus in Infinite Dimensions

We present a “classical” construction of the Itô stochastic integral, for processes evolving in a Hilbert space. This is introduced first for a one dimensional driving Brownian motion, before generalizations to other one dimensional martingales and, further, to cylindrical Brownian motion. The construction is direct and designed to be familiar to a reader who has undertaken the study of integration with respect to a real valued Brownian motion. In addition, we offer a thorough introduction to martingale theory in Hilbert spaces.