Non-smooth Brownian Martingales and stochastic integral representations

In this dissertation we explore aspects of Ito's formula and the Martingale Representation Theorem with relaxed smoothness assumptions. For an L2 functional of Brownian motion the Martingale Representation Theorem provides the existence of an associated Ito integrand. Under certain smoothness assumptions we may compute this integrand by the Clark-Ocone formula. We partially bridge this gap between the existence of Ito integrands for L2 functionals of Brownian motion, and the smoothness required to explicitly compute them via the Clark-Ocone formula. Then for various examples we reverse the steps in the Clark-Ocone formula in order to obtain stochastic integral representations. We will also examine a class of local martingale functionals and determine the explicit form they must have.