UC San Diego

In this dissertation we explore aspects of Itô'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 Itô integrand. Under certain smoothness assumptions we may compute this integrand by the Clark-Ocone formula. We partially bridge this gap between the existence of Itô 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