The Brownian Frame Process as a Rough Path

The Brownian frame process T B is defined as T B t := (Bt−1+u)0≤u≤1 , t ∈ [0, 1] , where B is a real-valued Brownian motion with parameter set [−1, 1]. This thesis investigates properties of the path-valued Brownian frame process relevant to establishing an integration theory based on the theory of rough paths ([Lyons, 1998]). The interest in studying this object comes from its connection with Gaussian Volterra processes (e.g. [Decreusefond, 2005]) and stochastic delay differential equations (e.g. [Mohammed, 1984]). Chapter 2 establishes the existence of T . We then examine the convergence of dyadic polygonal approximations to T B if the path-space V where T B takes its values is equipped with first the p-variation norm (p > 2) and second the sup-norm. In the case of the p-variation norm, the Brownian frame process is shown to have finite ṕ-variation for ṕ > 2p p−2 . In the case of the sup-norm, it is shown to have finite ṕ-variation for ṕ > 2. Chapter 3 provides a tail estimate for the probability that two evaluations of the Brownian frame process are far apart in the p-variation norm. Chapter 4 shows that T B does not have a Lévy area if V ⊗ V is equipped with the injective tensor product norm (where V = C ([0, 1])).