This thesis is comprised of six distinct research projects which share the theme of rough and stochastic integration theory.
Chapter 1 deals with the problem of approximating an SDE X in R^d with one Y defined on a specified submanifold, so as to minimise quantities such as E[|Y_t − X_t|^2] for small t: this is seen to be best performed when using Itô instead of Stratonovich calculus.
Chapter 2 develops the theory of not necessarily geometric 3 > p-rough paths on manifolds. Drawing on [FH14, É89, É90] we define controlled rough integration and RDEs both in the local and extrinsic framework, with the latter generalising [CDL15]. Finally, we lay out the theory of parallel transport and Cartan development, for which non-geometricity results in second-order conditions and corrections to the classical formulae.
In Chapter 3 we treat the theory of geometric rough paths of arbitrary roughness in the framework of controlled paths of [Gub04], from an algebraic and combinatorial point of view, and avoiding the smooth approximation arguments used in [FV10b]. As an application, we show how our emphasis on functoriality allows for a simple transposition of the theory to the manifold setting.
The goal of Chapter 4 is to treat the theory of branched rough paths on manifolds. Drawing on [HK15, Kel12], we show how to lift a controlled path to a rough path. The “transfer principle”, intended in the sense of Malliavin and Emery, refers to the expression of a connection-dependent “intrinsic differential” d_∇X that defines integration in a coordinate-invariant manner, which we derive by combining Kelly’s bracket corrections with certain higher-order Christoffel symbols. In reviewing branched rough paths, special attention is given to those that can be defined on Hoffman’s quasi-shuffle algebra [Hof00], for which some of the relations simplify.
The final two chapters do not involve any differential geometry. Chapter 5 is a report on work in progress, the aim of which is to compute the Wiener chaos decomposition (and in particular the expectation) of the signature of certain multidimensional Gaussian processes such as 1/3 < H-fractional Brownian motion (fBm). This generalises the results of [BC07], arrived at through a piecewise-linear approximation argument which fails when 1/4 < H ≤ 1/2. Furthermore, our calculation restricts to that of [Bau04] in the case of Brownian motion, and can be applied to other semimartingales, such as the Brownian bridge. Our novel approach makes use of Malliavin calculus and the recent rough-Skorokhod conversion formula of [CL19].
Finally, in Chapter 6 we combine the topics of the previous two to define a branched rough path above multidimensional 1/4 < H-fBm, and compute its terms and correction terms. Our rough path is defined intrinsically and canonically in terms of the stochastic process, restricts to the Itô rough path when H = 1/2, has the property that its integrals of one-forms vanish in mean, and is not quasi-geometric when H ∈ (1/4, 1/3].