We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Itô‐vector and Itô‐jet projections. This allows one to systematically develop low‐dimensional approximations to high‐dimensional SDEs using differential geometric techniques. The approach generalizes the notion of projecting a vector field onto a submanifold in order to derive approximations to ordinary differential equations, and improves the previous Stratonovich projection method by adding optimality analysis and results. Indeed, just as in the case of ordinary projection, our definitions of projection are based on optimality arguments and give in a well‐defined sense ‘optimal’ approximations to the original SDE in the mean‐square sense over small times. We also explain how the Stratonovich projection satisfies an optimality criterion that is more ad hoc and less appealing than the criteria satisfied by the Itô projections we introduce. As an application, we consider approximating the solution of the non‐linear filtering problem with a Gaussian distribution. We show how the newly introduced Itô projections lead to optimal approximations in the Gaussian family and briefly discuss the optimal approximation for more general families of distributions. We perform a numerical comparison of our optimally approximated filter with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.

We provide a theory of manifold-valued rough paths of bounded 3 > p-variation, which we do not assume to be geometric. Rough paths are defined in charts, and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of RDEs driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale we recover the theory of Itô integration and SDEs on manifolds [É89]. We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in [CDL15] to the setting of non-geometric rough paths and controlled integrands more general than 1-forms. In the last section we turn to parallel transport and Cartan development: the lack of geometricity leads us to make the choice of a connection on the tangent bundle of the manifold T M, which figures in an Itô correction term in the parallelism RDE; such connection, which is not needed in the geometric/Stratonovich setting, is required to satisfy properties which guarantee well-definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing numerous examples, some accompanied by numerical simulations, which explore the additional subtleties introduced by our change in perspective.

We develop the structure theory for transformations of weakly geometric rough paths of bounded 1 < p-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We derive an explicit combinatorial expression for the rough path lift of a controlled path, and use it to obtain fundamental identities such as the associativity of the rough integral, the adjunction between pushforwards and pullbacks, and a change of variables formula for rough differential equations (RDEs). As applications we define rough paths, rough integration and RDEs on manifolds, extending the results of [ ] to the case of arbitrary p.

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We provide a theory of manifold‐valued rough paths of bounded 3>p$3 > p$‐variation, which we do not assume to be geometric. Rough paths are defined in charts, relying on the vector space‐valued theory of Friz and Hairer (A course on rough paths, 2014), and coordinate‐free (but connection‐dependent) definitions of the rough integral of cotangent bundle‐valued controlled paths, and of rough differential equations driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale, we recover the theory of Itô integration and stochastic differential equations on manifolds (Émery, Stochastic calculus in manifolds, 1989). We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in Cass et al. (Proc. Lond. Math. Soc. (3) 111 (2015) 1471–1518) to the setting of non‐geometric rough paths and controlled integrands more general than 1‐forms. In the last section, we turn to parallel transport and Cartan development: the lack of geometricity leads us to make the choice of a connection on the tangent bundle of the manifold TM$TM$, which figures in an Itô correction term in the parallelism rough differential equation; such connection, which is not needed in the geometric/Stratonovich setting, is required to satisfy properties which guarantee well‐definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing a few examples that explore the additional subtleties introduced by our change in perspective.