Coordinate-free Stochastic Differential Equations as Jets

We explain how Itô Stochastic Differential Equations on manifolds may be defined as 2-jets of curves. We use jets as a natural language to express geometric properties of SDEs and show how jets can lead to intuitive representations of Itô SDEs, including three different types of drawings. We explain that the mainstream choice of Fisk-StratonovichMcShane calculus for stochastic differential geometry is not necessary and elaborate on the relationships with the jets approach. We consider the two calculi as being simply different coordinate systems for the same underlying coordinate-free stochastic differential equation. If the extrinsic approach to differential geometry is adopted, then Stratonovich calculus may appear to be necessary when studying SDEs on submanifolds but in fact one can use the Itô/2-jets framework proposed here by recalling that the curvature of the 2-jet follows the curvature of the manifold. We argue that the choice between Itô and Stratonovich is a modelling choice dictated by the type of problem one is facing and the related desiderata. We also discuss the forward Kolmogorov equation and the backward diffusion operator in geometric terms, and consider percentiles of the solutions of the SDE and their properties, leading to fan diagrams and their relationship with jets. In particular, the median of a SDE solution is associated to the drift of the SDE in Stratonovich form for small times. Finally, we prove convergence of the 2-jet scheme to classical Itô SDEs solutions.