We explain how Itˆo Stochastic Differential Equations on manifolds
may be defined as 2-jets of curves and show how this relationship
can be interpreted in terms of a convergent numerical scheme. We use
jets as a natural language to express geometric properties of SDEs. We
explain that the mainstream choice of Fisk-Stratonovich-McShane calculus
for stochastic differential geometry is not necessary. We give a new
geometric interpretation of the Itˆo–Stratonovich transformation in terms
of the 2-jets of curves induced by consecutive vector flows. We discuss
the forward Kolmogorov equation and the backward diffusion operator
in geometric terms. In the one-dimensional case we consider percentiles
of the solutions of the SDE and their properties. In particular the median
of a SDE solution is associated to the drift of the SDE in Stratonovich
form for small times.