We explain how Itô stochastic differential equations
(SDEs) on manifolds may be defined using 2-jets of
smooth functions. We show how this relationship can
be interpreted in terms of a convergent numerical
scheme. We also show how jets can be used to
derive graphical representations of Itô SDEs, and we
show how jets can be used to derive the differential
operators associated with SDEs in a coordinatefree
manner. We relate jets to vector flows, giving
a geometric interpretation of the Itô–Stratonovich
transformation. We show how percentiles can be used
to give an alternative coordinate-free interpretation of
the coefficients of one-dimensional SDEs. We relate
this to the jet approach. This allows us to interpret
the coefficients of SDEs in terms of ‘fan diagrams’. In
particular, the median of an SDE solution is associated
with the drift of the SDE in Stratonovich form for small
times.