Coordinate-free Stochastic Differential Equations as Jets

We explain how It\^o 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 show how jets can be used to derive graphical representations of It\^o SDEs. We show how jets can be used to derive the differential operators associated with SDEs in a coordinate free manner. We relate jets to vector flows, giving a geometric interpretation of the It\^o--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 a SDE solution is associated to the drift of the SDE in Stratonovich form for small times.