  While backward error analysis does not generalise straightforwardly to the
strong and weak approximation of stochastic differential equations, it extends
for the sampling of ergodic dynamics. The calculation of the modified equation
relies on tedious calculations and there is no expression of the modified
vector field, in opposition to the deterministic setting. We uncover in this
paper the Hopf algebra structures associated to the laws of composition and
substitution of exotic aromatic S-series, relying on the new idea of clumping.
We use these algebraic structures to provide the algebraic foundations of
stochastic numerical analysis with S-series, as well as an explicit expression
of the modified vector field as an exotic aromatic B-series.
