  In stochastic analysis, a standard method to study a path is to work with its
signature. This is a sequence of tensors of different order that encode
information of the path in a compact form. When the path varies, such
signatures parametrize an algebraic variety in the tensor space. The study of
these signature varieties builds a bridge between algebraic geometry and
stochastics, and allows a fruitful exchange of techniques, ideas, conjectures
and solutions. In this paper we study the signature varieties of two very
different classes of paths. The class of rough paths is a natural extension of
the class of piecewise smooth paths. It plays a central role in stochastics,
and its signature variety is toric. The class of axis-parallel paths has a
peculiar combinatoric flavour, and we prove that it is toric in many cases.
