  The signature of a path can be described as its full non-commutative
exponential. Following T. Lyons we regard its expectation, the expected
signature, as path space analogue of the classical moment generating function.
The logarithm thereof, taken in the tensor algebra, defines the signature
cumulant. We establish a universal functional relation in a general
semimartingale context. Our work exhibits the importance of Magnus expansions
in the algorithmic problem of computing expected signature cumulants, and
further offers a far-reaching generalization of recent results on
characteristic exponents dubbed diamond and cumulant expansions; with
motivation ranging from financial mathematics to statistical physics. From an
affine process perspective, the functional relation may be interpreted as
infinite-dimensional, non-commutative ("Hausdorff") variation of Riccati's
equation. Many examples are given.
