Unified signature cumulants and generalized Magnus expansions

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 a 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 motivations ranging from financial mathematics to statistical physics. From an affine semimartingale perspective, the functional relation may be interpreted as a type of generalized Riccati equation.