Unified signature cumulants and generalized Magnus expansions

Abstract The signature of a path can be described as its full non-commutative exponential. Following T. Lyons, we regard expectation, the expected , space analogue classical moment generating function. logarithm thereof, taken in tensor algebra, defines cumulant . We establish universal functional relation general semimartingale context. Our work exhibits importance Magnus expansions algorithmic problem computing cumulants and further offers far-reaching generalization recent results on characteristic exponents dubbed diamond with motivations ranging from financial mathematics to statistical physics. From an affine perspective, may interpreted type generalized Riccati equation.