WIAS Preprint No. 2814, (2021)

Unified signature cumulants and generalized Magnus expansions


  • Friz, Peter
  • Hager, Paul
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492

2020 Mathematics Subject Classification

  • 60L10 60L90 60E10


  • Signatures, Lévy processes, Markov processes, stochastic Volterra processes, universal signature relations for semimartingales, moment-cumulant relations, characteristic functions, diamond product




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.

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