Generalized Euler–Maclaurin Formula and Signatures.

The article focuses on a generalisation of the classical Euler–Maclaurin (EML) formula to Riemann–Stieltjes sums and integrals along rectifiable stochastic processes, whose paths are continuous with bounded variation. Introducing new tensor-valued functionals called flip and sawtooth signatures, the authors establish a deterministic identity relating discrete Riemann–Stieltjes sums to integrals, extending the EML formula to multidimensional Banach space-valued paths. A key contribution is an optimisation procedure that selects integration constants—generalising Bernoulli numbers—by minimising the remainder term in a Hilbert space setting, yielding an explicit generalised EML formula. Additionally, the work connects continuous path signatures with discrete sum signatures of time series, providing interpolation formulae and recovering known results on signatures of piecewise linear paths. This framework offers a novel perspective on the EML formula with potential applications in stochastic analysis and numerical methods.