Generalized Euler-Maclaurin formula and Signatures

The Euler-Maclaurin formula which relates a discrete sum with an integral, is generalised to the setting of Riemann-Stieltjes sums and integrals on stochastic processes whose paths are a.s. rectifiable, that is continuous and bounded variation. For this purpose, new variants of the signature are introduced, such as the flip and the sawtooth signature. The counterparts of the Bernoulli numbers that arise in the classical Euler-Maclaurin formula are obtained by choosing the appropriate integration constants in the repeated integration by parts to ``minimise the error'' of every truncation level.