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Generalized Euler-Maclaurin formula and Signatures

Carlo Bellingeri, Peter K. Friz, Sylvie Paycha

TL;DR

This work extends the Euler–Maclaurin formula to Riemann–Stieltjes integration against rectifiable paths, introducing flip and sawtooth signatures to bridge discrete and continuous signatures. By formulating a deterministic EML expansion and then optimizing the initial datum to minimize the remainder, the authors recover Bernoulli-number structures and derive generalized EML formulas with remainder terms. They also establish an interpolation framework between discrete time-series signatures and continuous path signatures via composition-based and sawtooth constructs, enabling computation of path signatures from time-series data. The results have potential implications for stochastic analysis, cubature on Wiener space, and connections between discrete and rough-path viewpoints on signatures.

Abstract

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, namely, continuous and with 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 shown to be the integration constants in the repeated integration by parts which ``recursively minimise the error'' at every truncation level.

Generalized Euler-Maclaurin formula and Signatures

TL;DR

This work extends the Euler–Maclaurin formula to Riemann–Stieltjes integration against rectifiable paths, introducing flip and sawtooth signatures to bridge discrete and continuous signatures. By formulating a deterministic EML expansion and then optimizing the initial datum to minimize the remainder, the authors recover Bernoulli-number structures and derive generalized EML formulas with remainder terms. They also establish an interpolation framework between discrete time-series signatures and continuous path signatures via composition-based and sawtooth constructs, enabling computation of path signatures from time-series data. The results have potential implications for stochastic analysis, cubature on Wiener space, and connections between discrete and rough-path viewpoints on signatures.

Abstract

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, namely, continuous and with 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 shown to be the integration constants in the repeated integration by parts which ``recursively minimise the error'' at every truncation level.
Paper Structure (5 sections, 13 theorems, 123 equations)

This paper contains 5 sections, 13 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Sawtooth and flip signature of a path
  3. A preliminary deterministic EML formula
  4. Optimal choice of initial datum and EML formula
  5. Interpolating discrete and continuous signature

Key Result

Theorem 1.1

For any rectifiable process $X\colon [0,N]\to V$ with values in a Banach space $V$, there exist two elements $b^{\pm,*}\in T_1((V))$, $b^{\pm,*}=(1, b^{\pm,*}_1, \ldots)$ called optimal backward and forward tensors and two rectifiable processes $Z^{\pm,*}(X)\colon [0, N]\to T_1((V))$, $Z^{\pm, *}(X) for any $f\in C^{m}(V, L(V,W))$ with $m\geq 1$, where the remainder is given by the Stieltjes integ

Theorems & Definitions (43)

  • Theorem 1.1
  • proof
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • proof
  • Definition 2.1
  • Proposition 2.2
  • ...and 33 more