Paper: arXiv 2504.06351

Authors: Federico M. Bandi, Roberto Renò, Sara Svaluto-Ferro

Abstract

We study the local (in time) expansion of a continuous-time process and its conditional moments, including the process’ characteristic function. The expansions are conducted by using the properties of the (time-extended) Ito signature, a tractable basis composed of iterated integrals of the driving deterministic and stochastic signals: time, multiple correlated Brownian motions and multiple correlated compound Poisson processes. We show that these properties are conducive to automated expansions to any order with explicit coefficients and, therefore, to stochastic representations in which asymptotics can be conducted for a shrinking time (t to 0), as in the extant continuous-time econometrics literature, but, also, for a fixed time (such that t smaller than 1) with a diverging expansion order. The latter design opens up novel opportunities for identifying deep characteristics of the assumed process.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly novel mathematical framework introducing time-extended Itô signatures to continuous-time econometrics for high-order local expansions. It is purely theoretical with dense stochastic calculus, lacking empirical backtesting or real-data applications in the excerpt, placing it squarely in the Lab Rats quadrant.

Research Flowchart

  flowchart TD
    A["Research Goal: Local Expansion of<br>Continuous-Time Processes & Conditional Moments"] --> B["Inputs: Time, Correlated Brownian Motions<br>& Compound Poisson Processes"]
    B --> C["Methodology: Construct Time-Extended<br>Ito Signature Basis"]
    C --> D["Computational Process: Automated Higher-Order<br>Expansions with Explicit Coefficients"]
    D --> E["Asymptotic Regimes: Shrinking Time (t -> 0)<br>& Fixed Time (t < 1) with Diverging Order"]
    E --> F["Outcomes: Exact Characteristic Function &<br>Identification of Deep Process Parameters"]