Paper: arXiv 2611.16087

Authors: Eduardo Abi Jaber, Dorinel Bastide, Ronan Pécheul, Clément Rey

Abstract

Linear expansions in Brownian Stratonovich signatures, which originate in stochastic Taylor expansions, are increasingly used to represent and model processes with memory. Their numerical use requires truncating the signature at a level $M$ and discretising the driving path along $N$ steps. Under a simple growth condition on the coefficients of the linear expansion, we establish strong and weak convergence rates for both approximations, uniformly in time. The strong truncation error decays like $(B^M/(M!)^\varepsilon)^a$ for some constant $B$, uniformly in $N$, where $\varepsilon$ measures the growth of the coefficients and $a$ the regularity of the test function; the weak error does not improve on it. The discretisation error is of order $N^{-a/2}$ in $L^p$, and $N^{-1}$ weakly for polynomial test functions, uniformly in $M$. All these rates are sharp for polynomial test functions, and $M$, $N$ can be chosen independently. We extend the estimates to errors in probability, on distribution functions and on risk measures, and illustrate them on stochastic delay equations, Volterra processes and option pricing in signature volatility models.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, focusing on rigorous mathematical derivations of convergence rates for signature expansions. While it illustrates applications, it lacks empirical backtesting or extensive data validation, placing it firmly in the ‘Lab Rats’ quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Establish Convergence Rates for Truncated & Discretised Signature Expansions] --> B{Key Methodology: Strong & Weak Error Analysis};
    B --> C[Inputs: Linear Expansions in Brownian Stratonovich Signatures];
    C --> D[Computational Processes: Truncation (level M) & Discretisation (N steps)];
    D --> E[Key Findings: Strong & Weak Convergence Rates for Truncation];
    E --> F[Key Findings: Strong & Weak Convergence Rates for Discretisation];
    F --> G[Outcomes: Extended Estimates & Applications (SDEs, Volterra Processes, Option Pricing)];

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5 of 46 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2570 linked papers.