Paper: arXiv 2604.22528

Authors: Eduardo Abi Jaber, Clément Rey, Dimitri Sotnikov

Abstract

Malliavin calculus is a powerful and general framework for the analysis of square-integrable random variables, but it often suffers from a lack of tractability and explicit representations. To address this limitation, we focus on a subclass of random variables given by finite linear combinations of time-extended Brownian motion signatures. The class remains rich due to the universal approximation properties of signatures. Leveraging the algebraic structure of signatures, we first derive explicit formulas for the Malliavin derivative of signatures of continuous Itô processes. As a consequence, we obtain closed-form expressions for the Clark–Ocone representation, the Ornstein–Uhlenbeck semigroup and its generator, as well as the integration-by-parts formula within the class of Brownian signature variables. These results provide purely algebraic formulations of the classical operators of Malliavin calculus. As an application, we compute Greeks for general path-dependent options under signature volatility models, and numerically compare different choices of Malliavin weights.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly rigorous theoretical framework connecting Malliavin calculus with rough path signatures to yield explicit algebraic representations. While it provides strong mathematical novelty and applications to path-dependent option Greeks, the empirical focus is largely limited to numerical illustrations rather than extensive market backtests.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Achieve explicit, tractable Malliavin calculus for path-dependent variables"] --> B["Input:<br>Time-extended Brownian motion signatures & continuous Itô processes"]
    
    B --> C["Algebraic Derivation:<br>Compute Malliavin derivatives using signature algebraic structure"]
    
    C --> D["Theoretical Formulations:<br>Establish closed-form Clark-Ocone, OU semigroup, & Integration-by-Parts"]
    
    D --> E["Financial Application:<br>Compute Greeks for path-dependent options in Signature Volatility Models"]
    
    E --> F["Numerical Analysis:<br>Compare efficiency & variance of different Malliavin weights"]
    
    F --> G["Key Outcome:<br>Purely algebraic, explicit Malliavin calculus framework for quantitative finance"]