Paper: arXiv 2610.35022

Authors: Dimitri Sotnikov, Adeline Viot

Abstract

We establish a martingality criterion for the Exponentially Fading Memory (EFM) signature volatility model, extending the recent result of Abi Jaber, Gassiat, and Sotnikov (2026). In the proposed framework, the volatility process is given by a linear functional of the EFM signature of a time-augmented Brownian motion, which yields a globally well-behaved, fully non-Markovian volatility model admitting a stationary distribution. The stationary EFM signature volatility model is also covered by allowing for a random initial value of the signature. Moreover, the volatility process can be adjusted to the initial forward variance curve. For this model, excluding trivial cases, we prove that the price process is a true martingale if and only if the truncation order is odd and a sign condition involving the correlation parameter holds.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, focusing on advanced mathematical proofs for martingality in a novel volatility model. While the mathematical framework is sophisticated, there is no empirical validation or backtesting presented. The novelty lies in extending the EFM signature volatility model and establishing a new martingality criterion.

Research Flowchart

  flowchart TD
    A[Research Goal: Establish Martingality Criterion for EFM Signature Volatility Model] --> B{Key Methodology: Extend Abi Jaber, Gassiat, Sotnikov (2026) Result};
    B --> C[Model Formulation: Volatility as Linear Functional of EFM Signature of Time-Augmented Brownian Motion];
    C --> D{Model Characteristics: Globally Well-Behaved, Fully Non-Markovian, Stationary Distribution};
    D --> E[Inputs: Truncation Order, Correlation Parameter, Initial Forward Variance Curve];
    E --> F{Computational Proof: Martingality Condition Analysis};
    F --> G[Key Finding: Price Process is True Martingale IFF Truncation Order is Odd AND Specific Sign Condition Holds];