Strong Solutions and Quantization-Based Numerical Schemes for a Class of Non-Markovian Volatility Models

Paper: arXiv 2503.00243

Authors: Martino Grasselli, Gilles Pagès

Abstract

We investigate a class of non-Markovian processes that hold particular relevance in the realm of mathematical finance. This family encompasses path-dependent volatility models, including those pioneered by [Platen and Rendek, 2018] and, more recently, by [Guyon and Lekeufack, 2023]. Our study unfolds in two principal phases. In the first phase, we introduce a functional quantization scheme based on an extended version of the Lamperti transformation that we propose to handle the presence of a memory term incorporated into the diffusion coefficient. In the second phase, we study the problem of existence and uniqueness of a strong solution for the SDEs related to the examples that motivate our study, in order to provide a theoretical basis to correctly apply the proposed numerical schemes.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents high-level mathematical concepts, extending the Lamperti transform and functional quantization to non-Markovian SDEs while establishing existence and uniqueness proofs for path-dependent volatility models. However, it is primarily a theoretical and numerical methodology paper with minimal empirical backtesting or real-world data application in the provided text.

Research Flowchart

  flowchart TD
    A["Research Goal: Numerical Schemes & Strong Solutions for Non-Markovian Volatility Models"] --> B["Inputs: Path-Dependent Volatility Models<br>(Platen & Rendek, Guyon & Lekeufack)"]
    
    subgraph Methodology & Computation
        B --> C["Phase 1: Extended Lamperti Transformation<br>(Handles Memory Term in Diffusion)"]
        C --> D["Construct Functional Quantization Scheme<br>for Numerical Approximation"]
        B --> E["Phase 2: Theoretical Analysis of SDEs"]
    end

    E --> F["Establish Existence & Uniqueness<br>of Strong Solutions"]
    D --> G["Key Outcomes: Validated Numerical Schemes &<br>Theoretical Basis for Path-Dependent Models"]
    F --> G