Paper: arXiv 2212.13628

Authors: Bruno Dupire, Valentin Tissot-Daguette

Abstract

Path dependence is omnipresent in many disciplines such as engineering, system theory and finance. It reflects the influence of the past on the future, often expressed through functionals. However, non-Markovian problems are often infinite-dimensional, thus challenging from a conceptual and computational perspective. In this work, we shed light on expansions of functionals. First, we treat static expansions made around paths of fixed length and propose a generalization of the Wiener series$-$the intrinsic value expansion (IVE). In the dynamic case, we revisit the functional Taylor expansion (FTE). The latter connects the functional Itô calculus with the signature to quantify the effect in a functional when a “perturbation” path is concatenated with the source path. In particular, the FTE elegantly separates the functional from future trajectories. The notions of real analyticity and radius of convergence are also extended to the path space. We discuss other dynamic expansions arising from Hilbert projections and the Wiener chaos, and finally show financial applications of the FTE to the pricing and hedging of exotic contingent claims.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a deeply rigorous theoretical framework using functional Itô calculus, roughness signatures, and functional Taylor expansions for non-Markovian systems. While highly innovative and mathematically dense, it primarily focuses on theoretical derivatives pricing and hedging formulations rather than extensive empirical backtesting or data validation.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Address Infinite-Dimensional Challenges in Non-Markovian & Path-Dependent Functionals"] --> B["Input Frameworks:<br>Path Space, Functional Itô Calculus, Rough Paths/Signature & Wiener Chaos"]
    
    B --> C["Develop Static Expansion:<br>Intrinsic Value Expansion (IVE) as a generalization of Wiener Series"]
    B --> D["Develop Dynamic Expansion:<br>Functional Taylor Expansion (FTE) via path concatenation"]
    
    C & D --> E["Mathematical Formulations:<br>Extend Real Analyticity & Radius of Convergence to Path Space"]
    
    E --> F["Financial Applications:<br>Pricing & Hedging of Exotic Contingent Claims"]
    
    F --> G["Key Outcome:<br>Elegantly separates functional operators from future trajectories for efficient computation"]