Multi-dimensional fractional Brownian motion in the G-setting

Paper: arXiv 2312.12139

Authors: Francesca Biagini, Andrea Mazzon, Katharina Oberpriller

Abstract

In this paper we introduce a definition of a multi-dimensional fractional Brownian motion of Hurst index $H \in (0, 1)$ under volatility uncertainty (in short G-fBm). We study the properties of such a process and provide first results about stochastic calculus with respect to a fractional G-Brownian motion for a Hurst index $H >\frac{1}{2}$ .

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework extending fractional Brownian motion to the G-setting to handle volatility uncertainty. It is highly advanced theoretical mathematics with zero empirical validation or backtesting, placing it squarely in the theoretical domain.

Research Flowchart

  flowchart TD
    A[Define Multi-Dimensional Fractional Brownian Motion<br>under Volatility Uncertainty] --> B[Integrate Fractional Brownian Motion H ∈ 0,1<br>into G-Expectation Framework]
    B --> C[Analyze Properties of Multi-Dimensional G-fBm]
    C --> D[Develop Stochastic Calculus for G-fBm<br>with Hurst Index H > 1/2]
    D --> E[Key Outcomes:<br>1. Rigorous G-fBm Definition<br>2. Structural Characterization<br>3. Foundation for Multi-Asset Derivatives]