Paper: arXiv 2405.09260

Authors: Roger J. A. Laeven, Emanuela Rosazza Gianin, Marco Zullino

Abstract

We introduce and develop the concepts of Geometric Backward Stochastic Differential Equations (GBSDEs, for short) and two-driver BSDEs. We demonstrate their natural suitability for modeling continuous-time dynamic return risk measures. We characterize a broad spectrum of associated, auxiliary ordinary BSDEs with drivers exhibiting growth rates involving terms of the form $y|\ln(y)|+|z|^2/y$. We establish the existence, regularity, uniqueness, and stability of solutions to this rich class of ordinary BSDEs, considering both bounded and unbounded coefficients and terminal conditions. We exploit these results to obtain corresponding results for the original two-driver BSDEs. Finally, we apply our findings within a GBSDE framework for representing the dynamics of return and star-shaped risk measures including (robust) $L^{p}$-norms, and analyze functional properties.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly rigorous theoretical analysis introducing Geometric BSDEs and two-driver BSDEs to model dynamic return risk measures. It solves a difficult open mathematical problem involving driver growth rates with logarithmic non-linearities and singularities, though it remains purely theoretical without empirical backtesting or real-world financial data applications.

Research Flowchart

  flowchart TD
    A[Goal: Model Dynamic Return Risk Measures via Geometric BSDEs] --> B[Inputs: Bounded & Unbounded Coefficients / Terminal Conditions]
    B --> C[Develop Framework: GBSDEs & Two-Driver BSDEs]
    C --> D[Characterize Auxiliary Ordinary BSDEs<br>Driver Growth: y|ln y| + |z|²/y]
    D --> E[Prove Existence, Regularity, Uniqueness & Stability]
    E --> F[Outcomes: Dynamic Representation of Return & Star-Shaped Risk Measures]