Paper: arXiv 2307.03447

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

Abstract

This paper establishes characterization results for dynamic return and star-shaped risk measures induced via backward stochastic differential equations (BSDEs). We first characterize a general family of static star-shaped functionals in a locally convex Fréchet lattice. Next, employing the Pasch-Hausdorff envelope, we build a suitable family of convex drivers of BSDEs inducing a corresponding family of dynamic convex risk measures of which the dynamic return and star-shaped risk measures emerge as the essential minimum. Furthermore, we prove that if the set of star-shaped supersolutions of a BSDE is not empty, then there exists, for each terminal condition, at least one convex BSDE with a non-empty set of supersolutions, yielding the minimal star-shaped supersolution. We illustrate our theoretical results in a few examples and demonstrate their usefulness in two applications, to capital allocation and portfolio choice.

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 rigorous theoretical exploration of star-shaped risk measures and dynamic return using BSDEs and advanced functional analysis. While highly novel and mathematically complex, it is purely theoretical with virtually no empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Characterize dynamic return & star-shaped risk measures via BSDEs] --> B[Static Foundations: Characterize star-shaped functionals in Fréchet lattice]
    B --> C[Methodology: Apply Pasch-Hausdorff envelope to construct convex BSDE drivers]
    C --> D[Inputs: Terminal conditions & non-empty set of star-shaped supersolutions]
    D --> E[Computation: Take essential minimum of dynamic convex risk measures]
    E --> F[Outcomes: Minimal star-shaped BSDE supersolutions & applications to capital allocation/portfolio choice]