Paper: arXiv 2505.07502

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

Abstract

We introduce the resilience rate as a measure of financial resilience. It captures the expected rate at which a dynamic risk measure recovers, i.e., bounces back, when the risk-acceptance set is breached. We develop the corresponding stochastic calculus by establishing representation theorems for expected time-derivatives of solutions to backward stochastic differential equations (BSDEs) with jumps, evaluated at stopping times. These results reveal that the resilience rate can be represented as a suitable expectation of the generator of a BSDE. We analyze the main properties of the resilience rate and the formal connection of these properties to the BSDE generator. We also introduce resilience-acceptance sets and study their properties in relation to both the resilience rate and the dynamic risk measure. We illustrate our results in several canonical financial examples and highlight their implications via the notion of resilience neutrality.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper establishes a rigorous mathematical foundation for financial resilience using BSDEs with jumps and stopping times, presenting strong theoretical novelty and formal proofs. However, as a purely theoretical contribution, it lacks empirical backtesting or real-world dataset validation, relying instead on canonical theoretical examples.

Research Flowchart

  flowchart TD
    A["Research Goal: Quantify Financial Resilience<br>via Dynamic Risk Measures & BSDEs"] --> B["Develop Stochastic Calculus:<br>Representation Theorems for BSDEs with Jumps"]
    C["Inputs: Multi-Asset Financial Models<br>& Breached Risk-Acceptance Sets"] --> D["Compute Resilience Rate:<br>Expected Time-Derivatives at Stopping Times"]
    B --> D
    D --> E["Analyze BSDE Generator<br>& Resilience-Acceptance Sets"]
    E --> F["Outcomes: Mathematical Framework for<br>Resilience Rate & Resilience Neutrality"]

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1 of 123 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.