Paper: arXiv 2611.16129

Authors: Tahir Choulli, Ella Elazkany

Abstract

This paper considers the setting constituted by the triplet $(S,\mathbb{F},τ)$. The pair $(S,\mathbb{F})$ is the initial model where $S$ is the discounted price process of risky assets (Stocks) and $\mathbb{F}$ is the ``public" flow of information that is available to all agents over time. $τ$ is an arbitrary random time, which might not be seen through the flow $\mathbb{F}$, and it represents a default time in credit risk and a death time of an insured in life insurance where mortality and/or longevity risks present serious challenges. For simplicity, we suppose $S$ follows a geometric jump-diffusion model with stochastic volatiles and $\mathbb{F}$ is generated by a Brownian motion and Poisson process. In this setting, we address how the Esscher densities and/or measures notions can evolves with the informational risk induced by $τ$. In fact, we introduce the notion of Esscher with degrees-of-freedom associated to the informational risks generated by $τ$. On the one hand, we give various and complete characterizations of these Esscher notions in our informational setting. On the other hand, we fully characterize the Escher-prices bounds, for vulnerable claims (claims built on the occurrence of $τ$), using constrained backward stochastic differential equations (CBSDEs for short hereafter) that appear to be novel as well.

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 is highly theoretical, focusing on advanced mathematical concepts like Esscher transforms and CBSDEs in a complex stochastic setting. While it introduces novel theoretical constructs for pricing vulnerable claims under informational risk, it lacks any empirical validation or data-driven backtesting. The methodology is rigorously mathematical but does not provide details for practical implementation or reproducibility of results beyond the theoretical framework.

Research Flowchart

  flowchart TD
    A[Research Goal: Model & Value Financial Risks under Random Horizon τ] --> B{Setting: (S, F, τ)<br>S: Geometric Jump-Diffusion<br>F: Brownian & Poisson Generated Info Flow<br>τ: Unseen Random Time (Default/Death)};
    B --> C[Methodology: Introduce Esscher Measures with Degrees-of-Freedom for Informational Risk];
    C --> D{Characterizations:<br>1. Esscher Notions in Informational Setting<br>2. Esscher-Price Bounds for Vulnerable Claims};
    D --> E[Computational Process: Solved using Constrained Backward Stochastic Differential Equations (CBSDEs)];
    E --> F[Key Findings: Novel Esscher Concepts & Price Bounds for Informational Risks];
    F --> G[Applications: Credit Risk & Life Insurance (Fixed Income/Insurance)];