Capturing cash non-additivity and horizon risk via BSDEs and generalized shortfall

2026-03-14

AI Summary

This study addresses the limitations of interest rate uncertainty, maturity risk, and the cash-additivity assumption in financial loss risk assessment across multiple time scales by introducing a class of cash-subadditive fully dynamic risk measures. By integrating backward stochastic differential equations (BSDEs) with Lipschitz and quadratic drivers, generalized shortfall functions, and their dual representation theory, the work incorporates cash non-additivity into the shortfall risk framework for the first time, proposing the h-generalized shortfall risk measure and the hq-entropic risk measure. The paper establishes the dual representation of the h-generalized shortfall risk measure and demonstrates that the hq-entropic risk measure, while belonging to this family, lies outside the class of certainty equivalents—thereby overcoming the constraints of classical entropic risk measures and extending the theoretical boundaries of dynamic risk measurement.

Technology Category

Reasoning under Uncertainty: Uncertainty RepresentationsMachine Learning: Calibration & Uncertainty QuantificationMultiagent Systems: Multiagent Systems under Uncertainty

Application Category

Security and Privacy: Large-scale security measurementsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Whenever dealing with horizons of different times scales, risk evaluation of losses may incur in both interest rate uncertainty and horizon risk as introduced in [11]. With the goal to capture both effects, we work with cash subadditive fully-dynamic risk measures. In this work we consider such measures obtained via the BSDE and the shortfall approaches. We stress that we consider BSDEs both with Lipschitz and quadratic drivers. We then introduce the hq-entropic risk measure on losses as an effective example of fully-dynamic risk measure serving the scope. Shortfall risk measures are extended to capture cash non-additivity. For our newly introduced h-generalized shortfall risk measures we provide a dual representation and we connect them to fully-dynamic certainty equivalent. To conclude, we can see that the hq-entropic risk measures on losses belong to the family h-generalized shortfall, but they are not of certainty equivalent type. We note that the classical entropic risk measure, besides being generated by a BSDE, is also both a shortfall and a certainty equivalent.

Problem

• cash non-additivity
• horizon risk
• risk measures
• BSDEs
• shortfall

Innovation

• BSDE
• cash non-additivity
• horizon risk
• generalized shortfall
• fully-dynamic risk measures