Financial Resilience Evaluation: From Conditional Expectations to Dynamic Convex Risk Measures

Abstract

Financial resilience concerns the rate at which a position recovers, or further deteriorates, in response to adverse conditions. As a first step, Laeven, Ferrari, Rosazza Gianin, and Zullino (arXiv:2505.07502) introduced the resilience rate, defined as the expected instantaneous rate of (favorable) change of a price or risk-assessment process. Since this quantity captures only the conditional mean of future increments, it cannot distinguish between positions having the same expected recovery but different conditional risk profiles. We obtain a richer characterization by evaluating such increments through a genuine, possibly nonlinear, dynamic risk measure. More precisely, for an Itô process $π$ and a normalized, cash-additive dynamic risk measure $ρ$, we define the resilience evaluation by \[\mathcal D_s^ρπ_t := L^1\text{-}\lim_{\varepsilon\to0^+} \frac{1}{\varepsilon}ρ_s(π_{t+\varepsilon}-π_t), \qquad 0\leq s\leq t<T,\] whenever the limit exists. When $ρ$ is a convex dynamic risk measure induced by a BSDE with a Lipschitz or quadratic driver, we prove that this limit is well-posed and admits an explicit dual representation. It is given by the worst-case conditional expectation, over a zero-penalty class of measure changes, of an effective drift combining the drift of $π$ with the risk adjustment assigned by $ρ$ to its volatility. We further establish attainment of the optimal scenario and illustrate the scope of the construction, as well as the role of the assumptions, through examples and counterexamples.