arXiv subjects

Marco Zullino

Publications and source records attributed to Marco Zullino.

6 recordsLinked to original sources

arXiv2026

Geometric BSDEs

We introduce Geometric Backward Stochastic Differential Equations (GBSDEs) and two-driver BSDEs, which arise naturally in the geometric dynamics of dynamic return risk measures and of recursive portfolio choice. Through a reduction to auxiliary ordinary BSDEs with logarithmic and singular quadratic (LN-Q) growth rate $y|\ln (y)|+|z|^2/y$, we establish existence, regularity, uniqueness and stability of solutions, under both bounded and unbounded driver coefficients and terminal conditions, and we transfer these results to the original two-driver equations. We then deploy the theory in two applications. We solve a portfolio optimization problem under stochastic differential utility, in which the opportunity process satisfies an endogenously derived two-driver BSDE and optimality is established via our two-driver comparison theorem. We further apply GBSDEs to dynamic return and star-shaped risk measures, including (robust) $L^p$-norms, and characterize their positive homogeneity, star-shapedness and multiplicative convexity.

math.PR↗

arXiv2026

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

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.

q-fin.MF↗

arXiv2026

Measuring Financial Resilience Using Backward Stochastic Differential Equations

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.

q-fin.MF↗

arXiv2023

Law-Invariant Return and Star-Shaped Risk Measures

This paper presents novel characterization results for classes of law-invariant star-shaped functionals. We begin by establishing characterizations for positively homogeneous and star-shaped functionals that exhibit second- or convex-order stochastic dominance consistency. Building on these characterizations, we proceed to derive Kusuoka-type representations for these functionals, shedding light on their mathematical structure and intimate connections to Value-at-Risk and Expected Shortfall. Furthermore, we offer representations of general law-invariant star-shaped functionals as robustifications of Value-at-Risk. Notably, our results are versatile, accommodating settings that may, or may not, involve monotonicity and/or cash-additivity. All of these characterizations are developed within a general locally convex topological space of random variables, ensuring the broad applicability of our results in various financial, insurance and probabilistic contexts.

q-fin.RM↗

arXiv2023

Dynamic Return and Star-Shaped Risk Measures via BSDEs

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.

q-fin.RM↗

arXiv2023

Capital allocation for cash-subadditive risk measures: from BSDEs to BSVIEs

In the context of risk measures, the capital allocation problem is widely studied in the literature where different approaches have been developed, also in connection with cooperative game theory and systemic risk. Although static capital allocation rules have been extensively studied in the recent years, only few works deal with dynamic capital allocations and its relation with BSDEs. Moreover, all those works only examine the case of an underneath risk measure satisfying cash-additivity and, moreover, a large part of them focuses on the specific case of the gradient allocation where Gateaux differentiability is assumed. The main goal of this paper is, instead, to study general dynamic capital allocations associated to cash-subadditive risk measures, generalizing the approaches already existing in the literature and motivated by the presence of (ambiguity on) interest rates. Starting from an axiomatic approach, we then focus on the case where the underlying risk measures are induced by BSDEs whose drivers depend also on the y-variable. In this setting, we surprisingly find that the corresponding capital allocation rules solve special kinds of Backward Stochastic Volterra Integral Equations (BSVIEs).

math.PR↗