arXiv subjects

Emanuela Rosazza Gianin

Publications and source records attributed to Emanuela Rosazza Gianin.

At least 19 recordsLinked to original sources

arXiv2026

Risk Measures under Paired-Ambiguity: A Deep Learning Reflected BSDE Framework

We study optimal stopping under dynamic risk measures with simultaneous ambiguity in the probability model and the discount rate. We introduce a paired ambiguity framework combining Girsanov model uncertainty with cash subadditive risk evaluation and characterize the stopping value by an upper reflected backward stochastic differential equation (BSDE). We establish structural properties of the resulting stopping operator and study quadratic drivers associated with entropic risk measures, obtaining explicit stopping rules in several benchmark cases. We then develop a deep learning scheme for the reflected quadratic BSDE. The convergence analysis uses discrete reflection and truncation to reduce the quadratic problem to a globally Lipschitz system and combines reflected BSDE discretization estimates with neural network approximation errors. Numerical experiments for American options illustrate the effects of discount rate and entropic ambiguity on stopping values and exercise decisions.

q-fin.RM↗

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

Robust quasi-convex risk measures and applications

This paper develops a unified framework for the robustification of risk measures beyond the classical convex and cash-additive setting. We consider general risk measures on Lp spaces and construct their robust counterparts through families of uncertainty sets that capture ambiguity. Two complementary mechanisms generate robust quasi-convex measures: in the first, quasi-convexity is inherited from the initial risk measure under convex uncertainty sets; in the second it comes from the quasi-convex (or c-quasi-convex) structure of the uncertainty sets themselves. Building on Cerreia-Vioglio et al. (2011); Frittelli and Maggis (2011), we derive dual (penalty-type) representations for robust quasi-convex and cash-subadditive risk measures, showing that the classical convex cash-additive case arises as a special instance. We further analyze acceptance families and capital allocation rules under robustification, highlighting how ambiguity affects acceptability and the distribution of capital.

q-fin.RM↗

arXiv2026

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

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.

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↗

arXiv2024

SIG-BSDE for Dynamic Risk Measures

In this paper, we consider dynamic risk measures induced by backward stochastic differential equations (BSDEs). We discuss different examples that come up in the literature, including the entropic risk measure and the risk measure arising from the ambiguous interest rate problem. We develop a numerical algorithm for solving a BSDE using the backward Euler-Maruyama scheme and the universal approximation theorem for the signature of a path. We prove the convergence theorem and use the algorithm to solve some examples of dynamic risk measures induced by BSDEs. At last a deep learning approach is included for solving the ambiguous interest rate problem as well.

math.PR↗

arXiv2024

Cash non-additive risk measures: horizon risk and generalized entropy

Horizon risk (see arXiv:2301.04971) is studied in the context of cash non-additive fully-dynamic risk measures induced by BSDEs. Furthermore, we introduce a risk measure based on generalized Tsallis entropy which can dynamically evaluate the riskiness of losses considering both horizon risk and interest rate uncertainty. The new q-entropic risk measure on losses can be used as a quantification of capital requirement.

q-fin.RM↗

arXiv2023

Fully-dynamic risk measures: horizon risk, time-consistency, and relations with BSDEs and BSVIEs

In a dynamic framework, we identify a new concept associated with the risk of assessing the financial exposure by a measure that is not adequate to the actual time horizon of the position. This will be called horizon risk. We clarify that dynamic risk measures are subject to horizon risk, so we propose to use the fully-dynamic version. To quantify horizon risk, we introduce h-longevity as an indicator. We investigate these notions together with other properties of risk measures as normalization, restriction property, and different formulations of time-consistency. We also consider these concepts for fully-dynamic risk measures generated by backward stochastic differential equations (BSDEs), backward stochastic Volterra integral equations (BSVIEs), and families of these. Within this study, we provide new results for BSVIEs such as a converse comparison theorem and the dual representation of the associated risk measures.

math.PR↗

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

Are Shortfall Systemic Risk Measures One Dimensional?

Shortfall systemic (multivariate) risk measures $ρ$ defined through an $N$-dimensional multivariate utility function $U$ and random allocations can be represented as classical (one dimensional) shortfall risk measures associated to an explicitly determined $1$-dimensional function constructed from $U$. This finding allows for simplifying the study of several properties of $ρ$, such as dual representations, law invariance and stability.

q-fin.MF↗

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↗

arXiv2022

Quasi-Logconvex Measures of Risk

This paper introduces and fully characterizes the novel class of quasi-logconvex measures of risk, to stand on equal footing with the rich class of quasi-convex measures of risk. Quasi-logconvex risk measures naturally generalize logconvex return risk measures, just like quasi-convex risk measures generalize convex monetary risk measures. We establish their dual representation and analyze their taxonomy in a few (sub)classification results. Furthermore, we characterize quasi-logconvex risk measures in terms of properties of families of acceptance sets and provide their law-invariant representation. Examples and applications to portfolio choice and capital allocation are also discussed.

q-fin.RM↗

arXiv2022

Dynamic capital allocation rules via BSDEs: an axiomatic approach

In this paper, we study capital allocation for dynamic risk measures, with an axiomatic approach but also by exploiting the relation between risk measures and BSDEs. Although there is a wide literature on capital allocation rules in a static setting and on dynamic risk measures, only a few recent papers on capital allocation work in a dynamic setting and, moreover, those papers mainly focus on the gradient approach. To fill this gap, we then discuss new perspectives to the capital allocation problem going beyond those already existing in the literature. In particular, we introduce and investigate a general axiomatic approach to dynamic capital allocations as well as an approach suitable for risk measures induced by g-expectations under weaker assumptions than Gateaux differentiability.

math.PR↗

arXiv2020

Risk measures and progressive enlargement of filtration: a BSDE approach

We consider dynamic risk measures induced by Backward Stochastic Differential Equations (BSDEs) in enlargement of filtration setting. On a fixed probability space, we are given a standard Brownian motion and a pair of random variables $(τ, ζ) \in (0,+\infty) \times E$, with $E \subset \mathbb{R}^m$, that enlarge the reference filtration, i.e., the one generated by the Brownian motion. These random variables can be interpreted financially as a default time and an associated mark. After introducing a BSDE driven by the Brownian motion and the random measure associated to $(τ, ζ)$, we define the dynamic risk measure $(ρ_t)_{t \in [0,T]}$, for a fixed time $T > 0$, induced by its solution. We prove that $(ρ_t)_{t \in [0,T]}$ can be decomposed in a pair of risk measures, acting before and after $τ$ and we characterize its properties giving suitable assumptions on the driver of the BSDE. Furthermore, we prove an inequality satisfied by the penalty term associated to the robust representation of $(ρ_t)_{t \in [0,T]}$ and we discuss the dynamic entropic risk measure case, providing examples where it is possible to write explicitly its decomposition and simulate it numerically.

q-fin.RM↗

arXiv2015

Time-consistency of cash-subadditive risk measures

The main goal of this paper is to investigate under which conditions cash-subadditive convex dynamic risk measures are time-consistent. Proceeding as in Detlefsen and Scandolo \cite{detlef-scandolo} and inspired by their result, we give a dual representation of dynamic cash-subadditive convex risk measures (that can also be seen as particular case of the dual quasiconvex representation). The main result of the paper consists in providing, in the cash-subadditive case, a sufficient condition for strong time-consistency (or recursivity) in terms of a generalized cocycle condition. On one hand, our result can be seen as an extension to cash-subadditive convex dynamic risk measures of Theorem 2.5 in Bion-Nadal \cite{bion-nadal-FS}; on the other hand, it is weaker since strong time-consistency is not fully characterized. Finally, we exploit the relation between different notions of time-consistency.

q-fin.RM↗

arXiv2014

Dual Representation of Minimal Supersolutions of Convex BSDEs

We give a dual representation of minimal supersolutions of BSDEs with non-bounded, but integrable terminal conditions and under weak requirements on the generator which is allowed to depend on the value process of the equation. Conversely, we show that any dynamic risk measure satisfying such a dual representation stems from a BSDE. We also give a condition under which a supersolution of a BSDE is even a solution.

math.PR↗