Risk Management & Tail Risk

We establish a profound connection between coherent risk measures, a prominent object in quantitative finance, and uniform integrability, a fundamental concept in probability theory. Instead of working with absolute values of random variables, which is convenient in studying integrability, we work d

Geometrically convex functions constitute an interesting class of functions obtained by replacing the arithmetic mean with the geometric mean in the definition of convexity. As recently suggested, geometric convexity may be a sensible property for financial risk measures ([7,13,4]). We introduce a n

In this paper, we introduce a new class of set-valued risk measures, named set-valued star-shaped risk measures. Motivated by the results of scalar monetary and star-shaped risk measures, this paper investigates the representation theorems in the set-valued framework. It is demonstrated that set-val

We construct Nash equilibria in feedback form for a class of two-person stochastic games of singular control with absorption, arising from a stylized model for corporate finance. More precisely, the paper focusses on a strategic dynamic game in which two financially-constrained firms operate in the

In this paper we study two important representations for extreme value distributions and their max-domains of attraction (MDA), namely von Mises representation (vMR) and variation representation (VR), which are convenient ways to gain limit results. Both VR and vMR are defined via so-called auxiliar

In this paper, we explore the aftermath of the Silicon Valley Bank (SVB) collapse, with a particular focus on its impact on crypto markets. We conduct a multi-dimensional investigation, which includes a factual summary, analysis of user sentiment, and examination of market performance. Based on such

The aim of this work is to study risk measures generated by distortion functions in a dynamic discrete time setup, and to investigate the corresponding dynamic coherent acceptability indices (DCAIs) generated by families of such risk measures. First we show that conditional version of Choquet integr

We introduce a framework for quantifying propagation of uncertainty arising in a dynamic setting. Specifically, we define dynamic uncertainty sets designed explicitly for discrete stochastic processes over a finite time horizon. These dynamic uncertainty sets capture the uncertainty surrounding stoc

This paper provides an introductory overview of how one may employ importance sampling effectively as a tool for solving stochastic optimization formulations incorporating tail risk measures such as Conditional Value-at-Risk. Approximating the tail risk measure by its sample average approximation, w

Motivated by the results of static monetary or star-shaped risk measures, the paper investigates the representation theorems in the dynamic framework. We show that dynamic monetary risk measures can be represented as the lower envelope of a family of dynamic convex risk measures, and normalized dyna

As a counterpart to the (static) risk measures of generalized quantiles and motivated by Bellini et al. (2018), we propose a new kind of conditional risk measure called conditional generalized quantiles. We first show their well-definedness and they can be equivalently characterised by a conditional

The tail behavior of aggregates of heavy-tailed random vectors is known to be determined by the so-called principle of “one large jump’’, be it for finite sums, random sums, or, Lévy processes. We establish that, in fact, a more general principle is at play. Assuming that the random vectors are mult

In this paper, we propose a novel axiomatic approach to evaluating the joint risk of multiple insurance risks under dependence uncertainty. Motivated by both the theory of expected utility and the Cobb-Dauglas utility function, we establish a joint risk measure for non-negative multivariate risks, w

In this paper we introduce a generalization of classical risk measures in which the risk is represented by a step function taking two values, corresponding to two endogenously determined market regimes. This extends the traditional framework where risk measures map random variables to single real nu

Reinforcement learning (RL) is an innovative approach to financial decision making, offering specialized solutions to complex investment problems where traditional methods fail. This review analyzes 167 articles from 2017–2025, focusing on market making, portfolio optimization, and algorithmic tradi

In this paper, we investigate the Lambda Value-at-Risk ($Λ$VaR) under ambiguity, where the ambiguity is represented by a family of probability measures. We establish that for increasing Lambda functions, the robust (i.e., worst-case) $Λ$VaR under such an ambiguity set is equivalent to $Λ$VaR compute

We consider the optimal risk sharing problem with a continuum of agents, modeled via a non-atomic measure space. Individual preferences are not assumed to be convex. We show the multiplicity of agents induces the value function to be convex, allowing for the application of convex duality techniques

We introduce a new paradigm for risk sharing that generalizes earlier models based on discrete agents and extends them to allow for sharing risk within a continuum of agents. Agents are represented by points of a measure space and have potentially heterogeneous risk preferences modeled by risk measu

We study risk measures $\varphi:E\longrightarrow\mathbb{R}\cup{\infty}$, where $E$ is a vector space of random variables which a priori has no lattice structure$\unicode{x2014}$a blind spot of the existing risk measures literature. In particular, we address when $\varphi$ admits a tractable dual r

This paper considers the setting governed by $(\mathbb{F},τ)$, where $\mathbb{F}$ is the “public” flow of information, and $τ$ is a random time which might not be $\mathbb{F}$-observable. This framework covers credit risk theory and life insurance. In this setting, we assume $\mathbb{F}$ being gener