The quantification of downside risk in terms of capital requirements is a key issue for both
regulators and the financial industry. This review presents the axiomatic approach, which is …

When estimating the risk of a P&L from historical data or Monte Carlo simulation, the
robustness of the estimate is important. We argue here that Hampel's classical notion of …

We present various versions of the maximum principle for optimal control of forward-
backward stochastic differential equations (SDE) with jumps. Our study is motivated by risk …

This paper presents novel characterization results for classes of law-invariant star-shaped
functionals. We begin by establishing characterizations for positively homogeneous and star …

The ongoing concern about systemic risk since the outburst of the global financial crisis has
highlighted the need for risk measures at the level of sets of interconnected financial …

This paper establishes characterization results for dynamic return and star-shaped risk
measures induced via backward stochastic differential equations (BSDEs). We first …

We provide a variety of results for quasiconvex, law-invariant functionals defined on a
general Orlicz space, which extend well-known results from the setting of bounded random …

We discuss risk measures representing the minimum amount of capital a financial institution
needs to raise and invest in a pre-specified eligible asset to ensure it is adequately …

Expected risk minimization (ERM) is at the core of many machine learning systems. This
means that the risk inherent in a loss distribution is summarized using a single number-its …

We present a general framework for a comparative theory of variability measures, with a
particular focus on the recently introduced one-parameter families of inter-Expected Shortfall …