This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions.
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 representa…
Since the quasiconvex risk measures is a bigger class than the well known convex risk measures, the study of quasiconvex risk measures makes sense especially in the financial markets with volatility. In this paper, we will study the quasiconvex risk measures defined on a special space $L^{p(\cdot)}$ where the variable exponent $p(\cdot)$ is no longer a given real number like the space $L^{p}$, but a random variable, which reflects the possible volatility of the financial markets. The dual representation for this quasiconvex risk measures will also provided.
Informally, a risk measure is said to be elicitable if there exists a suitable scoring function such that minimizing its expected value recovers the risk measure. In this paper, we analyze the elicitability properties of the class of return risk measures (i.e., normalized, monotone and positively homogeneous risk measures). First, we provide dual representation results for convex and geometrically convex return risk measures. Next, we establish new axiomatic characterizations of Orlicz premia (i.e., Luxemburg norms). More specifically, we prove, under different sets of conditions, that Orlicz …
Gaussian random vectors exhibit the loss of dimension phenomena, which relate to their joint survival tail behaviour. Besides, the fact that the components of such vectors are light-tailed complicates the approximations of various multivariate risk measures significantly. In this contribution we derive precise approximations of marginal mean excess, marginal expected shortfall and multivariate conditional tail expectation of Gaussian random vectors and highlight links with conditional limit theorems. Our study indicates that similar results hold for elliptical and Gaussian like multivariate ri…
Motivated by liquidity risk in mathematical finance, D. Lacker introduced concentration inequalities for risk measures, i.e. upper bounds on the \emph{liquidity risk profile} of a financial loss. We derive these inequalities in the case of time-consistent dynamic risk measures when the filtration is assumed to carry a Brownian motion. The theory of backward stochastic differential equations (BSDEs) and their dual formulation plays a crucial role in our analysis. Natural by-products of concentration of risk measures are a description of the tail behavior of the financial loss and transport-type…
In this paper we analyze a dynamic recursive extension of the (static) notion of a deviation measure and its properties. We study distribution invariant deviation measures and show that the only dynamic deviation measure which is law invariant and recursive is the variance. We also solve the problem of optimal risk-sharing generalizing classical risk-sharing results for variance through a dynamic inf-convolution problem involving a transformation of the original dynamic deviation measures.
We give an axiomatic framework for conditional generalized deviation measures. Under financially reasonable assumptions, we give the correspondence between conditional coherent risk measures and generalized deviation measures. Moreover, we establish the notion of continuous-time risk contribution for conditional coherent risk measures and generalized deviation measures. With the help of the correspondence between these two different types of risk measures, we give a microscopic interpretation of their risk contributions. Particularly, we show that the risk contributions of time-consistent risk…
In our previous paper, "A Unified Approach to Systemic Risk Measures via Acceptance Set" (\textit{Mathematical Finance, 2018}), we have introduced a general class of systemic risk measures that allow for random allocations to individual banks before aggregation of their risks. In the present paper, we prove the dual representation of a particular subclass of such systemic risk measures and the existence and uniqueness of the optimal allocation related to them. We also introduce an associated utility maximization problem which has the same optimal solution as the systemic risk measure. In addit…
To find a trade-off between profitability and prudence, financial practitioners need to choose appropriate risk measures. Two key points are: Firstly, investors' risk attitudes under uncertainty conditions should be an important reference for risk measures. Secondly, risk attitudes are not absolute. For different market performance, investors have different risk attitudes. We proposed a new risk measure named SlideVaR which sufficiently reflects the different subjective attitudes of investors and the impact of market changes on investors' attitudes. We proposed the concept of risk-tail region …
When estimating the risk of a financial position with empirical data or Monte Carlo simulations via a tail-dependent law invariant risk measure such as the Conditional Value-at-Risk (CVaR), it is important to ensure the robustness of the statistical estimator particularly when the data contain noise. Kratscher et al. [1] propose a new framework to examine the qualitative robustness of estimators for tail-dependent law invariant risk measures on Orlicz spaces, which is a step further from earlier work for studying the robustness of risk measurement procedures by Cont et al. [2]. In this paper,…
在本论文中，我们刻画了在分布水平上具有凸水平集的律确定的凸风险测度。通过放松Weber（2006）的假设，我们证明了这些风险度量可以被识别为一类广义短缺风险度量。作为一个直接的结果，我们能够扩展Ziegel（2014）和Bellini and Biggini（2014）关于凸可引出风险度量的结果，并确认期望值是唯一可引出的一致性风险度量。此外，我们提供了一个简单的表征的鲁棒性凸风险措施的弱概念的混合连续性。
Risk measures connect probability theory or statistics to optimization, particularly to convex optimization. They are nowadays standard in applications of finance and in insurance involving risk aversion. This paper investigates a wide class of risk measures on Orlicz spaces. The characterizing function describes the decision maker's risk assessment towards increasing losses. We link the risk measures to a crucial formula developed by Rockafellar for the Average Value-at-Risk based on convex duality, which is fundamental in corresponding optimization problems. We characterize the dual and prov…
The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk measure. Examples of multivariate risk measures satisfying the supermartingale property are given. Crucial for obtaining the results are dual representations of scalarizations of set-valued dynamic risk measures, which are of independent interest in the fast growing literature on …
The inf-convolution of risk measures is directly related to risk sharing and general equilibrium, and it has attracted considerable attention in mathematical finance and insurance problems. However, the theory is restricted to finite sets of risk measures. This study extends the inf-convolution of risk measures in its convex-combination form to a countable (not necessarily finite) set of alternatives. The intuitive meaning of this approach is to represent a generalization of the current finite convex weights to the countable case. Subsequently, we extensively generalize known properties and re…
An Integrated Approach to Importance Sampling and Machine Learning for Efficient Monte Carlo Estimation of Distortion Risk Measures in Black Box Models
This survey gives an introduction to monetary measures of risk as monotone and cash additive functions on spaces of univariate random variables. Primal and dual representation results as well as several examples are discussed. Principal ways to construct risk measures are given and extensions to more general situations indicated.
The purpose of the research presented in this article is to develop a dynamic system for forecasting and minimizing the risks of an industrial company based on their quantitative assessment. The article considers the conceptual apparatus of the essential content of risk management of an industrial enterprise, reviews the theoretical aspects of risk management systems and the most significant risk management methods from a practical point of view. The methodological apparatus of qualitative and quantitative analysis and risk assessment has been expanded on the basis of some conditionality of ri…
The intuition of risk is based on two main concepts: loss and variability. In this paper, we present a composition of risk and deviation measures, which contemplate these two concepts. Based on the proposed Limitedness axiom, we prove that this resulting composition, based on properties of the two components, is a coherent risk measure. Similar results for the cases of convex and co-monotone risk measures are exposed. We also provide examples of known and new risk measures constructed under this framework in order to highlight the importance of our approach, especially the role of the Limitedn…
