ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES

In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps.Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.

Risk measures with comonotonic subadditivity or convexity on product spaces

In this paper, by an axiomatic approach, we propose the concepts of comonotonic subadditivity and comonotonic convex risk measures for portfolios, which are extensions of the ones introduced by Song and Yan （2006）. Representation results for these new introduced risk measures for portfolios are given in terms of Choquet integrals. Links of these newly introduced risk measures to multi-period comonotonic risk measures are represented. Finally, applications of the newly introduced comonotonic coherent risk measures to capital allocations are provided.

CASH SUBADDITIVE RISK MEASURES FOR PORTFOLIO VECTORS

In this paper, from the viewpoint of the time value of money, we study the risk measures for portfolio vectors with discount factor. Cash subadditive risk measures for portfolio vectors are proposed. Representation results are given by two different methods which are convex analysis and enlarging space. Especially, the method of convex analysis make the line of reasoning and the representation result be simpler. Meanwhile, spot and forward risk measures for portfolio vectors are also introduced, and the relationships between them are investigated.

On the generalized risk measures

In this paper, new risk measures are introduced, tation results are also given. These newly introduced risk introduced by Song and Yan （2009） and Karoui （2009）. and the corresponding represen- measures are extensions of those

Risk Measure and Premium Distribution on Catastrophe Reinsurance

In this paper, we propose a new risk measure which is based on the Or- licz premium principle to characterize catastrophe risk premium. The intention is to develop a formulation strategy for Catastrophe Fund. The logarithm equivalent form of reinsurance premium is regarded as the retention of reinsurer, and the differential earnings between the reinsurance premium and the reinsurer＇s retention is accumu- lated as a part of Catastrophe Fund. We demonstrate that the aforementioned risk measure has some good properties, which are further confirmed by numerical simu- lations in R environment.

Capital allocation for cash-subadditive risk measures:From BSDEs toBSVIEs

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 yvariable.In this setting,we surprisingly find that the corresponding capital allocation rules solve special kinds of Backward Stochastic Volterra Integral Equations(BSVIEs).

A New Risk Measure MMVaR:Properties and Empirical Research

The paper presents the properties of an alternative method,which measures market risk over time-horizon exceeding one day:Mark to market value at risk(MMVaR).Relying on the minimal returns during the time interval,this method not only considers the non-normality of data and information about sample size,but also meets the requirement of increasing the minimal capital ratio in BaselⅢ,basically.The authors theoretically prove the translation invariance,monotonicity and subadditivity of MMVaR as a risk measure under some conditions,and study its finite sample properties through Monte Carlo simulations.The empirical analysis shows that MMVaR can measure multi-period risk accurately.

A Novel DBN-EFA-CFA-Based Dimensional Reduation for Credit Risk Measurement

Affected by the Federal Reserve's interest rate hike and the downward pressure on the domestic economy,the phenomenon of default is still prominent.The credit risk of the listed companies has become a growing concern of the community.In this paper we present a novel credit risk measurement method based on a dimensional reduation technique.The method first extracts the risk measure indexes from the basal financial data via dimensional reduation by using deep belief network(DBN),exploratory factor analysis(EFA)and confirmatory factor analysis(CFA)in turn.And then the credit risk is measured by a systemic structural equation model(SEM)and logistic distribution.To validate the proposed method,we employ the financial data of the listed companies from Q12019 to Q22022.The empirical results show its effectiveness on statistical evaluation,assessment on testing samples and credit risk forecasting.

Mean-field BSDEs with jumps and dual representation for global risk measures

We study mean-field BSDEs with jumps and a generalized mean-field operator that can capture higher-order interactions.We interpret the BSDE solution as a dynamic risk measure for a representative bank whose risk attitude is influenced by the system.This influence can come in a wide class of choices,including the average system state or average intensity of system interactions.Using Fenchel−Legendre transforms,our main result is a dual representation for the expectation of the risk measure in the convex case.In particular,we exhibit its dependence on the mean-field operator.

The relations among the three kinds of conditional risk measures

Let（Ω , E, P） be a probability space, F a sub-σ-algebra of E, L^p（E）（1 p ＋∞） the classical function space and LF^p（E） the L^0（F）-module generated by L^p（E）, which can be made into a random normed module in a natural way. Up to the present time, there are three kinds of conditional risk measures, whose model spaces are L^∞（E）, L^p（E）（1 p ＋∞） and LF^p（E）（1 p ＋∞） respectively, and a conditional convex dual representation theorem has been established for each kind. The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems. We first establish the relation between L^p（E） and LF^p（E）, namely LF^p（E） = Hcc（L^p（E））, which shows that LF^p（E）is exactly the countable concatenation hull of L^p（E）. Based on the precise relation, we then prove that every L^0（F）-convex L^p（E）-conditional risk measure（1 p ＋∞） can be uniquely extended to an L^0（F）-convex LF^p（E）-conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter, which shows that the study of L^p-conditional risk measures can be incorporated into that of LF^p（E）-conditional risk measures. In particular, in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L^0-convex conditional risk measures.

Construction Technology and Safety Risk Control Measures of Deep Foundation Pit Excavation

Deep foundation pit excavation is a basic and key step involved in modern building construction.In order to ensure the construction quality and safety of deep foundation pits,this paper takes a project as an example to analyze deep foundation pit excavation technology,including the nature of this construction project,the main technical measures in the construction of deep foundation pit,and the analysis of the safety risk prevention and control measures.The purpose of this analysis is to provide scientific reference for the construction quality and safety of deep foundation pits.

Recent progress in random metric theory and its applications to conditional risk measures

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures.This paper includes eight sections.Section 1 is a longer introduction,which gives a brief introduction to random metric theory,risk measures and conditional risk measures.Section 2 gives the central framework in random metric theory,topological structures,important examples,the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals.Section 3 gives several important representation theorems for random conjugate spaces.Section 4 gives characterizations for a complete random normed module to be random reflexive.Section 5 gives hyperplane separation theorems currently available in random locally convex modules.Section 6 gives the theory of random duality with respect to the locally L0-convex topology and in particular a characterization for a locally L0-convex module to be L0-pre-barreled.Section 7 gives some basic results on L0-convex analysis together with some applications to conditional risk measures.Finally,Section 8 is devoted to extensions of conditional convex risk measures,which shows that every representable L∞-type of conditional convex risk measure and every continuous Lp-type of convex conditional risk measure(1 ≤ p < +∞) can be extended to an L∞F(E)-type of σ,λ(L∞F(E),L1F(E))-lower semicontinuous conditional convex risk measure and an LpF(E)-type of T,λ-continuous conditional convex risk measure(1 ≤ p < +∞),respectively.

Optimal Reinsurance Under Distortion Risk Measures and Expected Value Premium Principle for Reinsurer

This paper discusses optimal reinsurance strategy by minimizing insurer's risk under one general risk measure:Distortion risk measure.The authors assume that the reinsurance premium is determined by the expected value premium principle and the retained loss of the insurer is an increasing function of the initial loss.An explicit solution of the insurer's optimal reinsurance problem is obtained.The optimal strategies for some special distortion risk measures,such as value-at-risk(VaR) and tail value-at-risk(TVaR),are also investigated.

First and Second Order Asymptotics of the Spectral Risk Measure for Portfolio Loss Under Multivariate Regular Variation

In the context of multivariate regular variation,the authors establish the first-order asymptotics of the spectral risk measure of portfolio loss.Furthermore,by the notion of second-order regular variation,the second-order asymptotics of the spectral risk measure of portfolio loss is also presented.In order to illustrate the derived results,a numerical example with Monte Carlo simulation is carried out.

An overview of representation theorems for static risk measures

In this paper,we give an overview of representation theorems for various static risk measures:coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity, law-invariant coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity and respecting stochastic orders.

A Worst-Case Risk Measure by G-VaR

G-VaR,which is a type of worst-case value-at-risk(VaR),is defined as measuring risk incorporating model uncertainty.Compared with most extant notions of worst-case VaR,G-VaR can be computed using an explicit formula,and can be applied to large portfolios of several hundred dimensions with low computational cost.We also apply G-VaR to robust portfolio optimization,thereby providing a tractable means to facilitate optimal allocations under the condition of market ambiguity.

Time Consistent Multi-period Worst-Case Risk Measure in Robust Portfolio Selection

In this paper,we first construct a time consistent multi-period worst-case risk measure,which measures the dynamic investment risk period-wise from a distributionally robust perspective.Under the usually adopted uncertainty set,we derive the explicit optimal investment strategy for the multi-period robust portfolio selection problem under the multi-period worst-case risk measure.Empirical results demonstrate that the portfolio selection model under the proposed risk measure is a good complement to existing multi-period robust portfolio selection models using the adjustable robust approach.

Optimal reinsurance designs based on risk measures:a review

Reinsurance is an effective way for an insurance company to control its risk.How to design an optimal reinsurance contract is not only a key topic in actuarial science,but also an interesting research question in mathematics and statistics.Optimal reinsurance design problems can be proposed from different perspectives.Risk measures as tools of quantitative risk management have been extensively used in insurance and finance.Optimal reinsurance designs based on risk measures have been widely studied in the literature of insurance and become an active research topic.Different research approaches have been developed and many interesting results have been obtained in this area.These approaches and results have potential applications in future research.In this article,we review the recent advances in optimal reinsurance designs based on risk measures in static models and discuss some interesting problems on this topic for future research.

Stock Efficiency Evaluation Based on Multiple Risk Measures:A DEA-Like Envelopment Approach

This paper proposes a new approach for stock efficiency evaluation based on multiple risk measures. A derived programming model with quadratic constraints is developed based on the envelopment form of data envelopment analysis(DEA). The derived model serves as an input-oriented DEA model by minimizing inputs such as multiple risk measures. In addition, the Russell input measure is introduced and the corresponding efficiency results are evaluated. The findings show that stock efficiency evaluation under the new framework is also effective. The efficiency values indicate that the portfolio frontier under the new framework is more externally enveloped than the DEA efficient surface under the standard DEA framework.

Conditional coherent risk measures and regime-switching conic pricing

This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distortion function.A model is then developed for the bid and ask prices of a European-type asset by a conic formulation.The price process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain.The bid and ask prices of a European-type asset are then characterized using conic quantization.