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This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean-variance models are reinterpreted as conditional location-scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location-scale time series models under mild conditions. In the application of this result to ARMA-generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.

Recent studies in Lee and Prékopa (Oper Res Lett 45:19–24, 2017) and Lee (Oper Res Lett 45:1204–1220, 2017) showed that a union of partially ordered orthants in Rn can be decomposed only into the largest and the second largest chains. This allows us to calculate the probability of the union of such events in a recursive manner. If the vertices of such orthants designate p-level efficient points, i.e., the multivariate quantile or the multivariate value-at-risk (MVaR) in Rn, then the number of them, say N, is typically very large, which makes it almost impossible to calculate the multivariate conditional value-at-risk (MCVaR) introduced by Prékopa (Ann Oper Res 193(1):49–69, 2012). This is because it takes O(2N) in case of N MVaRs in Rn to find the exact value of MCVaR. In this paper, upon the basis of ideas in Lee and Prékopa (Oper Res Lett 45:19–24, 2017) and Lee (Oper Res Lett 45:1204–1220, 2017), together with proper adjustments, we study efficient methods for the calculation of the MCVaR without resorting to an approximation. In fact, the proposed methods not only have polynomial time complexity but also computes the exact value of MCVaR. We also discuss additional benefits MCVaR has to offer over its univariate counter part, the conditional value-at-risk, by providing numerical results. Numerical examples are presented with computing time in both cases of given population and sample data sets.

In safety‐critical control systems such as autonomous vehicles and medical devices, managing the risk of rare but severe tail events under uncertainty is crucial. This paper addresses this challenge by proposing a risk‐aware control framework that integrates the worst‐case conditional value‐at‐risk (CVaR) with control barrier functions (CBFs). Specifically, we formulate risk‐aware safety constraints based on the worst‐case CVaR, and show that the resulting risk‐aware controllers can be computed via quadratic programs (for half‐space and polytopic safe sets) or a semidefinite program (for ellipsoidal safe sets). Numerical simulations on an inverted pendulum illustrate that the proposed approach ensures safety under various scenarios and significantly reduces the safety constraint violation compared to existing CBF approaches. Overall, we show that incorporating worst‐case CVaR into CBF design offers a tractable solution for safety‐critical applications under uncertainty. Overview of the proposed risk‐aware control. The proposed approach combines control barrier functions with worst‐case conditional value‐at‐risk to design optimization‐based controllers for safety‐critical systems under stochastic uncertainties.

We consider a road-ban problem in hazardous material (hazmat) transportation. We formulate the problem as a network design problem to select a set of closed road segments for hazmat traffic and obtain a bilevel optimization problem. While modeling probabilistic route choices of hazmat carriers by the random utility model (RUM) in the lower level, we consider a risk-averse measure called conditional value at risk (CVaR) in the upper level, instead of the widely used expected risk measure. Using the RUM and CVaR, we quantify the risk of having hazmat accidents and large consequences and design the network policy for road bans accordingly. Although CVaR has been used in hazmat routing problems, this paper is the first attempt to apply CVaR in risk averse hazmat network design problems considering stochastic route choices of hazmat carriers. The resulting problem is a mixed integer nonlinear programming problem, for which we devise a line search approach combined with Benders decomposition. We demonstrate the efficiency of the proposed computational method with case studies. The average computation time for a network with 105 nodes and 268 arcs is three hours. By applying CVaR to the route-choice behavior of hazmat carriers, we protect the road network from undesirable route choices that may lead to severe consequences. We define the value of RUM-CVaR solutions (VRCS) over the deterministic model based on shortest-path problems and the expected risk measure. Our case study shows that the VRCS can range from 4.9% to 64.1% depending on the probability threshold used in the CVaR measure.

This study took online lending as the main research object to quantitatively measure FinTech risk in China. A theoretical model was built to analyse the relationship between the assets and liabilities of peer-to-peer platforms and the risk of the entire online lending industry. The conditional value at risk method was used to measure the risk spillover effect of the online lending industry and the different types of platforms. Index smoothing and moving average were used to examine risk contagion. When the risk rate of the portfolio of the platform generally increases, the systemic risk of the whole industry also increases, and if the systemic risk of the industry spreads to the portfolio of the platform, it could affect the stability of the capital flow of the platform, and then affect the risk expectation of the platform itself. Most platforms with risk concerns were private. Banking platforms and public platforms were greatly affected by market risk; however, their own risks had relatively little impact on the overall market. The greater the market risk, the more platforms become risk platforms, and the smaller the impact on transition platforms. Regulatory sandboxes may be an effective means of preventing and controlling Fintech risks.

We propose an enhanced indexing portfolio optimization model that not only seeks to maximize the excess returns over and above the benchmark index but simultaneously control the risk by introducing a constraint on the weighted conditional value at risk (WCVaR) of the portfolio. The constraint in the proposed model can be seen as hedging the risk described by WCVaR of the portfolio. To carry out a comparative analysis of the proposed model, we also suggest an enhanced indexing CVaR model. We analyze the performance of the proposed model at various risk levels on eight publicly available financial data sets from Beasley OR library, and S&P 500, S&P BSE 500, NASDAQ composite, FTSE 100 index, and their constituents, for average returns, Sharpe ratio, and upside potential ratio. Empirical analysis exhibits superior performance of the portfolios from the proposed WCVaR model over the respective benchmark indices and additionally the optimal portfolios obtained from various other enhanced indexing models that exist in the literature. Furthermore, we present evidence of better performance of WCVaR model over the CVaR model for long-term investment horizons.

Summary In wind‐integrated power systems, the stochastic and random wind power brings more uncertainties to power system operation. To address multiple uncertain factors including wind power forecast error, generator failure, and load forecast error, a conditional value‐at‐risk (CVaR)‐based optimal spinning reserve determination method for wind‐integrated power system is proposed in this paper with consideration of both reliability and economy of system operation under a certain level of risk. The CVaR model of spinning reserve considering different uncertain factors is built and then incorporated into the coordinated scheduling model of power generation and spinning reserve in wind integrated power system. The optimization model is solved by mix integer linear programming, and the CVaR‐based spinning reserve allocation and the power output of each unit are optimally determined at a certain confidence level. The proposed method can provide a trade‐off between the economy and reliability of system operation with respect to the confidence level of risk. Simulation results on 10‐unit system and 100‐unit system demonstrate the effectiveness of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.

The portfolio optimization model has limited impact in practice because of estimation issues when applied to real data. To address this, we adapt two machine learning methods, regularization and cross-validation, for portfolio optimization. First, we introduce performance-based regularization (PBR), where the idea is to constrain the sample variances of the estimated portfolio risk and return, which steers the solution toward one associated with less estimation error in the performance. We consider PBR for both mean-variance and mean-conditional value-at-risk (CVaR) problems. For the mean-variance problem, PBR introduces a quartic polynomial constraint, for which we make two convex approximations: one based on rank-1 approximation and another based on a convex quadratic approximation. The rank-1 approximation PBR adds a bias to the optimal allocation, and the convex quadratic approximation PBR shrinks the sample covariance matrix. For the mean-CVaR problem, the PBR model is a combinatorial optimization problem, but we prove its convex relaxation, a quadratically constrained quadratic program, is essentially tight. We show that the PBR models can be cast as robust optimization problems with novel uncertainty sets and establish asymptotic optimality of both sample average approximation (SAA) and PBR solutions and the corresponding efficient frontiers. To calibrate the right-hand sides of the PBR constraints, we develop new, performance-based k -fold cross-validation algorithms. Using these algorithms, we carry out an extensive empirical investigation of PBR against SAA, as well as L1 and L2 regularizations and the equally weighted portfolio. We find that PBR dominates all other benchmarks for two out of three Fama–French data sets. This paper was accepted by Yinyu Ye, optimization .

Summary This paper presents a new stochastic operation scheduling model for microgrid in day‐ahead electricity market considering the forecast uncertainties of wind generations, microgrid's load, and locational marginal price of the point of common coupling, as well as the uncertainties pertaining to availability of generation units of the microgrid. The financial risk caused by these uncertainties is modeled by conditional value‐at‐risk criterion. The proposed approach is formulated as a risk‐minimizing two‐stage stochastic model based on mixed‐integer linear programming framework. This model minimizes the expected scheduling cost together with the cost of financial risk. The impacts of different uncertainty sources on the model's results in both grid‐connected and islanded operation modes are extensively studied. Additionally, higher effectiveness of the proposed stochastic model compared with common deterministic approach, regarding total operation cost and convergence behavior, is illustrated through an out‐of‐sample analysis.