Risk-coherent H2-optimal disturbance rejection under model uncertainty

This paper introduces the concept of entropic value-at-risk (EVaR), a new coherent risk measure that corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the value-at-risk (VaR) as well as the conditional value-at-risk (CVaR). We show that a broad class of stochastic optimization problems that are computationally intractable with the CVaR is efficiently solvable when the EVaR is incorporated. We also prove that if two distributions have the same EVaR at all confidence levels, then they are identical at all points. The dual representation of the EVaR is closely related to the Kullback-Leibler divergence, also known as the relative entropy. Inspired by this dual representation, we define a large class of coherent risk measures, called g-entropic risk measures. The new class includes both the CVaR and the EVaR.

In this paper, we address a basic production planning problem with price dependent demand and stochastic yield of production. We use price and target quantity as decision variables to lower the risk of low yield. The value of risk control becomes more important especially for products with short life cycle. This is because, the profit implications of low yield might be unbearable in the short run. We apply Conditional Value at Risk (CVaR) to model the risk. CVaR measure is a coherent risk measure and thereby having nice conceptual and mathematical underpinnings. It is also widely used in practice. We consider the problem under general demand function and general distribution function of yield and find sufficient conditions under which the problem has a unique local maximum. We also both analytically and numerically analyze the impact of parameter change on the optimal solution. Among our results, we analytically show that with increasing risk aversion, the optimal price increases. This relation is opposite to that of in Newsvendor problem where the uncertainty lies in demand side.

In this chapter, the authors explore how to apply the Youla parameterization method to MIMO systems. Similar to the SISO feedback loop, they discuss the role of neoclassical control for MIMO feedback control design. The authors present equivalent MIMO feedback loops which Youla parameterization technique could be utilized either directly (Internal Model Control) or utilized to compute a MIMO feedback controller. In neoclassical control, the performance and robustness criteria are achieved by direct frequency loop-shaping of three different closed-loop transfer function matrices, Youla, sensitivity, and complementary sensitivity. The authors then investigate the performance and robustness criteria for MIMO feedback controller design. They also extend their MIMO control design approach to unstable plants.

This chapter introduces the neoclassical method of controller design. It will discuss measures of stability and robustness followed by algebraic and analytical constraints in feedback design, revealing how Youla parameter could be utilized in designing a feedback control by considering the constraints and trade-offs in SISO control design.

This letter proposes a novel Risk-Perception-Aware (RPA) control design using non-rational perception of risks associated with uncertain dynamic spatial costs. We use Cumulative Prospect Theory (CPT) to model the risk perception of a Decision Maker (DM) and use it to construct perceived risk functions that transform the uncertain dynamic spatial cost to deterministic perceived risks of a DM. These risks are then used to build safety sets which can represent risk-averse to risk-insensitive perception. Using these sets, we define novel notions of “inclusiveness” and “versatility” which can be employed to compare and evaluate any risk models in the context of RPA safety-critical controls. We then prove that CPT is the most “inclusive” and “versatile” model w.r.t. Conditional Value at Risk (CVaR) and Expected Risk (ER). Given a RPM, we construct a class of Control Barrier Functions (CBFs) and generate perceived-safety-critical controls using a Quadratic Program (QP) to guide an agent safely to a goal. For a class of truncated-Gaussian costs, we provide sufficient geometric conditions for the above QP to be feasible. We also prove that CPT-equipped RPA controller has both a larger feasible control set and more accurate stabilization w.r.t CVaR and ER models. We present simulations in a 2D environment to illustrate the proposed controller.

This paper presents a decentralized controller design for formation and traction control of multiple robots with collision avoidance. The proposed control design consists of two steps: formation control design with collision avoidance and traction control design for movement of robot formation along the desired trajectory. In the first step, a formation controller is designed based on the formation graph model of multiple robots and potential function approach. Then, the traction control input is derived in the second step by locating the virtual leader at the central position of robot formation which leads mobile robots to follow the assigned trajectory in formation. In this distributed control design, parameter uncertainty in kinematic system model is also accommodated by deriving an adaptive rule for estimation of uncertain system parameters. Together with the stability analysis of the proposed controller, simulation and experiment results verify the feasibility and effectiveness of the distributed controller for multi-robot coordination.

This paper is concerned with control design for a generalized discrete-time Takagi-Sugeno fuzzy system. The Takagi-Sugeno fuzzy system generally describes nonlinear systems by employing local linear system representations. It is a great tool to analyze a nonlinear system and to design a stabilizing controller for it. However, there is a case where a controller design for a nonlinear system based on the Takagi-Sugeno fuzzy system becomes difficult due to its representation. A generalized fuzzy system to be considered in this paper describes even a wider class of nonlinear systems by representing locally nonlinear systems, and gives a different representation from the standard Takagi-Sugeno fuzzy system, which leads to a rather easy controller design. For such a generalized system, a stabilizing controller design method is proposed by introducing a new class of nonlinear controllers. Then, the resulting closed-loop system becomes a fuzzy system with linear system structures locally, which makes the control design easier. Stabilizing controller design conditions are given in terms of a set of linear matrix inequalities(LMIs), which are easily numerically solvable. A relaxation method is used to reduce the conservatism of design conditions. Finally, numerical examples are given to illustrate our nonlinear control design and to show the effectiveness over other existing results.

A spectrum of upper bounds (Qα(X ; p) αε[0,∞] on the (largest) (1-p)-quantile Q(X;p) of an arbitrary random variable X is introduced and shown to be stable and monotonic in α, p, and X , with Q0(X ;p) = Q(X;p). If p is small enough and the distribution of X is regular enough, then Qα(X ; p) is rather close to Q(X ; p). Moreover, these quantile bounds are coherent measures of risk. Furthermore, Qα(X ; p) is the optimal value in a certain minimization problem, the minimizers in which are described in detail. This allows of a comparatively easy incorporation of these bounds into more specialized optimization problems. In finance, Q0(X;p) and Q1(X ; p) are known as the value at risk (VaR) and the conditional value at risk (CVaR). The bounds Qα(X ; p) can also be used as measures of economic inequality. The spectrum parameter α plays the role of an index of sensitivity to risk. The problems of the effective computation of the bounds are considered. Various other related results are obtained.

In the financial industry, the trading of multiple portfolios is usually aggregated and optimized simultaneously. When multiple portfolios are managed together, unique issues such as market impact costs must be dealt with properly. Conditional Value-at-Risk (CVaR) is a coherent risk measure with the computationally friendly feature of convexity. In this study, we propose the new combination of CVaR with the multiportfolio optimization (MPO) problem and develop optimization models with using CVaR to measure risks in MPO problems. A five-step scheme is presented for practical operations with considering the impact costs caused by aggregating the trading of multiple portfolios. The impact of CVaR on returns and utility in MPO environment is studied, and the comparisons with existing methods and sensitivity analysis are reported.

In this chapter, we consider approaches to express the risk preferences of a newsvendor by means of the risk measures value at risk (VaR), conditional value at risk (CVaR), and the mean-CVaR rule, which usually is defined as a convex combination of expected profit and CVaR. With these risk measures the decision maker can exploit risk-averse or risk-neutral behavior. In addition, we introduce a more general mean-CVaR measure where also risk- taking behavior can be expressed. The overall goal of the paper is a comparative analysis of these risk measures in the newsvendor framework. On the one hand VaR, CVaR and the (general) mean-CVaR, measures are used as objective functions to derive the respective optimal order quantity. Extensions of the basic models are reviewed. On the other hand the risk measures, especially VaR, are constraints of the model. We first review models with the expected profit as objective. Then the general mean-CVaR measure is taken as objective function and a service constraint and a loss constraint are added. In this framework, the risk attitudes of the newsvendor can be deduced from the characteristics of a product together with the specified service target and loss target.

PurposeThe aim is to study jump liquidity risk and its impact on risk measures: value at risk (VaR) and conditional VaR (CVaR).Design/methodology/approachThe liquidity discount factor is modelled with mean revision jump diffusion processes and the liquidity risk is integrated in the framework of VaR and CVaR.FindingsThe standard VaR, CVaR, and the liquidity adjusted VaR can seriously underestimate the potential loss over a short holding period for rare jump liquidity events. A better risk measure is the liquidity adjusted CVaR which gives a more realistic loss estimation in the presence of the liquidity risk. An efficient Monte Carlo method is also suggested to find approximate VaR and CVaR of all percentiles with one set of samples from the loss distribution, which applies to portfolios of securities as well as single securities.Originality/valueThe paper offers plausible stochastic processes to model liquidity risk.

The purpose of this study is to compare two major risk measurement models in financial markets—the Value at Risk (VaR) model and the Conditional Value at Risk (CVaR) model—and to evaluate their effectiveness and limitations in stock market risk management. First, through an in-depth analysis of the theoretical foundations of stock market risk, the paper clarifies the importance of risk for investment decision-making, in particular its impact on asset allocation and risk management strategies. The findings show that VaR, as a traditional risk measurement tool, is simple and convenient and enables a rapid assessment of market risk; however, when confronted with extreme tail events it exhibits certain shortcomings and cannot fully capture the true risk of extreme losses. By contrast, CVaR, by focusing on tail risk, makes it possible to identify potential extreme losses more accurately; especially during periods of sharp market volatility, its early-warning function substantially outperforms VaR. In the empirical part of the article, historical data for the Shanghai A-share index are used. The GARCH-M model is applied to estimate VaR and CVaR, and their values are compared when measuring stock market risk. The results indicate that, at the same confidence level, CVaR is always greater than VaR, which further confirms the advantage of CVaR in reflecting risk losses more accurately, especially tail risk. Overall, this study not only provides a theoretical basis for risk management in financial markets, but also offers recommendations for improving risk measurement methods in the stock market, emphasizing the need to account for tail risk when making investment decisions. The conclusions have important reference value for investors seeking to develop more scientifically grounded risk control strategies in a dynamic market environment.

Bayes Risk, Elicitability, and the Expected Shortfall

The article is dedicated to the optimization of credit risk through the application of Conditional Value at Risk (CVaR). CVaR is a risk measure, the expected loss exceeding Value-at-Risk and is also known as Mean Excess, Mean Shortfall, or Tail VaR. The link between credit risk and the current financial crisis accentuates the importance of measuring and predicting extreme credit risk. Conditional Value at Risk has become an increasingly popular method for measurement and optimization of extreme market risk. The use of model can regulate all positions in a portfolio of financial instruments in order to minimize CVaR subject to trading and return constraints at the same time. The credit risk distribution is created by Monte Carlo simulations and the optimization problem is solved effectively by linear programming. We apply these CVaR techniques to the optimization of credit risk on portfolio of selected bonds.                  Keywords: value at risk; conditional value at risk; credit risk; portfolio

This paper presents a method that reduces the conservatism inherent in the disturbance representation in standard H ∞ theory. It addresses several discrete-time control synthesis problems where the disturbances acting on the system are structured by a convex family of linear filters (‘signal polytope’) and where the system disturbance attenuation level attained over the signal polytope is ensured to (just) a prescribed probability. This setup enables realistic multi-feature disturbance delimiting while accounting for uncertainty in the disturbance model itself by allowing probability waivers. The core of the probability aspects of the proposed solutions is the search for a truncated signal polytope which provides both the required probability and the best robust disturbance attenuation level. Many examples are given, including one of an aircraft output-feedback control with a polytope of low-pass filters representing different wind phenomena. The examples demonstrate that addressing realistic disturbances results in better control designs (hence better performance) and that a small certainty waiver can yield a large performance gain.