MULTIDIMENSIONAL DYNAMIC RISK MEASURE VIA CONDITIONAL g‐EXPECTATION

This paper deals with multidimensional dynamic risk measures induced by conditional g‐expectations. A notion of multidimensional g‐expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem, and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, and translatability properties of multidimensional conditional g‐expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g‐risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi‐monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g‐risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacting subsidiaries; optimal risk sharing for ‐tolerant g‐risk measures, and risk contribution for coherent g‐risk measures are investigated. Insurance g‐risk measure and other ways to induce g‐risk measures are also studied at the end of the paper.     

Nonparametric Evaluation of Economies of Scope in the Context of Technical Efficiency: The Case of Rice and Vegetable Farms in Korea

This paper provides a nonparametric evaluation of economies of scope in the context of technical efficiency allowing for non‐convexity, with an application to Korean rice and vegetable farms. Relying on non‐parametric‐free disposal hull and data envelopment analysis approaches and input–output data for rice and vegetable farms from the Korea Farm Household Economy Survey data collected in 2007, this article examines technical inefficiency and its decomposition under non‐convexity and convexity. Empirical measures of technical inefficiency and its decomposition results are provided, with a focus on a component of technical inefficiency associated with economies of scope (i.e. diversification benefits) under non‐convexity in production technology. The decomposition measures of technical inefficiency include pure technical inefficiency, size inefficiency and diversification inefficiency. The results provide empirical evidence supporting size‐dependent diversification strategies, where benefits of diversification are larger for small farms on average compared to those of large farms.     

Convex risk measures for the aggregation of multiple information sources and applications in insurance

We propose a novel class of convex risk measures, based on the concept of the Fréchet mean, designed in order to handle uncertainty which arises from multiple information sources regarding the risk factors of interest. The proposed risk measures robustly characterize the exposure of the firm, by filtering out appropriately the partial information available in individual sources into an aggregate model for the risk factors of interest. Importantly, the proposed risks can be expressed in closed analytic forms allowing for interesting qualitative interpretations as well as comparative statics and thus facilitate their use in the everyday risk management process of the insurance firms. The potential use of the proposed risk measures in insurance is illustrated by two concrete applications, capital risk allocation and premia calculation under uncertainty.     

无穷范数松弛的低复杂度超奈奎斯特检测

超奈奎斯特(Faster-than-Nyquist,FTN)速率传输可以有效提高频谱效率，但这种非正交传输方式引入的严重码间串扰相应提高了接收端的处理难度。针对该问题，设计了一种基于循环成块传输的低复杂度检测算法。最优检测被建模为无约束的二元二次规划(Boolean Quadratic Program,BQP)问题，为了求解该NP-hard问题，采用无穷范数约束松弛原问题的非凸可行解集，并基于次梯度下降法提出松弛问题的有效优化算法。数值仿真结果表明，所提算法在误比特率(Bit Error Rate,BER)性能上优于频域均衡，且在可接受的性能损失范围内算法执行效率远高于理论最优的最大似然序列估计(Maximum Likelihood Sequence Estimation,MLSE)。     

CONVEX RISK MEASURES FOR GOOD DEAL BOUNDS

We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no‐arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized by several equivalent properties and in particular, we see that a convex risk measure is a good deal valuation only if it is given as a risk indifference price. An application to shortfall risk measure is given. In addition, we show that the no‐free‐lunch (NFL) condition is equivalent to the existence of a relevant convex risk measure, which is a good deal valuation. The relevance turns out to be a condition for a good deal valuation to be reasonable. Further, we investigate conditions under which any good deal valuation is relevant.     

General convex order on risk aggregation

Using a general notion of convex order, we derive general lower bounds for risk measures of aggregated positions under dependence uncertainty, and this in arbitrary dimensions and for heterogeneous models. We also prove sharpness of the bounds obtained when each marginal distribution has a decreasing density. The main result answers a long-standing open question and yields an insight in optimal dependence structures. A numerical algorithm provides bounds for quantities of interest in risk management. Furthermore, our numerical results suggest that the bounds obtained in this paper are generally sharp for a broader class of models.     

INDIFFERENCE PRICE WITH GENERAL SEMIMARTINGALES

Using duality methods, we prove several key properties of the indifference price π for contingent claims. The underlying market model is very general and the mathematical formulation is based on a duality naturally induced by the problem. In particular, the indifference price π turns out to be a convex risk measure on the Orlicz space induced by the utility function.     

ROBUST ASSET ALLOCATION WITH BENCHMARKED OBJECTIVES

In this paper, we introduce a new approach for finding robust portfolios when there is model uncertainty. It differs from the usual worst‐case approach in that a (dynamic) portfolio is evaluated not only by its performance when there is an adversarial opponent (“nature”), but also by its performance relative to a stochastic benchmark. The benchmark corresponds to the wealth of a fictitious benchmark investor who invests optimally given knowledge of the model chosen by nature, so in this regard, our objective has the flavor of min–max regret. This relative performance approach has several important properties: (i) optimal portfolios seek to perform well over the entire range of models and not just the worst case, and hence are less pessimistic than those obtained from the usual worst‐case approach; (ii) the dynamic problem reduces to a convex static optimization problem under reasonable choices of the benchmark portfolio for important classes of models including ambiguous jump‐diffusions; and (iii) this static problem is dual to a Bayesian version of a single period asset allocation problem where the prior on the unknown parameters (for the dual problem) correspond to the Lagrange multipliers in this duality relationship. This dual static problem can be interpreted as a less pessimistic alternative to the single period worst‐case Markowitz problem. More generally, this duality suggests that learning and robustness are closely related when benchmarked objectives are used.     

STABILITY OF THE UTILITY MAXIMIZATION PROBLEM WITH RANDOM ENDOWMENT IN INCOMPLETE MARKETS

We perform a stability analysis for the utility maximization problem in a general semimartingale model where both liquid and illiquid assets (random endowments) are present. Small misspecifications of preferences (as modeled via expected utility), as well as views of the world or the market model (as modeled via subjective probabilities) are considered. Simple sufficient conditions are given for the problem to be well posed, in the sense that the optimal wealth and the marginal utility‐based prices are continuous functionals of preferences and probabilistic views.     

A constraint-free approach to optimal reinsurance

Reinsurance is available for a reinsurance premium that is determined according to a convex premium principle H. The first insurer selects the reinsurance coverage that maximizes its expected utility. No conditions are imposed on the reinsurer's payment. The optimality condition involves the gradient of H. For several combinations of H and the first insurer's utility function, closed-form formulas for the optimal reinsurance are given. If H is a zero utility principle (for example, an exponential principle or an expectile principle), it is shown, by means of Borch's Theorem, that the optimal reinsurer's payment is a function of the total claim amount and that this function satisfies the so-called 1-Lipschitz condition. Frequently, authors impose these two conclusions as hypotheses at the outset.