Stochastic Dominance Research Articles

Adjustable light robust optimization with second order stochastic dominance constraints

Robust optimization suffers from excessive conservatism and infeasibility due to stringent requirements on all constraints within the uncertainty set, rendering robust solutions impractical. The light robust approach has been proposed to mitigate these challenges. This method aims to identify a solution that minimizes the weighted sum of all constraint violations while adhering to a predetermined limit on the deterioration of expected return. Drawing inspiration from light robustness, our paper introduces the adjustable light robust optimization models with second-order stochastic dominance constraints. We empirically examine their feasibility, out-of-sample performance, and the dynamic trade-off between return and robustness.

ODC and ROC Curves, Comparison Curves and Stochastic Dominance†

SummaryWe discuss two novel approaches to inter‐distributional comparisons in the classical two‐sample problem. Our starting point is properly standardised and combined, very popular in several areas of statistics and data analysis, ordinal dominance and receiver characteristic curves, denoted by ODC and ROC, respectively. The proposed new curves are termed the comparison curves. Their estimates, being weighted rank processes on (0,1), form the basis of inference. These weighted processes are intuitive, well‐suited for visual inspection of data at hand and are also useful for constructing some formal inferential procedures. They can be applied to several variants of two‐sample problem. Their use can help improve some existing procedures both in terms of power and the ability to identify the sources of departures from the postulated model. To simplify interpretation of finite sample results, we restrict attention to values of the processes on a finite grid of points. This results in the so‐called bar plots (B‐plots), which readably summarise the information contained in the data. What is more, we show that B‐plots along with adjusted simultaneous acceptance regions provide principled information about where the model departs from the data. This leads to a framework that facilitates identification of regions with locally significant differences.We show an implementation of the considered techniques to a standard stochastic dominance testing problem. Some min‐type statistics are introduced and investigated. A simulation study compares two tests pertinent to the comparison curves to well‐established tests in the literature and demonstrates the strong and competitive performance of the former in many typical situations. Some real data applications illustrate simplicity and practical usefulness of the proposed approaches. A range of other applications of considered weighted processes is briefly discussed too.

Law-invariant return and star-shaped risk measures

This paper presents novel characterization results for classes of law-invariant star-shaped functionals. We begin by establishing characterizations for positively homogeneous and star-shaped functionals that exhibit second- or convex-order stochastic dominance consistency. Building on these characterizations, we proceed to derive Kusuoka-type representations for these functionals, shedding light on their mathematical structure and intimate connections to Value-at-Risk and Expected Shortfall. Furthermore, we offer representations of general law-invariant star-shaped functionals as robustifications of Value-at-Risk. Notably, our results are versatile, accommodating settings that may, or may not, involve monotonicity and/or cash-additivity. All of these characterizations are developed within a general locally convex topological space of random variables, ensuring the broad applicability of our results in various financial, insurance and probabilistic contexts.

Conditions for extrapolating differences in consumption to differences in welfare

AbstractWe characterize conditions under which a better consumption distribution implies higher welfare. Specifically, here “better consumption” means first‐order stochastic dominance, and “higher welfare” means higher expected utility for every subpopulation of individuals with the same utility function. Although this implication seems natural, we first provide a counterexample wherein better consumption risk allocation outweighs lower consumption. We then show that higher expected utility results from higher consumption in different settings, including fixed dependence (fixed copula) between consumption and individual risk preferences, or alternatively using the rank invariance assumption from the treatment effects literature. These are discussed in several real‐world examples.

Investments in transmission lines and storage units considering second-order stochastic dominance constraints

Increasing intermittent renewable generation to meet the climate goals entails a deep transformation of current power systems. The transmission system must adapt to ensure a rapid and flexible response to the changes in the energy flows. Flexibility can be provided by reinforcing the interconnection among all the market agents and/or by installing facilities with fast response to the changes. In this work, we study the investment problem of a central planner that seeks to expand the transmission network and install storage units considering decarbonizing measures. We propose a two-stage stochastic problem with uncertainty on the demand growth and including representative days to characterize the hourly demand and renewable power variability. To obtain better expansion strategies according to a limit on the carbon emissions, second-order stochastic dominance constraints are imposed. Numerical analyses based on the power system of the Canary island (Spain) are provided. The results show that to install storage units is key to efficiently integrate renewable generation. The investments in the transmission system are mostly in lower-voltage lines. The formulation with stochastic dominance constraints results in higher second-stage investments, allowing a better adaptation to the demand growth evolution.

Estimating Probability Weighting Functions through Option Pricing Bounds

Abstract This paper proposes a novel approach to estimating the probability weighting function (PWF) of investors in the option market. We match observed option prices to the option pricing bounds under stochastic dominance rules. Using 1-month S&P 500 index option data, we find that investors could subjectively employ an inverse S-shaped probability weighting function, which increases the weights on extreme returns and asymmetrically assigns greater weights to extremely low returns than to extremely high returns. Our findings suggest that the inverse S-shaped nature of the PWFs is robust across various estimation specifications, such as adopting an alternative methodology to construct the return distribution, and employing option data with different times to maturity. (JEL G12)

Exploring skill distribution tails through stochastic dominance

SummaryLocation choices of differently skilled workers are analyzed in previous work on labor mobility, which proposes a model that suggests thicker tails in the skill distributions of large cities. This paper replicates the empirical findings of this work by using quantile regression and density plots as employed in the existing study, while also suggesting an alternative testing method for thick tails in the form of an initial stochastic dominance test. The test reveals clear evidence of a thicker lower tail, but the results are less clear for the upper tail, which raises some questions on how to best handle extreme upper tails of skill distributions.

Probabilistic linguistic prospect outranking risk decision making method based on stochastic dominance and application in emergency plan evaluation

In the decision-making process, probabilistic linguistic term sets (PLTSs) could represent decision-makers’ uncertain evaluation information. The combination of probabilistic and linguistic information assists decision-makers to depict their evaluation information more flexibly. In this paper, a prospect outranking decision-making method based on stochastic dominance is proposed for solving the multi-attribute risk decision-making problem under the probabilistic linguistic information environment. Firstly, to fully preserve the original probabilistic linguistic information, we give the definitions of the generalized probability distribution and the corresponding generalized cumulative distribution function of a PLTS for defining the distance measure between PLTSs. Secondly, based on the positive and negative ideal cumulative distribution functions, we propose the probabilistic linguistic prospect value functions. After that, considering the uncertainty of the relationship between alternatives, several kinds of novel prospect stochastic dominance and almost stochastic dominance relations are defined, which consider the decision makers’ distinction between gains and losses to measure the subtle relationship between alternatives. The stochastic dominance approach and the almost stochastic dominance approach under the probabilistic linguistic decision-making environment are put forward. Besides, we use the superiority and inferiority ranking method to solve the dominance ranking problem of the uncertainty relationship between any two alternatives. Finally, the proposed method is applied in the evaluation of emergency plans to show the accuracy and reliability, and comparative analyses are made to illustrate its availability.

Exact Characterization of the Jointly Optimal Restocking and Auditing Policy in Inventory Systems with Record Inaccuracy

We present a continuous-time stochastic model of an inventory system with record inaccuracy. In this formulation, demand is modeled by a point process and is observable only when it leads to sales. In addition to demand that can reduce the stock, an unobservable stochastic loss process can also reduce the stock. The retailer’s goal is to identify the restocking and auditing policy that minimizes the expected discounted cost of carrying a product over an infinite horizon. We analytically characterize the optimal restocking and jointly optimal auditing policy. We prove that the optimal restocking policy is a threshold policy. Our proof of this result is based on a coupling argument that is valid for any demand and loss model. Unlike the optimal restocking policy, the jointly optimal auditing policy is not of threshold type. We show that a complete proof of this statement cannot be obtained by solely resorting to the first-order stochastic dominance property of the Bayesian shelf stock distribution induced by the demand and loss process. Instead, our characterization of the jointly optimal auditing policy is based on proving that the dynamics of the shelf stock distribution constitute a (strictly) sign-regular kernel. To our knowledge, this is the first paper that characterizes the optimal policy of a complex control problem by establishing sign regularity of its underlying Markovian dynamics. Our theoretical results lead to a fast algorithm for computing the exact jointly optimal auditing/restocking policy over the problem’s entire state space. This enables comparative statics analysis, which allows us to determine how inventory record inaccuracy affects the economic significance of various cost drivers. This, in turn, allows us to determine when or, better, under what conditions auditing can be an effective tool for reducing the total cost.

An Unexpected Stochastic Dominance: Pareto Distributions, Dependence, and Diversification

Diversification is generally regarded as an efficient tool to reduce portfolio risks. In “An unexpected stochastic dominance: Pareto distributions, dependence, and diversification,” Chen, Embrechts, and Wang showed that the weighted average of independent and identically distributed (i.i.d.) Pareto random variables with infinite mean is larger than one such random variable in the sense of first-order stochastic dominance, and thus diversification is, surprisingly, worse than no diversification. The relation implies superadditivity of value-at-risk, a regulatory risk measure used in the finance and insurance sectors. The obtained relation also holds under some form of negative dependence.

Choice under uncertainty and cognitive load

Does exposure to cognitive load affect key properties of economic behavior? In this experiment, subjects face a series of simple binary decision tasks between prospects, testing for monotonicity in monetary payments, consistency with (first-order) stochastic dominance, reduction of compound lotteries, risk attitudes, and ambiguity attitudes. Cognitive load is manipulated via simultaneous memory tasks. Our data show treatment differences resulting from cognitive load for decision tasks with risky outcomes. However, cognitive load has no impact on monotonicity and ambiguity attitudes. Under a dual-process view of human decision-making, our findings suggest that ambiguity attitudes and preferences for “more certain money” are intuitive, not reasoned.

Microeukaryotic plankton community dynamics under ecological water replenishment: Insights from eDNA metabarcoding

Ecological water replenishment (EWR) is an important strategy for river restoration globally, but timely evaluation of its ecological effects at a large spatiotemporal scale to further adjust the EWR schemes is of great challenge. Here, we examine the impact of EWR on microeukaryotic plankton communities in three distinct river ecosystems through environmental DNA (eDNA) metabarcoding. The three ecosystems include a long-term cut-off river, a short-term connected river after EWR, and long-term connected rivers. We analyzed community stability by investigating species composition, stochastic and deterministic dynamics interplay, and ecological network robustness. We found that EWR markedly reduced the diversity and complexity of microeukaryotic plankton, altered their community dynamics, and lessened the variation within the community. Moreover, EWR disrupted the deterministic patterns of community organization, favoring dispersal constraints, and aligning with trends observed in naturally connected rivers. The shift from an isolated to a temporarily connected river appeared to transition community structuring mechanisms from deterministic to stochastic dominance, whereas, in permanently connected rivers, both forces concurrently influenced community assembly. The ecological network in temporarily connected rivers post-EWR demonstrated significantly greater stability and intricacy compared to other river systems. This shift markedly bolstered the resilience of the ecological network. The eDNA metabarcoding insights offer a novel understanding of ecosystem resilience under EWR interventions, which could be critical in assessing the effects of river restoration projects throughout their life cycle.

Quasi Second-Order Stochastic Dominance Model for Balancing Wildfire Risks and Power Outages due to Proactive Public Safety De-Energizations

Quasi Second-Order Stochastic Dominance Model for Balancing Wildfire Risks and Power Outages due to Proactive Public Safety De-Energizations

How do price (risk) changes influence farmers’ preferences to reduce fertilizer application?

AbstractThe decision of farmers to reduce fertilizer applications and, thus, the achievement of agri‐environmental policy goals interacts with market price developments. In this study, we analyze how changes in price levels and volatility over time (i.e., 1991–2006 vs. 2007–2022) affected farmers’ preferences to reduce fertilizer application using statistical inferences of stochastic dominances. The analysis considers two cropping systems and fertilizer reduction measures: (i) grassland‐based milk production and the use of legumes and (ii) wheat production and the use of variable rate application. We show that the economic value of reducing fertilizer increased over time in both grassland‐based milk and wheat production. However, only in the case of wheat production was the reduction in fertilizer application observed as more risk‐reducing over time. In contrast, in grassland‐based milk production, the co‐movement of fertilizer and milk prices canceled out the increase in risk reduction. We conclude that changes in market price, along with agri‐environmental subsidies, can increasingly incentivize the reduction of fertilizer use.

Risk Sharing with Lambda Value at Risk

In this paper, we study the risk-sharing problem among multiple agents using lambda value at risk ([Formula: see text]) as their preferences via the tool of inf-convolution, where [Formula: see text] is an extension of value at risk ([Formula: see text]). We obtain explicit formulas of the inf-convolution of multiple [Formula: see text] with monotone Λ and explicit forms of the corresponding optimal allocations, extending the results of the inf-convolution of [Formula: see text]. It turns out that the inf-convolution of several [Formula: see text] is still a [Formula: see text] under some mild condition. Moreover, we investigate the inf-convolution of one [Formula: see text] and a general monotone risk measure without cash additivity, including [Formula: see text], expected utility, and rank-dependent expected utility as special cases. The expression of the inf-convolution and the explicit forms of the optimal allocation are derived, leading to some partial solution of the risk-sharing problem with multiple [Formula: see text] for general Λ functions. Finally, we discuss the risk-sharing problem with [Formula: see text], another definition of lambda value at risk. We focus on the inf-convolution of [Formula: see text] and a risk measure that is consistent with the second-order stochastic dominance, deriving very different expression of the inf-convolution and the forms of the optimal allocations.

Random distortion risk measures

This paper presents one type of random risk measures, named as the random distortion risk measure. The random distortion risk measure is a generalization of the traditional deterministic distortion risk measure by randomizing the deterministic distortion function and the risk distribution respectively, where a stochastic distortion is introduced to randomize the distortion function, and a sub-σ-algebra is introduced to illustrate the influence of the known information on the risk distribution. Some theoretical properties of the random distortion risk measure are provided, such as normalization, conditional positive homogeneity, conditional comonotonic additivity, monotonicity in stochastic dominance order, and continuity from below, and a method for specifying the stochastic distortion and the sub-σ-algebra is provided. Based on some stochastic axioms, a representation theorem of the random distortion risk measure is proved. For considering the randomization of a given deterministic distortion risk measure, some families of random distortion risk measures are introduced with the stochastic distortions constructed from a Poisson process, a Brownian motion, and a Dirichlet process, respectively. A numerical analysis is carried out for showing the influence of the stochastic distortion and the sub-σ-algebra by focusing on the sample statistics, empirical distributions, and tail behavior of the random distortion risk measures.

Portfolio reshaping under 1st-order stochastic dominance constraints by the exact penalty function methods

The paper studies a financial portfolio selection problem under 1st-order stochastic dominance constraints. These constraints constitute lower bounds on the return profile of the portfolio. In particular, they allow searching for a better portfolio than some reference portfolio by comparing their cumulative distribution functions. Candidate objective functions are the average return, a value at risk, or the average value at risk. The optimization problems obtained are computationally hard because of possibly non-convex constraints and possibly discontinuous objective functions. In the case of a discrete distribution of the return, we develop numerical procedures to solve the problem. The proposed approach uses new exact penalty functions to tackle with the 1st-order stochastic dominance constraints. The resulting penalized objective function is further optimized be the stochastic successive smoothing method as a local optimizer within some branch and bound global optimization scheme. The approach is numerically and graphically illustrated on small portfolio selection problems up to dimension 10.

Weighted Utility and Optimism/Pessimism: A Decision-Theoretic Foundation of Various Stochastic Dominance Orders

Weighted Utility and Optimism/Pessimism: A Decision-Theoretic Foundation of Various Stochastic Dominance Orders

Two New Tests for the Relationships of Stochastic Dominance

Two New Tests for the Relationships of Stochastic Dominance

Retrieving almost stochastic Dominance momentum in Taiwan stock market

We propose new momentum strategies based on the Almost Stochastic Dominance rules. Relative to classic momentum, our novel strategy achieves better risk-adjusted performance, and exhibits lower volatility and reduced negative skewness in returns. The abnormal returns are statistically and economically significant when testing against alternative common risk factors. Most interestingly, the strategy's ability to generate excess returns is particularly pronounced when using shorter-term ranking and holding periods. Notable improvements in computational efficiency suggest that practical implementability of the strategy shall prevail in cases where a large span of assets is considered. In empirical studies, we apply the new momentum strategies to the Taiwan stock market and compare them with some existing momentum strategies.