PSD准则与累积展望理论不一致的讨论

Levy M和Levy H设计了一些实验,应用展望随机占优准则(PSD)给出方案的偏好序,实验结果拒绝了累积展望理论;而Wakker认为实验的数据恰恰支持了累积展望理论.试图讨论PSD准则拒绝累积展望理论的原因,PSD准则以概率的线性加权为基础,累积展望理论的重要假设是对概率进行了编译,二者存在着差异,据此,提出广义展望随机占优准则,认为在此准则下Levy M和Levy H的实验数据无法给出方案的偏好序,因此不能否认累积展望理论的正确性.     

Characterizations on Almost Stochastic Dominance Revisited

Almost stochastic dominance has been receiving more attention in the financial and economic literature. In this short note, we characterize the almost first- and second-degree stochastic dominance by requiring one distribution to be ``close to' a new distribution that dominates or is dominated by another distribution in the traditional sense of the first- and second-order stochastic dominance, respectively. We also investigate the concept of almost stochastic dominance for unbounded random variables.     

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??Almost stochastic dominance has been receiving more attention in the financial and economic literature. In this short note, we characterize the almost first- and second-degree stochastic dominance by requiring one distribution to be ``close to' a new distribution that dominates or is dominated by another distribution in the traditional sense of the first- and second-order stochastic dominance, respectively. We also investigate the concept of almost stochastic dominance for unbounded random variables.     

On the relative efficiency of nth order and DARA stochastic dominance rules

It is known that third order stochastic dominance implies DARA dominance while no implications exist between higher orders and DARA dominance. A recent contribution points out that, with regard to the problem of determining lower and upper bounds for the price of a financial option, the DARA rule turns out to improve the stochastic dominance criteria of any order. In this paper the relative efficiency of the ordinary stochastic dominance and DARA criteria for alternatives with discrete distributions are compared, in order to see if the better performance of DARA criterion is also suitable for other practical applications. Moreover, the operational use of the stochastic dominance techniques for financial choices is deepened.     

二阶随机控制变点的Kolmogrov型检验和估计

基于Kolmogrov型统计量和Kiefer过程,对一样本情形,我们讨论了二阶随机控制变点的检验和估计到了检验统计量的渐近分布且用模拟方法给出了其有限样本的分位数,并证明了变点的估计为强相合的。     

Lévy processes in semisimple Lie groups and stability of stochastic flows

We study the asymptotic stability of stochastic flows on compact spaces induced by Levy processes in semisimple Lie groups. It is shown that the Lyapunov exponents can be determined naturally in terms of root structure of the Lie group and there is an open subset whose complement has a positive codimension such that, after a random rotation, each of its connected components is shrunk to a single moving point exponentially under the flow.

     

Fractional Levy processes on Gel'fand triple and stochastic integration

In this paper, we investigate the long-range dependence of fractional Levy processes on Gel’fand triple and construct stochastic integral with respect to fractional Levy processes for a class of deterministic integrands.     

Multivariate robust second-order stochastic dominance and resulting risk-averse optimization

ABSTRACT

By utilizing a min-biaffine scalarization function, we define the multivariate robust second-order stochastic dominance relationship to flexibly compare two random vectors. We discuss the basic properties of the multivariate robust second-order stochastic dominance and relate it to the nonpositiveness of a functional which is continuous and subdifferentiable everywhere. We study a stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and develop the necessary and sufficient conditions of optimality in the convex case. After specifying an ambiguity set based on moments information, we approximate the ambiguity set by a series of sets consisting of discrete distributions. Furthermore, we design a convex approximation to the proposed stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and establish its qualitative stability under Kantorovich metric and pseudo metric, respectively. All these results lay a theoretical foundation for the modelling and solution of complex stochastic decision-making problems with multivariate robust second-order stochastic dominance constraints.     

Sufficient conditions under which SSD- and MR-efficient sets are identical

Three approaches are commonly used for analyzing decisions under uncertainty: expected utility (EU), second-degree stochastic dominance (SSD), and mean-risk (MR) models, with the mean–standard deviation (MS) being the best-known MR model. Because MR models generally lead to different efficient sets and thus are a continuing source of controversy, the specific concern of this article is not to suggest another MR model. Instead, we show that the SSD- and MR-efficient sets are identical, as long as (a) the risk measure satisfies both positive homogeneity and consistency with respect to the Rothschild and Stiglitz (1970) definition(s) of increasing risk and (b) the choice set includes the riskless asset and satisfies a generalized location and scale property, which can be interpreted as a market model. Under these conditions, there is no controversy among MR models and they all have a decision-theoretic foundation. They also offer a convenient way to compare the estimation error related to the empirical implementation of different MR models.     

Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).     

Relaxations of linear programming problems with first order stochastic dominance constraints

Linear stochastic programming problems with first order stochastic dominance (FSD) constraints are non-convex. For their mixed 0-1 linear programming formulation we present two convex relaxations based on second order stochastic dominance (SSD). We develop necessary and sufficient conditions for FSD, used to obtain a disjunctive programming formulation and to strengthen one of the SSD-based relaxations.     

Robust stochastic dominance and its application to risk-averse optimization

We introduce a new preference relation in the space of random variables, which we call robust stochastic dominance. We consider stochastic optimization problems where risk-aversion is expressed by a robust stochastic dominance constraint. These are composite semi-infinite optimization problems with constraints on compositions of measures of risk and utility functions. We develop necessary and sufficient conditions of optimality for such optimization problems in the convex case. In the nonconvex case, we derive necessary conditions of optimality under additional smoothness assumptions of some mappings involved in the problem.     

On capacity expansion planning under strategic and operational uncertainties based on stochastic dominance risk averse management

A new scheme for dealing with uncertainty in scenario trees is presented for dynamic mixed 0–1 optimization problems with strategic and operational stochastic parameters. Let us generically name this type of problems as capacity expansion planning (CEP) in a given system, e.g., supply chain, production, rapid transit network, energy generation and transmission network, etc. The strategic scenario tree is usually a multistage one, and the replicas of the strategic nodes root structures in the form of either a special scenario graph or a two-stage scenario tree, depending on the type of operational activity in the system. Those operational scenario structures impact in the constraints of the model and, thus, in the decomposition methodology for solving usually large-scale problems. This work presents the modeling framework for some of the risk neutral and risk averse measures to consider for CEP problem solving. Two types of risk averse measures are considered. The first one is a time-inconsistent mixture of the chance-constrained and second-order stochastic dominance (SSD) functionals of the value of a given set of functions up to the strategic nodes in selected stages along the time horizon, The second type is a strategic node-based time-consistent SSD functional for the set of operational scenarios in the strategic nodes at selected stages. A specialization of the nested stochastic decomposition methodology for that problem solving is outlined. Its advantages and drawbacks as well as the framework for some schemes to, at least, partially avoid those drawbacks are also presented.     

Impact of noise in a phytoplankton-zooplankton system

In this paper, we investigate the dynamics of a delayed toxic phytoplankton-two zooplankton system incorporating the effects of Levy noise and white noise. The value of this study lies in two aspects: Mathematically, we first prove the existence of a unique global positive solution of the system, and then we investigate the sufficient conditions that guarantee the stochastic extinction and persistence in the mean of each population. Ecologically, via numerical simulations, we find that the effect of white noise or Levy noise on the stochastic extinction and persistence of phytoplankton and zooplankton are similar, but the synergistic effects of the two noises on the stochastic extinction and persistence of these plankton are stronger than that of single noise. In addition, an increase in the toxin liberation rate or the intraspecific competition rate of zooplankton was found to be capable to increase the biomass of the phytoplankton but decrease the biomass of zooplankton. These results may help us to better understand the phytoplankton-zooplankton dynamics in the fluctuating environments.     

Gains from diversification on convex combinations: A majorization and stochastic dominance approach

By incorporating both majorization theory and stochastic dominance theory, this paper presents a general theory and a unifying framework for determining the diversification preferences of risk-averse investors and conditions under which they would unanimously judge a particular asset to be superior. In particular, we develop a theory for comparing the preferences of different convex combinations of assets that characterize a portfolio to give higher expected utility by second-order stochastic dominance. Our findings also provide an additional methodology for determining the second-order stochastic dominance efficient set.     

Optimization with multivariate stochastic dominance constraints

We consider stochastic optimization problems where risk-aversion is expressed by a stochastic ordering constraint. The constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector. We identify a suitable multivariate stochastic order and describe its generator in terms of von Neumann–Morgenstern utility functions. We develop necessary and sufficient conditions of optimality and duality relations for optimization problems with this constraint. Assuming convexity we show that the Lagrange multipliers corresponding to dominance constraints are elements of the generator of this order, thus refining and generalizing earlier results for optimization under univariate stochastic dominance constraints. Furthermore, we obtain necessary conditions of optimality for non-convex problems under additional smoothness assumptions.     

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??In this paper, we prove the existence and uniqueness of solutions for reflected backward stochastic differential equations driven by a Levy process, in which the reflecting barriers are just right continuous with left limits whose jumps are arbitrary. To derive the above results, the monotonic limit theorem of Backward SDE associated with Levy process is established.     

Random unfriendly seating arrangement in a dining table

A detailed study is made of the number of occupied seats in an unfriendly seating scheme with two rows of seats. An unusual identity is derived for the probability generating function, which is itself an asymptotic expansion. The identity implies particularly a local limit theorem with optimal convergence rate. Our approach relies on the resolution of Riccati equations. We also clarify some simple yet delicate stochastic dominance relations.     

服务系统中位相型分布的随机比较

本文运用应用概率中的随机占优研究位相型(PH)分布的随机比较问题,具体给出在一阶、二阶随机占优下比较两个离散PH分布或两个连续PH分布的充分条件及充分必要条件。研究表明,比较两个离散PH分布可变性的条件与比较两个连续PH分布可变性的条件不同,在二阶随机占优意义下比较两个连续PH分布的条件与均值无关,而比较两个离散PH分布的条件与均值有关。本文的结果可用于研究PH分布的最小变异系数问题和可变性问题,也可用于研究带有PH到达间隔或PH服务的排队系统中到达过程或服务时间可变性对系统队长或等待时间的影响。     

Modeling Stochastic Dominance as Infinite-Dimensional Constraint Systems via the Strassen Theorem

We use the Strassen theorem to solve stochastic optimization problems with stochastic dominance constraints. First, we show that a dominance-constrained problem on general probability spaces can be expressed as an infinite-dimensional optimization problem with a convenient representation of the dominance constraints provided by the Strassen theorem. This result generalizes earlier work which was limited to finite probability spaces. Second, we derive optimality conditions and a duality theory to gain insight into this optimization problem. Finally, we present a computational scheme for constructing finite approximations along with a convergence rate analysis on the approximation quality.