Random attractors for stochastic differential equations driven by two-sided Lévy processes

Abstract

In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.     

Stochastic optimal control and algorithm of the trajectory of horizontal wells

This paper presents a nonlinear, multi-phase and stochastic dynamical system according to engineering background. We show that the stochastic dynamical system exists a unique solution for every initial state. A stochastic optimal control model is constructed and the sufficient and necessary conditions for optimality are proved via dynamic programming principle. This model can be converted into a parametric nonlinear stochastic programming by integrating the state equation. It is discussed here that the local optimal solution depends in a continuous way on the parameters. A revised Hooke–Jeeves algorithm based on this property has been developed. Computer simulation is used for this paper, and the numerical results illustrate the validity and efficiency of the algorithm.     

基于批发价契约的多零售商横向转载的供应链协调

研究了多零售商横向转载的供应链批发价契约协调问题。以包含一个制造商和多个零售商的供应链系统为研究对象,基于批发价契约研究了零售商转载下的供应链协调问题,获取了批发价契约可以协调零售商存在横向转载的供应链的理论证据,并给出了相应的产生供应链协调的条件,详细分析了协调情形下供应链系统最优订货量与多零售商无转载及单报童等情形下的最优订货量之间的关系。进一步研究了制造商与零售商在Stackelberg博弈下,零售商横向转载对制造商收益的影响,并提出了在Stackelberg博弈模型中,批发价契约也有可能促使制造商选择供应链系统最优订货量所对应的批发价格,使得供应链协调,且给出了此种协调产生的具体条件。数值算例则对两种供应链协调情形下的订货量、批发价格及期望收益进行了计算与仿真。研究表明,批发价契约可能会使得多零售商存在转载的供应链实现协调,传统的双重边际化效应将会由于制造商和零售商的理性而被弱化。     

Coordination mechanisms of supply chain systems

Supply chain management (SCM) has become an important management paradigm. As supply chain members are often separate and independent economic entities, a key issue in SCM is to develop mechanisms that can align their objectives and coordinate their activities so as to optimize system performance. In this paper, we provide a review of coordination mechanisms of supply chain systems in a framework that is based on supply chain decision structure and nature of demand. This framework highlights the behavioral aspects and information need in the coordination of a supply chain. The identification of these issues points out several directions of future research in this area.     

突发事件下需求信息不对称时的供应链协调应对

协调供应链系统使其具有抗突发事件性的研究是供应链管理的核心议题之一。为了分析突发事件下需求信息不对称时的供应链协调问题,考虑由一个制造商和一个零售商组成的供应链,在随机市场需求下,首先分析了数量折扣契约对供应链的协调作用;然后探讨了突发事件导致市场需求发生变化且变化后的需求信息是不对称信息时数量折扣契约对供应链的协调作用,研究表明:基准的数量折扣契约对突发事件下的供应链不再发挥协调作用,为此,给出了供应链应对突发事件的最优应对策略,并调整了原来的数量折扣契约使其具有抗突发事件性。最后,应用一个算例对比加以说明。     

随机非线性物价模型的最优控制

将政府对价格系统的宏观调控作为外部控制力,建立受控的随机非线性物价模型;利用拟Hamilton系统随机平均法和随机动态规划原理的非线性随机控制策略对系统实施最优控制,控制目标是实现系统的稳定性变大;并通过对比控制前后的Lyapunov指教值说明了控制的有效性.     

Stochastic optimal control of first-passage failure for coupled Duffing–van der Pol system under Gaussian white noise excitations

In this paper, the stochastic averaging method of quasi-non-integrable-Hamiltonian systems is applied to Duffing–van der Pol system to obtain partially averaged Ito stochastic differential equations. On the basis of the stochastic dynamical programming principle and the partially averaged Ito equation, dynamical programming equations for the reliability function and the mean first-passage time of controlled system are established. Then a non-linear stochastic optimal control strategy for coupled Duffing–van der Pol system subject to Gaussian white noise excitation is taken for investigating feedback minimization of first-passage failure. By averaging the terms involving control forces and replacing control forces by the optimal ones, the fully averaged Ito equation is derived. Thus, the feedback minimization for first-passage failure of controlled system can be obtained by solving the final dynamical programming equations. Numerical results for first-passage reliability function and mean first-passage time of the controlled and uncontrolled systems are compared in illustrative figures to show effectiveness and efficiency of the proposed method.     

Stabilization of a stochastic nonlinear economy

Methodology for dynamical systems described by nonlinear stochastic differential equations is applied to economic systems, particularly the Phillips-Turnovsky model, generalizing the model further to nonlinear terms and stochastic coefficients and inputs, to determine expectations and correlation matrices for the variables which represent solution processes of a set of coupled equations. The theory is quite general and will apply if the model is changed to use different nonlinear forms, e.g., for taxes as a function of income, or different statistics for fluctuations (which need not be stationary or Gaussian or small), or, to include retarded effects.     

Stochastic optimal time-delay control of quasi-integrable Hamiltonian systems

A nonlinear stochastic optimal time-delay control strategy for quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal control problem of quasi-integrable Hamiltonian system with time-delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed feedback control forces are approximated by the control forces without time-delay and the original problem is converted into a stochastic optimal control problem without time-delay. After that, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. As an example, the stochastic time-delay optimal control of two coupled van der Pol oscillators under stochastic excitation is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.     

随机弹性需求下供应链销售价格与提前期联合决策及协调机制

研究了市场需求随机且对销售价格及提前期敏感的条件下,供应链如何制定最优的销售价格、提前期以及库存因子使得利润最大化。首先,分析了集中决策与分散决策供应链的最优决策,发现集中决策与分散决策模式下的提前期与库存因子是相同的,但分散决策下的销售价格更高、期望利润更低,且分散决策与集中决策供应链的利润之比随着需求对价格的敏感程度的增大而增大。然后建立了收入共享与成本分担的协调机制,并分析了其最优决策。研究发现,协调决策机制可以有效地压缩提前期,并能同时使供应链达到帕累托改进,但不能达到完美协调,而且还发现协调决策下制定的最优销售价格比分散决策要低。最后通过数值计算对三种供应链决策模式下的绩效进行了比较分析,结果表明供应链利润随着需求方差增大而减小,提前期压缩程度随着材料成本承担比例增大而增大。上述结论可以为企业制定销售价格与提前期决策以及企业间的协调提供有益的指导。     

Performance metrics in supply chain management

This survey paper starts with a critical analysis of various performance metrics for supply chain management (SCM), used by a specific manufacturing company. Then it summarizes how economic theory treats multiple performance metrics. Actually, the paper proposes to deal with multiple metrics in SCM via the balanced scorecard — which measures customers, internal processes, innovations, and finance. To forecast how the values of these metrics will change — once a supply chain is redesigned — simulation may be used. This paper distinguishes four simulation types for SCM: (i) spreadsheet simulation, (ii) system dynamics, (iii) discrete-event simulation, and (iv) business games. These simulation types may explain the bullwhip effect, predict fill rate values, and educate and train users. Validation of simulation models requires sensitivity analysis; a statistical methodology is proposed. The paper concludes with suggestions for a possible research agenda in SCM. A list with 50 references for further study is included.     

Market selection decisions for inventory models with price-sensitive demand

In the majority of classical inventory theory literature, demand arises from exogenous sources upon which the firm has little or no control. In many practical contexts, however, aggregate demand is comprised of individual demands from a number of distinct customers or markets. This introduces new dimensions to supply chain planning problems involving the selection of markets or customers to include in the demand portfolio. We present a nonlinear, combinatorial optimization model to address planning decisions in both deterministic and stochastic settings, where a firm constructs a demand portfolio from a set of potential markets having price-sensitive demands. We first consider a pricing strategy that dictates a single price throughout all markets and provide an efficient algorithm for maximizing total profit. We also analyze the model under a market-specific pricing policy and describe its optimal solution. An extensive computational study characterizes the effects of key system parameters on the optimal value of expected profit, and provides some interesting insights on how a given market’s characteristics can affect optimal pricing decisions in other markets.     

Uncertainty investigations in nonlinear aeroelastic systems

In this paper, the stochastic collocation method (SCM) is applied to investigate the nonlinear behavior of an aeroelastic system with uncertainties in the system parameter and the initial condition. Numerical case studies for problems with uncertainties are carried out. In particular, the performance of the SCM is compared with solutions based on other computational techniques such as Monte Carlo simulation, Wiener chaos expansion and wavelet chaos expansion. From the computational results, we conclude that the SCM is an effective tool to study a nonlinear aeroelastic system with random parameters.     

An optimization approach for supply chain management models with quantity discount policy

Owing to the difficulty of treating nonlinear functions, many supply chain management (SCM) models assume that the average prices of materials, production, transportation, and inventory are constant. This assumption, however, is not practical. Vendors usually offer quantity discounts to encourage the buyers to order more, and the producer intends to discount the unit production cost if the amount of production is large. This study solves a nonlinear SCM model capable of treating various quantity discount functions simultaneously, including linear, single breakpoint, step, and multiple breakpoint functions. By utilizing the presented linearization techniques, such a nonlinear model is approximated to a linear mixed 0–1 program solvable to obtain a global optimum.     

随机需求下有产能约束的技术创新策略研究

在随机需求和技术变革的环境下,基于有产能约束的单供应商-单零售商的供应链结构,研究供应商分销价格决策和技术创新策略以及零售商订货决策。建立了三阶段Stackelberg博弈模型,通过逆推方法求得了供应商最优分销价格和技术创新策略以及零售商最优订货量,深入探讨了供应商产能、新技术出现概率以及市场需求期望与波动分别对供应商、零售商和供应链整体利润的影响。结果表明当供应商产能不足时进行技术创新会提高供应商和供应链的利润,但零售商因间接承担供应商技术创新的投资成本而利润下降;当供应商产能过剩时进行技术创新则会降低供应商及供应链的利润,而零售商的利润增加。新技术出现概率增加会提高供应链各成员的利润;提高市场需求期望并减小市场波动对供应商及供应链有利,但可能会降低零售商的利润。     

Uncertainty Quantification for Systems with Random Initial Conditions Using Wiener–Hermite Expansions

A number of engineering problems, including laminar-turbulent transition in convectively unstable flows, require predicting the evolution of a nonlinear dynamical system under uncertain initial conditions. The method of Wiener–Hermite expansion is an attractive alternative to modeling methods, which solve for the joint probability density function of the stochastic amplitudes. These problems include the "curse of dimensionality" and closure problems. In this paper, we apply truncated Wiener–Hermite expansions with both fixed and time-varying bases to a model stochastic system with three degrees of freedom. The model problem represents the combined effects of quadratic nonlinearity and stochastic initial conditions in a generic setting and occurs in related forms in both classical dynamics, turbulence theory, and the nonlinear theory of hydrodynamic stability. In this problem, the truncated Wiener–Hermite expansions give a good account of short-time behavior, but not of the long-time relaxation characteristic of this system. It is concluded that successful application of truncated Wiener–Hermite expansions may require special adaptations for each physical problem.     

碳减排补贴与碳税下的供应链碳减排决策优化与协调

针考虑碳税政策和政府对制造商的碳减排技术创新投入补贴,构建了供应链博弈模型,根据协调条件给出了相应的契约形式及参数。通过对供应链模型的求解,得到了碳税和碳减排技术创新投入补贴下的最优碳减排量和最优订货量。数值分析结果表明:碳减排技术创新投入的补贴比例变化对供应链的最优碳减排量、订货量及期望利润的影响有较大影响,而且是成正向变化关系,而且在其他参数不变及供应链碳减排量一定的情况下,碳税与碳减排补贴比例呈反向变化关系,进而可为政府碳减排补贴政策制定及供应链碳减排优化提供科学的决策依据。     

基于时间可控和随机损失的生鲜农产品供应链协调

生鲜农产品不仅是易腐的,其寿命还具有随机性,在订货和由产地运往远方销地的过程中,面临随机供给与需求的匹配、变质损失与物流成本的平衡这两重难题。基于现代物流条件下物流时间的可控性,分别建立了分散控制和集中控制供应链的决策模型,分析得到了它们的最优订货量和最优物流时间。在此基础上,设计了一组联合契约,推导出了实现生鲜农产品供应链协调时契约参数取值范围的计算公式。研究结果表明:通过选择合理的契约参数φ,联合契约能够完美协调生鲜农产品供应链、实现供需双方帕累托最优。进一步,对变质率参数和需求价格弹性系数进行了敏感性分析,为生鲜农产品供应链实践提供了一些管理启示。     

资金成本下延迟支付期限的收益分享契约

针对由单一制造商和单一零售商组成的供应链系统,建立了随机需求下考虑有资金成本延迟支付期限的收益分享契约模型,研究两种决策(分散化和集中化)下,延迟支付期限是如何影响资金约束零售商的最优订货数量以及供应链系统的利润.分析表明,不管是分散化还是集中化决策,考虑有资金成本延迟支付期限的收益分享契约使供应链系统的最优订货量和利润都增大.最后,运用数值分析验证了结论.     

Interpreting supply chain dynamics: A quasi-chaos perspective

Chaotic phenomena, chaos amplification and other interesting nonlinear behaviors have been observed in supply chain systems. Chaos can be defined theoretically if the dynamics under study are produced only by deterministic factors. However, deterministic settings rarely present themselves in reality. In fact, real data are typically unknown. How can the chaos theory and its related methodology be applied in the real world? When the demand is stochastic, the interpretation and distribution of the Lyapunov exponents derived from the effective inventory at different supply chain levels are not similar to those under deterministic demand settings. Are the observed dynamics of the effective inventory random, chaotic, or simply quasi-chaos? In this study, we investigate a situation whereby the chaos analysis is applied to a time series as if its underlying structure, deterministic or stochastic, is unknown. The result shows clear distinction in chaos characterization between the two categories of demand process, deterministic vs. stochastic. It also highlights the complexity of the interplay between stochastic demand processes and nonlinear dynamics. Therefore, caution should be exercised in interpreting system dynamics when applying chaos analysis to a system of unknown underlying structure. By understanding this delicate interplay, decision makers have the better chance to tackle the problem correctly or more effectively at the demand end or the supply end.