随机微分方程数值解在泄洪风险分析中的应用

根据泄洪过程中库水位过程的随机微分方程,利用数值解方法,模拟了随机干扰下的库水位及其波动状况.采用相应公式计算了洪水漫越坝顶事件的概率以及库水位过程在不同时刻的样本均值.并通过比较在同样强度的随机干扰下库水位的高低状况,确定出各种泄洪方案的优劣,从而对防洪工作具有重要的指导意义.     

Sensitivity Statistical Estimates for Local A Posteriori Inference Matrix-Vector Equations in Algebraic Bayesian Networks over Quantum Propositions

An approach to the sensitivity analysis of local a posteriori inference equations in algebraic Bayesian networks is proposed in this paper. Some basic definitions and formulations are briefly given and the development of the matrix-vector a posteriori inference approach is considered. Some cases of the propagation of deterministic and stochastic evidence in a knowledge pattern with scalar estimates of component truth probabilities over quantum propositions are described. For each of the considered cases, the necessary metrics are introduced, and some transformations resulting in four linear programming problems are performed. The solution of these problems gives the required estimates. In addition, two theorems postulating the covering estimates for the considered parameters are formulated. The results obtained in this work prove the correct application of models and create a basis for the sensitivity analysis of local and global probabilistic-logic inference equations.     

Estimating the bias on the logdeterminant transformation for evolutionary trees

The logarithm of the determinant of a contingency table allows evolutionary information to be recovered from data generated under very general stochastic models. The variance of the estimate of the dissimilarity between taxa is generally estimated with the help of resampling methods. We show that this technique leads to biased estimates, and we derive exact formula. Practical implications are considered.     

Stochastic modelling of reservoir operations

The nature of hydrologic parameters in reservoir management models is uncertain. In mathematical programming models the uncertainties are dealt with either indirectly (sensitivity analysis of a deterministic model) or directly by applying a chance-constrained type of formulation or some of the stochastic programming techniques (LP and DP based models). Various approaches are reviewed in the paper. Moran's theory of storage is an alternative stochastic modelling approach to mathematical programming techniques. The basis of the approach and its application is presented. Reliability programming is a stochastic technique based on the chance-constrained approach, where the reliabilities of the chance constraints are considered as extra decision variables in the model. The problem of random event treatment in the reservoir management model formulation using reliability programming is addressed in this paper.     

Stochastic dual dynamic programming applied to nonconvex hydrothermal models

In this paper we apply stochastic dual dynamic programming decomposition to a nonconvex multistage stochastic hydrothermal model where the nonlinear water head effects on production and the nonlinear dependence between the reservoir head and the reservoir volume are modeled. The nonconvex constraints that represent the production function of a hydro plant are approximated by McCormick envelopes. These constraints are split into smaller regions and the McCormick envelopes are used for each region. We use binary variables for this disjunctive programming approach and solve the problem with a decomposition method. We resort to a variant of the L-shaped method for solving the MIP subproblem with binary variables at any stage inside the stochastic dual dynamic programming algorithm. A realistic large-scale case study is presented.     

Modelling of hydrological persistence for hidden state Markov decision processes

A reservoir in south east Queensland can supply irrigators, industry or domestic users. Stochastic inflow is modelled by a hidden state Markov chain, with three hidden states corresponding to prevailing climatic conditions. A stochastic dynamic program that relies on estimation of the hidden state is implemented. The optimal decisions are compared with those obtained if the hidden state Markov chain model is replaced with a model that relies on the Southern Oscillation Index to define prevailing climatic conditions.     

Dynamic programming for the stochastic burgers equation

We solve a control problem for the stochastic Burgers equation using the dynamic programming approach. The cost functional involves exponentially growing functions and the analog of the kinetic energy; the case of a distributed parameter control is considered. The Hamilton-Jacobi equation is solved by a compactness method and a-priori estimates are obtained thanks to the regularizing properties of the transition semigroup associated to the stochastic Burgers equation; a fixed point argument does not seem to apply here. Entrata in Redazione il 10 dicembre 1998.     

The First Attempt on the Stochastic Calculus on Time Scale

The aim of this article is to study the Doob–Meyer decomposition theorem, ?-stochastic integration and Ito's formula for stochastic processes defined on time scale. The obtained results can be considered as a first attempt on the stochastic calculus on time scale.     

防洪风险分析中的一个二元标值犘狅犻狊狊狅狀点过程模型

该文将洪水的大小和持续时间作为防洪设施的工程风险中不可忽略的因素,提出了以洪水的大小和持续时间为标值的二元标值Ｐｏｉｓｓｏｎ点过程模型,给出了防洪综合风险率的计算公式,并进行了实例计算．     

An enhanced decomposition algorithm for multistage stochastic hydroelectric scheduling

Handling uncertainty in natural inflow is an important part of a hydroelectric scheduling model. In a stochastic programming formulation, natural inflow may be modeled as a random vector with known distribution, but the size of the resulting mathematical program can be formidable. Decomposition-based algorithms take advantage of special structure and provide an attractive approach to such problems. We develop an enhanced Benders decomposition algorithm for solving multistage stochastic linear programs. The enhancements include warm start basis selection, preliminary cut generation, the multicut procedure, and decision tree traversing strategies. Computational results are presented for a collection of stochastic hydroelectric scheduling problems.     

A finite volume method for scalar conservation laws with stochastic time–space dependent flux functions

We propose a new finite volume method for scalar conservation laws with stochastic time–space dependent flux functions. The stochastic effects appear in the flux function and can be interpreted as a random manner to localize the discontinuity in the time–space dependent flux function. The location of the interface between the fluxes can be obtained by solving a system of stochastic differential equations for the velocity fluctuation and displacement variable. In this paper we develop a modified Rusanov method for the reconstruction of numerical fluxes in the finite volume discretization. To solve the system of stochastic differential equations for the interface we apply a second-order Runge–Kutta scheme. Numerical results are presented for stochastic problems in traffic flow and two-phase flow applications. It is found that the proposed finite volume method offers a robust and accurate approach for solving scalar conservation laws with stochastic time–space dependent flux functions.     

Existence,Uniqueness, and Upper Estimates for Solutions of Mcshane Type Stochastic Differential Systems

A system of stochastic differential equations of McShane type is studied. The Schauder fixed point theorem is applied to obtain existence of solutions and Osgood type existence and uniqueness results are derived using successive approximations. The noise processes are not required to be martingales or quasimartingales. As a byproduct of our approach, upper estimates for solutions of McShane type stochastic differential systems are obtained     

Generalized estimates for performance sensitivities of stochastic systems

Three kinds of estimates for performance sensitivities (gradients, Hessians etc.) of stochastic systems are introduced. These estimates are given in general operator form. Their convergence conditions and rate of convergence are presented. Particular attention is given to estimates obtained from a single sample path. Various examples of estimates are considered.     

On randomization of Halton quasi-random sequences

The problem of estimating the error of quasi-Monte Carlo methods by means of randomization is considered. The well known Koksma–Hlawka inequality enables one to estimate asymptotics for the error, but it is not useful in computational practice, since computation of the quantities occurring in it, the variation of the function and the discrepancy of the sequence, is an extremely timeconsuming and impractical process. For this reason, there were numerous attempts to solve the problem mentioned above by the probability theory methods. A common approach is to shift randomly the points of quasi-random sequence. There are known cases of the practical use of this approach, but theoretically it is scantily studied. In this paper, it is shown that the estimates obtained this way are upper estimates. A connection with the theory of cubature formulas with one random node is established. The case of Halton sequences is considered in detail. The van der Corput transformation of a sequence of natural numbers is studied, and the Halton points are constructed with its help. It is shown that the cubature formula with one free node corresponding to the Halton sequence is exact for some class of step functions. This class is explicitly described. The obtained results enable one to use these sequences more effectively for calculating integrals and finding extrema and can serve as a starting point for further theoretical studies in the field of quasi-random sequences.     

Unbiased estimates for gradients of stochastic network performance measures

Three classes of stochastic networks and their performance measures are considered. These performance measures are defined as the expected value of some random variables and cannot normally be obtained analytically as functions of network parameters in a closed form. We give similar representations for the random variables to provide a useful way of analytical study of these functions and their gradients. The representations are used to obtain sufficient conditions for the gradient estimates to be unbiased. The conditions are rather general and usually met in simulation study of the stochastic networks. Applications of the results are discussed and some practical algorithms of calculating unbiased estimates of the gradients are also presented.     

Plastic Structural Analysis Under Stochastic Uncertainty

Problems from limit load or shakedown analysis are based on the convex, linear or linearized yield/strength condition and the linear equilibrium equation for the generic stress vector. Having to take into account, in practice, stochastic variations of the model parameters (e.g., yield stresses, plastic capacities) and external loadings, the basic stochastic plastic analysis problem must be replaced by an appropriate deterministic substitute problem. Instead of calculating approximatively the probability of failure based on a certain choice of failure modes, here, a direct approach is presented based on the costs for missing carrying capacity and the failure costs (e.g., costs for damage, repair, compensation for weakness within the structure, etc.). Based on the basic mechanical survival conditions, the failure costs may be represented by the minimum value of a convex and often linear program. Several mathematical properties of this program are shown. Minimizing then the total expected costs subject to the remaining (simple) deterministic constraints, a stochastic optimization problem is obtained which may be represented by a “Stochastic Convex Program (SCP) with recourse”. Working with linearized yield/strength conditions, a “Stochastic Linear Program (SLP) with complete fixed recourse” is obtained. In case of a discretely distributed probability distribution or after the discretization of a more general probability distribution of the random structural parameters and loadings as well as certain random cost factors one has a linear program (LP) with a so-called “dual decomposition data” structure. For stochastic programs of this type many theoretical results and efficient numerical solution procedures (LP-solver) are available. The mathematical properties of theses substitute problems are considered. Furthermore approximate analytical formulas for the limit load factor are given.     

Discontinuous periodic behaviour in a looped macroscopic model of traffic dynamics

A whole link model of traffic flow in which the link trip time varies linearly with link volume is investigated in the case where a fraction of the flow exiting the link is ‘re-injected’ as inflow, thus forming the link into a loop. The scenario where the fraction of flow re-injected linearly decreases as the volume of traffic on the link increases is considered. It is found that for certain parameter choices a continuous periodic inflow can lead to sharp post-transient discontinuities in periodic outflow. The conditions under which this type of behaviour can occur are considered.     

非仿射随机波动率的欧式障碍期权定价

研究非仿射随机波动率模型的欧式障碍期权定价问题时,首先介绍了非仿射随机波动率模型,其次利用投资组合和It^o引理,得到了该模型下扩展的Black-Schole偏微分方程.由于这个方程没有显示解,因此采用对偶蒙特卡罗模拟法计算欧式障碍期权的价格.最后,通过数值实例验证了算法的可行性和准确性.     

The Numerical Approximation of Stochastic Partial Differential Equations

The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s, when stochastic Taylor schemes based on an iterated application of the Itô formula were introduced and used to derive higher order numerical schemes. An Itô formula in the generality needed for Taylor expansions of the solution of a SPDE is however not available. Nevertheless, it was shown recently how stochastic Taylor expansions for the solution of a SPDE can be derived from the mild form representation of the SPDE, which avoid the need of an Itô formula. A brief review of the literature is given here and the new stochastic Taylor expansions are discussed along with numerical schemes that are based on them. Both strong and pathwise convergence are considered.