A gaussian upper bound for gaussian multi-stage stochastic linear programs

This paper deals with two-stage and multi-stage stochastic programs in which the right-hand sides of the constraints are Gaussian random variables. Such problems are of interest since the use of Gaussian estimators of random variables is widespread. We introduce algorithms to find upper bounds on the optimal value of two-stage and multi-stage stochastic (minimization) programs with Gaussian right-hand sides. The upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size. The algorithm for the two-stage problem involves the solution of a deterministic linear program and a simple semidefinite program. The algorithm for the multi-stage problem invovles the solution of a quadratically constrained convex programming problem.     

A mathematical framework for stochastic climate models

There has been a recent burst of activity in the atmosphere‐ocean sciences community in utilizing stable linear Langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction. Here a systematic mathematical strategy for stochastic climate modeling is developed, and some of the new phenomena in the resulting equations for the climate variables alone are explored. The new phenomena include the emergence of both unstable linear Langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances. All of these phenomena are derived from a systematic self‐consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit. The theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated. The feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography. Explicit concrete examples with the new phenomena are presented for the stochastically forced three‐mode interaction equations. The conjecture of Smith and Waleffe [Phys. Fluids 11 (1999), 1608–1622] for stochastically forced three‐wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach. Examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self‐consistency of the mathematical approach with the predictions of equilibrium statistical mechanics. In particular, for these examples, the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables. © 2001 John Wiley & Sons, Inc.     

The Stratonovich Interpretation of Quantum Stochastic Approximations

The Stratonovich version of non-commutative stochastic calculus is introduced and shown to be equivalent to the Itô version developed by Hudson and Parthasarathy [1]. The conversion from Stratonovich to Itô version is shown to be implemented by a stochastic form of Wick's theorem: that is, involving the normal ordering of time-dependent noise fields. It is shown for a model of a quantum mechanical system coupled to a Bosonic field in a Gaussian state that under suitable scaling limits, in particular the weak coupling limit (for linear interactions) and low density limit (for scattering interactions), the limit form of the dynamical equation of motion is most naturally described as a quantum stochastic differential equation of Stratonovich form. We then convert the limit dynamical equations from Stratonovich to Itô form. Thermal Stratonovich noises are also presented.     

Computing Bounds on the Expected Maximum of Correlated Normal Variables

We compute upper and lower bounds on the expected maximum of correlated normal variables (up to a few hundred in number) with arbitrary means, variances, and correlations. Two types of bounding processes are used: perfectly dependent normal variables, and independent normal variables, both with arbitrary mean values. The expected maximum for the perfectly dependent variables can be evaluated in closed form; for the independent variables, a single numerical integration is required. Higher moments are also available. We use mathematical programming to find parameters for the processes, so they will give bounds on the expected maximum, rather than approximations of unknown accuracy. Our original application is to the maximum number of people on-line simultaneously during the day in an infinite-server queue with a time-varying arrival rate. The upper and lower bounds are tighter than previous bounds, and in many of our examples are within 5% or 10% of each other. We also demonstrate the bounds’ performance on some PERT models, AR/MA time series, Brownian motion, and product-form correlation matrices.     

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the non-linear stochastic heat equation in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.     

一种考虑属性具有关联性的正态随机多属性决策方法

针对属性具有关联性的正态随机多属性决策问题,给出一种决策方法。首先,将正态随机变量形式的属性值进行规范化;然后,在考虑属性之间具有关联性的情况下,运用正态随机变量计算公式和Choquet积分计算公式,对规范化后的属性值进行集结,得到各方案的正态随机变量形式的综合评价值。进一步地,依据事先定义的正态随机变量的序关系确定规则,对各方案的综合评价值进行排序,进而确定方案的排序结果。最后,通过一个算例说明了本文给出方法的可行性和有效性。     

Doubling-Time Probability Densities For Growth Processes

The doubling-time probability density of a growth process is the probability density for the time it takes for the size to double. Doubling-time probability densities are useful in studying growth rates, for example, of organisms, populations, financial products, or chemical reactions. Three fundamental stochastic models of growth are investigated for their doubling-time probability densities. It is shown that two of the stochastic models have doubling-time probability densities which are inverse Gaussian. Although the third stochastic model’s doubling-time density does not have a simple analytical form, it is shown to be approximately inverse Gaussian under a reasonable hypothesis on the model’s parameters. Two data sets for doubling time, spruce seedling size and Texas Mega Millions Lottery jackpot, are fit to inverse Gaussian distributions.     

Large buffer asymptotics for generalized processor sharing queues with Gaussian inputs

In this paper we derive large-buffer asymptotics for a two-class Generalized Processor Sharing (GPS) model. We assume both classes to have Gaussian characteristics. We distinguish three cases depending on whether the GPS weights are above or below the average rate at which traffic is sent. First, we calculate exact asymptotic upper and lower bounds, then we calculate the logarithmic asymptotics, and finally we show that the decay rates of the upper and lower bound match. We apply our results to two special Gaussian models: the integrated Gaussian process and the fractional Brownian motion. Finally we derive the logarithmic large-buffer asymptotics for the case where a Gaussian flow interacts with an on-off flow. AMS Subject Classification Primary—60K25; Secondary—68M20, 60G15     

Upper bounds for Gaussian stochastic programs

We present a construction which gives deterministic upper bounds for stochastic programs in which the randomness appears on the right–hand–side and has a multivariate Gaussian distribution. Computation of these bounds requires the solution of only as many linear programs as the problem has variables. Received December 2, 1997 / Revised version received January 5, 1999? Published online May 12, 1999     

Convergent Bounds for Stochastic Programs with Expected Value Constraints

This article describes a bounding approximation scheme for convex multistage stochastic programs (MSP) that constrain the conditional expectation of some decision-dependent random variables. Expected value constraints of this type are useful for modelling a decision maker’s risk preferences, but they may also arise as artifacts of stage-aggregation. We develop two finite-dimensional approximate problems that provide bounds on the (infinite-dimensional) original problem, and we show that the gap between the bounds can be made smaller than any prescribed tolerance. Moreover, the solutions of the approximate MSPs give rise to a feasible policy for the original MSP, and this policy’s optimality gap is shown to be smaller than the difference of the bounds. The considered problem class comprises models with integrated chance constraints and conditional value-at-risk constraints. No relatively complete recourse is assumed.     

Large deviations for diffusion processes in duals of nuclear spaces

Motivated by applications to neurophysiological problems, various authors have studied diffusion processes in duals of countably Hilbertian nuclear spaces governed by stochastic differential equations. In these models the diffusion coefficients describe the random stimuli received by spatially extended neurons. In this paper we present a large deviation principle for such processes when the diffusion terms tend to zero in terms of a small parameter. The lower bounds are established by making use of the Girsanov formula in abstract Wiener space. The upper bounds are obtained by Gaussian approximation of the diffusion processes and by taking advantage of the nuclear structure of the state space to pass from compact sets to closed sets.This research was partially supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620-92-J-0154 and the Army Research Office Grant No. DAAL03-92-G-0008.     

A class of non-Gaussian second order random fields

Non-Gaussian stochastic fields are introduced by means of integrals with respect to independently scattered stochastic measures distributed according to generalized Laplace laws. In particular, we discuss stationary second order random fields that, as opposed to their Gaussian counterpart, have a possibility of accounting for asymmetry and heavier tails. Additionally to this greater flexibility the models discussed continue to share most spectral properties with Gaussian processes. Their statistical distributions at crossing levels are computed numerically via the generalized Rice formula. The potential for stochastic modeling of real life phenomena that deviate from the Gaussian paradigm is exemplified by a stochastic field model with Matérn covariances.     

Solving stochastic complementarity problems in energy market modeling using scenario reduction

In this paper, we analyze market equilibrium models with random aspects that lead to stochastic complementarity problems. While the models presented depict energy markets, the results are believed to be applicable to more general stochastic complementarity problems. The contribution is the development of new heuristic, scenario reduction approaches that iteratively work towards solving the full, extensive form, stochastic market model. The methods are tested on three representative models and supporting numerical results are provided as well as derived mathematical bounds.     

Modelling Joint Behaviour of Asset Prices Using Stochastic Correlation

Association or interdependence of two stock prices is analyzed, and selection criteria for a suitable model developed in the present paper. The association is generated by stochastic correlation, given by a stochastic differential equation (SDE), creating interdependent Wiener processes. These, in turn, drive the SDEs in the Heston model for stock prices. To choose from possible stochastic correlation models, two goodness-of-fit procedures are proposed based on the copula of Wiener increments. One uses the confidence domain for the centered Kendall function, and the other relies on strong and weak tail dependence. The constant correlation model and two different stochastic correlation models, given by Jacobi and hyperbolic tangent transformation of Ornstein-Uhlenbeck (HtanOU) processes, are compared by analyzing daily close prices for Apple and Microsoft stocks. The constant correlation, i.e., the Gaussian copula model, is unanimously rejected by the methods, but all other two are acceptable at a 95% confidence level. The analysis also reveals that even for Wiener processes, stochastic correlation can create tail dependence, unlike constant correlation, which results in multivariate normal distributions and hence zero tail dependence. Hence models with stochastic correlation are suitable to describe more dangerous situations in terms of correlation risk.

     

Stochasticity in the image greenhouse model

The potential impacts of climate change are a major issue in the Greenhouse debate. Various models and particularly IMAGE (Integrated Model to Assess the Greenhouse Effect), are being used by the IPCC to investigate climate change. The IMAGE model has been reduced to a system of differential equations and incorporates various initial conditions and model parameters. These initial conditions and parameter values are not known precisely and are subject to variability of various forms. This paper briefly describes the mathematical form of IMAGE and then investigates the stochastic properties of the model. Particular attention is paid to the propagation and amplification of assumed distributions for the initial conditions and certain key physical parameters. The IMAGE model appears to be quite robust with respect to these stochastic characteristics.     

Asymptotic Bayesian structure learning using graph supports for Gaussian graphical models

The theory of Gaussian graphical models is a powerful tool for independence analysis between continuous variables. In this framework, various methods have been conceived to infer independence relations from data samples. However, most of them result in stepwise, deterministic, descent algorithms that are inadequate for solving this issue. More recent developments have focused on stochastic procedures, yet they all base their research on strong a priori knowledge and are unable to perform model selection among the set of all possible models. Moreover, convergence of the corresponding algorithms is slow, precluding applications on a large scale. In this paper, we propose a novel Bayesian strategy to deal with structure learning. Relating graphs to their supports, we convert the problem of model selection into that of parameter estimation. Use of non-informative priors and asymptotic results yield a posterior probability for independence graph supports in closed form. Gibbs sampling is then applied to approximate the full joint posterior density. We finally give three examples of structure learning, one from synthetic data, and the two others from real data.     

Geometric Ergodicity for Piecewise Contracting Processes with Applications for Tropical Stochastic Lattice Models

Stochastic lattice models are increasingly prominent as a way to capture highly intermittent unresolved features of moist tropical convection in climate science and as continuum mesoscopic models in material science. Stochastic lattice models consist of suitably discretized continuum partial differential equations interacting with Markov jump processes at each lattice site with transition rates depending on the local value of the continuum equation; they are a special case of piecewise deterministic Markov processes but often have an infinite state space and unbounded transition rates. Here a general theorem on geometric ergodicity for piecewise deterministic contracting processes is developed with full generality to apply to stochastic lattice models. A highly nontrivial application to the stochastic skeleton model for the Madden‐Julian oscillation (Thual et al., 2013) is developed here where there is an infinite state space with unbounded and also degenerate transition rates. Geometric ergodicity for the stochastic skeleton model guarantees exponential convergence to a unique invariant measure that defines the statistical tropical climate of the model. Another application of the general framework is developed here for stochastic lattice models designed to capture intermittent fluctuation in the simplest tropical climate models. Other straightforward applications to models motivated by material science are mentioned briefly here. © 2016 Wiley Periodicals, Inc.     

A second-order Monte Carlo method for the solution of the Ito stochastic differential equation

A difference approximation that is second-order accurate in the time step his derived for the general Ito stochastic differential equation. The difference equation has the form of a second-order random walk in which the random terms are non-linear combinations of Gaussian random variables. For a wide class of problems, the transition pdf is joint-normal to second order in h; the technique then reduces to a Gaussian random walk, but its application is not limited to problems having a Gaussian solution. A large number of independent sample paths are generated in a Monte Carlo solution algorithm; any statistical function of the solution (e.g., moments or pdf's) can be estimated by ensemble averaging over these paths     

海气耦合随机-动力气候模式的周期解问题

目前大多数对随机 动力气候模式的研究都是在随机强迫项为白噪声的假定下进行的,而实际上许多天气快变量往往表现为非线性的其它随机过程.该文运用Mawhin重合度理论,探讨了一类随机强迫项是其它随机过程,而非白噪声时的海气耦合随机 动力气候模式的周期解问题,得到了一定条件下该模型存在周期解的结果.     

Geometric partitioning and robust ad-hoc network design

We consider bounds for the price of a European-style call option under regime switching. Stochastic semidefinite programming models are developed that incorporate a lattice generated by a finite-state Markov chain regime-switching model as a representation of scenarios (uncertainty) to compute bounds. The optimal first-stage bound value is equivalent to a Value at Risk quantity, and the optimal solution can be obtained via simple sorting. The upper (lower) bounds from the stochastic model are bounded below (above) by the corresponding deterministic bounds and are always less conservative than their robust optimization (min-max) counterparts. In addition, penalty parameters in the model allow controllability in the degree to which the regime switching dynamics are incorporated into the bounds. We demonstrate the value of the stochastic solution (bound) and computational experiments using the S&P 500 index are performed that illustrate the advantages of the stochastic programming approach over the deterministic strategy.