Stochastic Galerkin techniques for random ordinary differential equations

Over the last decade the stochastic Galerkin method has become an established method to solve differential equations involving uncertain parameters. It is based on the generalized Wiener expansion of square integrable random variables. Although there exist very sophisticated variants of the stochastic Galerkin method (wavelet basis, multi-element approach) convergence for random ordinary differential equations has rarely been considered analytically. In this work we develop an asymptotic upper boundary for the L ₂-error of the stochastic Galerkin method. Furthermore, we prove convergence of a local application of the stochastic Galerkin method and confirm convergence of the multi-element approach within this context.     

Implicit Stochastic Runge–Kutta Methods for Stochastic Differential Equations

In this paper we construct implicit stochastic Runge–Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Case Study of Stochastic Optimization in Health Policy: Problem Formulation and Preliminary Results

We use Bayesian decision theory to address a variable selection problem arising in attempts to indirectly measure the quality of hospital care, by comparing observed mortality rates to expected values based on patient sickness at admission. Our method weighs data collection costs against predictive accuracy to find an optimal subset of the available admission sickness variables. The approach involves maximizing expected utility across possible subsets, using Monte Carlo methods based on random division of the available data into N modeling and validation splits to approximate the expectation. After exploring the geometry of the solution space, we compare a variety of stochastic optimization methods –- including genetic algorithms (GA), simulated annealing (SA), tabu search (TS), threshold acceptance (TA), and messy simulated annealing (MSA) –- on their performance in finding good subsets of variables, and we clarify the role of N in the optimization. Preliminary results indicate that TS is somewhat better than TA and SA in this problem, with MSA and GA well behind the other three methods. Sensitivity analysis reveals broad stability of our conclusions.     

Two-Stage Stochastic Variational Inequality Arising from Stochastic Programming

We consider a two-stage stochastic variational inequality arising from a general convex two-stage stochastic programming problem, where the random variables have continuous distributions. The equivalence between the two problems is shown under some moderate conditions, and the monotonicity of the two-stage stochastic variational inequality is discussed under additional conditions. We provide a discretization scheme with convergence results and employ the progressive hedging method with double parameterization to solve the discretized stochastic variational inequality. As an application, we show how the water resources management problem under uncertainty can be transformed from a two-stage stochastic programming problem to a two-stage stochastic variational inequality, and how to solve it, using the discretization scheme and the progressive hedging method with double parameterization.

     

Stochastic production frontiers and panel data: A latent variable framework

This paper deals with the issue of estimating production frontier and measuring efficiency from a panel data set. First, it proposes an alternate method for the estimation of a production frontier on a short panel data set. The method is based on the so-called mean-and-covariance structure analysis which is closely related to the generalized method of moments. One advantage of the method is that it allows us to investigate the presence of correlations between individual effects and exogenous variables without the requirement of some available instruments uncorrelated with the individual effects as in instrumental variable estimation. Another advantage is that the method is well suited to a panel data set with a short number of periods. Second, the paper considers the question of recovering individual efficiency levels from the estimates obtained from the mean-and-covariance structure analysis. Since individual effects are here viewed as latent variables, they can be estimated as factor scores, i.e., weighted sums of the observed variables. We illustrate the proposed methods with the estimation of a stochastic production frontier on a short panel data of French fruit growers.     

Stochastic programming problems with generalized integrated chance constraints

If the constraints in an optimization problem are dependent on a random parameter, we would like to ensure that they are fulfilled with a high level of reliability. The most natural way is to employ chance constraints. However, the resulting problem is very hard to solve. We propose an alternative formulation of stochastic programs using penalty functions. The expectations of penalties can be left as constraints leading to generalized integrated chance constraints, or incorporated into the objective as a penalty term. We show that the penalty problems are asymptotically equivalent under quite mild conditions. We discuss applications of sample-approximation techniques to the problems with generalized integrated chance constraints and propose rates of convergence for the set of feasible solutions. We will direct our attention to the case when the set of feasible solutions is finite, which can appear in integer programming. The results are then extended to the bounded sets with continuous variables. Additional binary variables are necessary to solve sample-approximated chance-constrained problems, leading to a large mixed-integer non-linear program. On the other hand, the problems with penalties can be solved without adding binary variables; just continuous variables are necessary to model the penalties. The introduced approaches are applied to the blending problem leading to comparably reliable solutions.     

A Comparison of Stochastic Systems with Different Types of Delays

In this article, we investigate the effects of different types of delays, a fixed delay and a random delay, on the dynamics of stochastic systems as well as their relationship with each other in the context of a just-in-time network model. The specific example on which we focus is a pork production network model. We numerically explore the corresponding deterministic approximations for the stochastic systems with these two different types of delays. Numerical results reveal that the agreement of stochastic systems with fixed and random delays depend on the population size and the variance of the random delay, even when the mean value of the random delay is chosen the same as the value of the fixed delay. When the variance of the random delay is sufficiently small, the histograms of state solutions to the stochastic system with a random delay are similar to those of the stochastic model with a fixed delay regardless of the population size. We also compared the stochastic system with a Gamma distributed random delay to the stochastic system constructed based on the Kurtz's limit theorem from a system of deterministic delay differential equations with a Gamma distributed delay. We found that with the same population size the histogram plots for the solution to the second system appear more dispersed than the corresponding ones obtained for the first case. In addition, we found that there is more agreement between the histograms of these two stochastic systems as the variance of the Gamma distributed random delay decreases.     

Sequential Order Statistics: Ageing and Stochastic Orderings

Ordered random variables play an important role in statistics, reliability theory, and many applied areas. Sequential order statistics provide a unified approach to a variety of models of ordered random variables. We investigate conditions on the underlying distribution functions on which the sequential order statistics are based, to obtain stochastic comparisons of sequential order statistics given some well known stochastic orderings, such as the usual stochastic, the hazard rate and the likelihood ratio orders, among others. Also, we derive sufficient conditions under which the sequential order statistics are increasing hazard rate, increasing hazard rate average or decreasing hazard rate average. Applications of the main results involving nonhomogeneous pure birth processes are also given.     

Term Structure Models in Multistage Stochastic Programming: Estimation and Approximation

This paper investigates some common interest rate models for scenario generation in financial applications of stochastic optimization. We discuss conditions for the underlying distributions of state variables which preserve convexity of value functions in a multistage stochastic program. One- and multi-factor term structure models are estimated based on historical data for the Swiss Franc. An analysis of the dynamic behavior of interest rates generated with these models reveals several deficiencies which have an impact on the performance of investment policies derived from the stochastic program. While barycentric approximation is used here for the generation of scenario trees, these insights may be generalized to other discretization techniques as well.     

Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDE) with random data, where the random coefficient is parametrized by means of a countable sequence of terms in a suitable expansion. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data, and naturally, the error analysis uses the joint regularity of the solution with respect to both the variables in the physical domain and parametric variables. In MISC, the number of problem solutions performed at each discretization level is not determined by balancing the spatial and stochastic components of the error, but rather by suitably extending the knapsack-problem approach employed in the construction of the quasi-optimal sparse-grids and Multi-index Monte Carlo methods, i.e., we use a greedy optimization procedure to select the most effective mixed differences to include in the MISC estimator. We apply our theoretical estimates to a linear elliptic PDE in which the log-diffusion coefficient is modeled as a random field, with a covariance similar to a Matérn model, whose realizations have spatial regularity determined by a scalar parameter. We conduct a complexity analysis based on a summability argument showing algebraic rates of convergence with respect to the overall computational work. The rate of convergence depends on the smoothness parameter, the physical dimensionality and the efficiency of the linear solver. Numerical experiments show the effectiveness of MISC in this infinite dimensional setting compared with the Multi-index Monte Carlo method and compare the convergence rate against the rates predicted in our theoretical analysis.     

Stochastic Perturbations of Line Solitons of KP

We consider the properties of localized solutions of the KP equation coupled to a stochastic noise. Corresponding to white noise, we find that the traveling waves are destroyed asymptotically, and we determine the distribution of the wave position and the arrival time. For generalized Ornstein–Uhlenbeck processes, we show that the only effect of noise is to render the asymptotic position random; in particular, when the noise has a sufficiently strong attenuation mechanism, the random wave coincides asymptotically with the unperturbed one. We also consider linearization of the corresponding Cauchy problem in the plane corresponding to this kind of initial data.     

Rooted Tree Analysis for Order Conditions of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations

Abstract

A general class of stochastic Runge-Kutta methods for the weak approximation of Itô and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an expansion of the solution process and of the approximation process calculated with the stochastic Runge-Kutta method. A theorem on general order conditions for the coefficients and the random variables of the stochastic Runge-Kutta method is proved by rooted tree analysis. This theorem can be applied for the derivation of stochastic Runge-Kutta methods converging with an arbitrarily high order.     

Stochastic response restrictions

This paper considers the implementation of prior stochastic information on unknown outcomes of the response variables into estimation and forecasting of systems of linear regression equations in the context of time series, cross sections, pooled and longitudinal data models. The established approach proves particularly useful when only aggregated information on the response variables is available, as is frequently the case in applied statistics. We address the combination of prior stochastic and sample information as an extension of standard Gauss-Markov theory. Prior stochastic information could be given in the form of experts' expectations, or from estimations and/or projections of other models. A classical (i.e. non-Bayesian) regression framework for the incorporation of prior knowledge in generalized least-squares estimation and prediction is developed.     

Stochastic binary problems with simple penalties for capacity constraints violations

This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly ${\mathcal{N}\mathcal{P}}$ -hard in general. We provide pseudo-polynomial algorithms for three special cases of the problem: constant weights and capacity uniformly distributed, subset sum with Gaussian weights and strictly positively distributed random capacity, and subset sum with constant weights and arbitrary random capacity. We then turn to a branch-and-cut algorithm based on the outer approximation of the objective function. We provide computational results for the stochastic knapsack problem (i) with Gaussian weights and constant capacity and (ii) with constant weights and capacity uniformly distributed, on randomly generated instances inspired by computational results for the knapsack problem.     

Stochastic dominance in an ordinal world

Existing stochastic dominance rules apply to variables such as income, wealth and rates of return, all of which are measured on cardinal scales. This study develops and applies stochastic dominance rules for ordinal data. It is shown that the new rules are consistent with the traditional von Neumann-Morgenstern expected utility approach, and that they are applicable and relevant in a wide variety of managerial decision making situations, where existing stochastic dominance rules fail to apply. We apply ordinal SD rules to the transformation of random variables.     

The convergence of the random lines defined by observations of a stochastic process

We deal with independent random variables which are the values of a stochastic process taken at random points in time. So we have random variables depending upon a random parameter. We obtain the conditions providing the weak convergence of random lines defined by sums or maxima or bilinear forms of these random variables for almost all values of the parameter, to one and the same stochastic process. These limit stochastic processes are described. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

Anticipative calculus for Lévy processes and stochastic differential equations*

We develop an anticipative calculus for Lévy processes with finite second moment for analysing anticipating stochastic differential equations. The calculus is based on the chaos expansion of square-integrable random variables in terms of iterated integrals with respect to the compensated Poisson random measure. We define a space of smooth and generalized random variables in terms of such chaos expansions, and present anticipative stochastic integration, the Wick product and the so-called 𝒮-transform. These concepts serve as tools for studying general Wick type stochastic differential equations with anticipative initial conditions. We apply the 𝒮-transform to find the unique solutions to a class of linear stochastic differential equations. The solutions can be expressed in terms of the Wick product.     

On the Pathwise Comparison of Jump Processes Driven by Stochastic Intensities

In this paper we present a pathwise comparison theorem for jump processes governed by stochastic intensities and taking values in an arbitrary partially ordered Polish space. This generalizes recent results for the real-valued case. The proof given here is based on competing risk arguments rather than on thinning and allows to avoid additional domination conditions. For real valued processes we obtain an almost sure pathwise representation of the comparison result based on an i.i.d. sequence of (0, 1)-uniformly distributed random variables. For Markov chains our conditions coincide with the classical comparison conditions.     

Stochastic Variational Inequality Approaches to the Stochastic Generalized Nash Equilibrium with Shared Constraints

In this paper, we consider the generalized Nash equilibrium with shared constraints in the stochastic environment, and we call it the stochastic generalized Nash equilibrium. The stochastic variational inequalities are employed to solve this kind of problems, and the expected residual minimization model and the conditional value-at-risk formulations defined by the residual function for the stochastic variational inequalities are discussed. We show the risk for different kinds of solutions for the stochastic generalized Nash equilibrium by the conditional value-at-risk formulations. The properties of the stochastic quadratic generalized Nash equilibrium are shown. The smoothing approximations for the expected residual minimization formulation and the conditional value-at-risk formulation are employed. Moreover, we establish the gradient consistency for the measurable smoothing functions and the integrable functions under some suitable conditions, and we also analyze the properties of the formulations. Numerical results for the applications arising from the electricity market model illustrate that the solutions for the stochastic generalized Nash equilibrium given by the ERM model have good properties, such as robustness, low risk and so on.     

The generalized well-posedness of the Cauchy problem for an abstract stochastic equation with multiplicative noise

We study the existence, uniqueness, and stability of a solution to the Cauchy problem for a stochastic differential equation with multiplicative noise in the spaces of generalized random variables with values in a Hilbert space.