Showing papers on "Stochastic process published in 1992"

Numerical Solution of Stochastic Differential Equations

Abstract: 1 Probability and Statistics- 2 Probability and Stochastic Processes- 3 Ito Stochastic Calculus- 4 Stochastic Differential Equations- 5 Stochastic Taylor Expansions- 6 Modelling with Stochastic Differential Equations- 7 Applications of Stochastic Differential Equations- 8 Time Discrete Approximation of Deterministic Differential Equations- 9 Introduction to Stochastic Time Discrete Approximation- 10 Strong Taylor Approximations- 11 Explicit Strong Approximations- 12 Implicit Strong Approximations- 13 Selected Applications of Strong Approximations- 14 Weak Taylor Approximations- 15 Explicit and Implicit Weak Approximations- 16 Variance Reduction Methods- 17 Selected Applications of Weak Approximations- Solutions of Exercises- Bibliographical Notes

Stochastic Equations in Infinite Dimensions

Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Wavelet analysis and synthesis of fractional Brownian motion

Abstract: Fractional Brownian motion (FBM) offers a convenient modeling for nonstationary stochastic processes with long-term dependencies and 1/f-type spectral behavior over wide ranges of frequencies. Statistical self-similarity is an essential feature of FBM and makes natural the use of wavelets for both its analysis and its synthesis. A detailed second-order analysis is carried out for wavelet coefficients of FBM. It reveals a stationary structure at each scale and a power-law behavior of the coefficients' variance from which the fractal dimension of FBM can be estimated. Conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed, consequences of discretization are considered, and some connections between the wavelet point of view and previous approaches based on length measurements (analysis) or dyadic interpolation (synthesis) are briefly pointed out. >

Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise.

Abstract: We use the influence functional path-integral method to derive an exact master equation for the quantum Brownian motion of a particle linearly coupled to a general environment (ohmic, subohmic, or supraohmic) at arbitrary temperature and apply it to study certain aspects of the loss of quantum coherence.

The integer l-shaped method for stochastic integer programs with complete recourse

Abstract: In this paper, a general branch-and-cut procedure for stochastic integer programs with complete recourse and first stage binary variables is presented. It is shown to provide a finite exact algorithm for a number of stochastic integer programs, even in the presence of binary variables or continous random variables in the second stage. (A)

Statistical analysis and simulation study of video teleconference traffic in ATM networks

Abstract: Source modeling and performance issues are studied using a long (30 min) sequence of real video teleconference data. It is found that traffic periodicity can cause different sources with identical statistical characteristics to experience differing cell-loss rates. For a single-stage multiplexer model, some of this source-periodicity effect can be mitigated by appropriate buffer scheduling and one effective scheduling policy is presented. For the sequence analyzed, the number of cells per frame follows a gamma (or negative binomial) distribution. The number of cells per frame is a stationary stochastic process. For traffic studies, neither an autoregressive model of order two nor a two-state Markov chain model is good because they do not model correctly the occurrence of frames with a large number of cells, which are a primary factor in determining cell-loss rates. The order two autoregressive model, however, fits the data well in a statistical sense. A multistate Markov chain model that can be derived from three traffic parameters is sufficiently accurate for use in traffic studies. >

Direct test for determinism in a time series

Abstract: A direct test for deterministic dynamics can be established by measurement of average directional vectors in a coarse-grained d-dimensional embedding of a time series. Theoretical analysis of the statistical properties of a random time series using the same embedding technique is possible by consideration of classical results concerning random walks in d dimensions. Examples are given to show the clear differences between deterministic dynamics, such as may be generated by chaotic systems, and stochastic dynamics.

Correlation structure of the discrete wavelet coefficients of fractional Brownian motion

Abstract: It is shown that the discrete wavelet coefficients of fractional Brownian motion at different scales are correlated and that their auto- and cross-correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional Brownian motion itself. The rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet. >

Modeling and estimation of multiresolution stochastic processes

Abstract: An overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. A natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image. In particular, it is shown how the wavelet transform directly suggests such a modeling paradigm. This perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time-like variable. The investigation of models on homogeneous trees is emphasized. The framework examined here allows for consideration, in a very natural way, of the fusion of data from sensors with differing resolutions. Also, thanks to the fact that wavelet transforms do an excellent job of 'compressing' large classes of covariance kernels, it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect. >

Stochastic monotonicity and stationary distributions for dynamic economies

Abstract: The existence and stability of invariant distributions for stochastically monotone processes is studied. The Knaster-Tarski fixed point theorem is applied to establish existence of fixed points of mappings on compact sets of measures that are increasing with respect to a stochastic ordering. Global convergence of a monotone Markov process to its unique invariant distribution is established under an easily verified assumption. Topkis' theory of supermodular functions is applied to stochastic dynamic optimization, providing conditions under which optimal stationary decisions are monotone functions of the state and induce a monotone Markov process. Applications of these results to investment theory, stochastic growth, and industry equilibrium dynamics are given.

On properties of stochastic inventory systems

Abstract: For most order quantity/reorder point inventory systems, the stochastic model, which specifies the demands as stochastic processes, is often more accurate than its deterministic counterpart—the EOQ model. However, the application of the stochastic model has been limited because of the absence of insightful analytical results on the model. This paper analyzes the stochastic order quantity/reorder point model in comparison with a corresponding deterministic EOQ model. Based on simple optimality conditions for the control variables derived in the paper, a sensitivity analysis is carried out, and a number of basic qualitative properties are established for the optimal control parameters. Our main results include the following: (1) in contrast to the deterministic EOQ model, the controllable costs of the stochastic model due to selection of the order quantity (assuming the reorder point is chosen optimally for every order quantity) are actually smaller, while the total costs are clearly larger; the optimal ord...

Planning under uncertainty solving large-scale stochastic linear programs

Abstract: For many practical problems, solutions obtained from deterministic models are unsatisfactory because they fail to hedge against certain contingencies that may occur in the future. Stochastic models address this shortcoming, but up to recently seemed to be intractable due to their size. Recent advances both in solution algorithms and in computer technology now allow us to solve important and general classes of practical stochastic problems. We show how large-scale stochastic linear programs can be efficiently solved by combining classical decomposition and Monte Carlo (importance) sampling techniques. We discuss the methodology for solving two-stage stochastic linear programs with recourse, present numerical results of large problems with numerous stochastic parameters, show how to efficiently implement the methodology on a parallel multi-computer and derive the theory for solving a general class of multi-stage problems with dependency of the stochastic parameters within a stage and between different stages.

Nonlinear prediction as a way of distinguishing chaos from random fractal sequences

Abstract: NONLINEAR forecasting has recently been shown to distinguish between deterministic chaos and uncorrelated (white) noise added to periodic signals1, and can be used to estimate the degree of chaos in the underlying dynamical system2. Distinguishing the more general class of coloured (autocorrelated) noise has proven more difficult because, unlike additive noise, the correlation between predicted and actual values measured may decrease with time—a property synonymous with chaos. Here, we show that by determining the scaling properties of the prediction error as a function of time, we can use nonlinear prediction to distinguish between chaos and random fractal sequences. Random fractal sequences are a particular class of coloured noise which represent stochastic (infinite-dimensional) systems with power-law spectra. Such sequences have been known to fool other procedures for identifying chaotic behaviour in natural time series9, particularly when the data sets are small. The recognition of this type of noise is of practical importance, as measurements from a variety of dynamical systems (such as three-dimensional turbulence, two-dimensional and geostrophic turbulence, internal ocean waves, sandpile models, drifter trajectories in large-scale flows, the motion of a classical electron in a crystal and other low-dimensional systems) may over some range of frequencies exhibit power-law spectra.

Method to distinguish possible chaos from colored noise and to determine embedding parameters

Abstract: We present a computational method to determine if an observed time series possesses structure statistically distinguishable from high-dimensional linearly correlated noise, possibly with a nonwhite spectrum. This method should be useful in identifying deterministic chaos in natural signals with broadband power spectra, and is capable of distinguishing between chaos and a random process that has the same power spectrum. The method compares nonlinear predictability of the given data to an ensemble of random control data sets

Mechanics of saltating grains. II

Abstract: A numerical model of saltation akin to those of previous researchers is presented to investigate the detailed nature of bed load motion in terms of the mechanics of saltation. The model is deterministic in the computation of particle trajectory, but probabilistic in terms of bed collision. This probabilistic element arises from the random structure of the bed. In the present analysis, a stochastic treatment of collision allows for a fairly complete model of saltation in water. Characteristic quantities, including the sediment transport rate itself, are derived and compared with experimental data. The model yields good agreement with data with a minimum of assumptions. The assumed three-dimensional random structure of the bed is a unique feature of the present model. It allows for the prediction of several phenomena that cannot be obtained via the assumption of a regular two-dimensional bed structure.

Low-dimensional behavior and symmetry breaking of stochastic systems near criticality-can these effects be observed in space and in the laboratory?

Abstract: It is demonstrated that nonlinear stochastic systems near criticality (including self-organized criticality) will generally exhibit low-dimensional behavior. A connection is given between the fractal dimensions of finite-dimensional chaotic systems and the anomalous dimensions in stochastic systems near criticality. The effect of additional random noise on stochastic systems will be delineated in terms of the crossover phenomenon between competing criticalities. The possibility of observing such effects in space (such as the onset of substorms) and in the laboratory (such as stochastic particle heating in 'noisy' magnetic fields) is discussed. >

Statistical properties of construction duration data

Abstract: Simulators of construction operations often must approximate the underlying distribution of a random process using a standard statistical distribution eg, lognormal, normal, and beta In many of

Probability via expectation

Abstract: 1 Uncertainty, Intuition, and Expectation.- 1 Ideas and Examples.- 2 The Empirical Basis.- 3 Averages over a Finite Population.- 4 Repeated Sampling: Expectation.- 5 More on Sample Spaces and Variables.- 6 Ideal and Actual Experiments: Observables.- 2 Expectation.- 1 Random Variables.- 2 Axioms for the Expectation Operator.- 3 Events: Probability.- 4 Some Examples of an Expectation.- 5 Moments.- 6 Applications: Optimization Problems.- 7 Equiprobable Outcomes: Sample Surveys.- 8 Applications: Least Square Estimation of Random Variables.- 9 Some Implications of the Axioms.- 3 Probability.- 1 Events, Sets and Indicators.- 2 Probability Measure.- 3 Expectation as a Probability Integral.- 4 Some History.- 5 Subjective Probability.- 4 Some Basic Models.- 1 A Model of Spatial Distribution.- 2 The Multinomial, Binomial, Poisson and Geometric Distributions.- 3 Independence.- 4 Probability Generating Functions.- 5 The St. Petersburg Paradox.- 6 Matching, and Other Combinatorial Problems.- 7 Conditioning.- 8 Variables on the Continuum: The Exponential and Gamma Distributions.- 5 Conditioning.- 1 Conditional Expectation.- 2 Conditional Probability.- 3 A Conditional Expectation as a Random Variable.- 4 Conditioning on a ? Field.- 5 Independence.- 6 Statistical Decision Theory.- 7 Information Transmission.- 8 Acceptance Sampling.- 6 Applications of the Independence Concept.- 1 Renewal Processes.- 2 Recurrent Events: Regeneration Points.- 3 A Result in Statistical Mechanics: The Gibbs Distribution.- 4 Branching Processes.- 7 The Two Basic Limit Theorems.- 1 Convergence in Distribution (Weak Convergence).- 2 Properties of the Characteristic Function.- 3 The Law of Large Numbers.- 4 Normal Convergence (the Central Limit Theorem).- 5 The Normal Distribution.- 6 The Law of Large Numbers and the Evaluation of Channel Capacity.- 8 Continuous Random Variables and Their Transformations.- 1 Distributions with a Density.- 2 Functions of Random Variables.- 3 Conditional Densities.- 9 Markov Processes in Discrete Time.- 1 Stochastic Processes and the Markov Property.- 2 The Case of a Discrete State Space: The Kolmogorov Equations.- 3 Some Examples: Ruin, Survival and Runs.- 4 Birth and Death Processes: Detailed Balance.- 5 Some Examples We Should Like to Defer.- 6 Random Walks, Random Stopping and Ruin.- 7 Auguries of Martingales.- 8 Recurrence and Equilibrium.- 9 Recurrence and Dimension.- 10 Markov Processes in Continuous Time.- 1 The Markov Property in Continuous Time.- 2 The Case of a Discrete State Space.- 3 The Poisson Process.- 4 Birth and Death Processes.- 5 Processes on Nondiscrete State Spaces.- 6 The Filing Problem.- 7 Some Continuous-Time Martingales.- 8 Stationarity and Reversibility.- 9 The Ehrenfest Model.- 10 Processes of Independent Increments.- 11 Brownian Motion: Diffusion Processes.- 12 First Passage and Recurrence for Brownian Motion.- 11 Action Optimisation Dynamic Programming.- 1 Action Optimisation.- 2 Optimisation over Time: the Dynamic Programming Equation.- 3 State Structure.- 4 Optimal Control Under LQG Assumptions.- 5 Minimal-Length Coding.- 6 Discounting.- 7 Continuous-Time Versions and Infinite-Horizon Limits.- 8 Policy Improvement.- 12 Optimal Resource Allocation.- 1 Portfolio Selection in Discrete Time.- 2 Portfolio Selection in Continuous Time.- 3 Multi-Armed Bandits and the Gittins Index.- 4 Open Processes.- 5 Tax Problems.- 13 Finance: 'Risk-Free' Trading and Option Pricing.- 1 Options and Hedging Strategies.- 2 Optimal Targeting of the Contract.- 3 An Example.- 4 A Continuous-Time Model.- 5 How Should it Be Done?.- 14 Second-Order Theory.- 1 Back to L2.- 2 Linear Least Square Approximation.- 3 Projection: Innovation.- 4 The Gauss-Markov Theorem.- 5 The Convergence of Linear Least Square Estimates.- 6 Direct and Mutual Mean Square Convergence.- 7 Conditional Expectations as Least Square Estimates: Martingale Convergence.- 15 Consistency and Extension: The Finite-Dimensional Case.- 1 The Issues.- 2 Convex Sets.- 3 The Consistency Condition for Expectation Values.- 4 The Extension of Expectation Values.- 5 Examples of Extension.- 6 Dependence Information: Chernoff Bounds.- 16 Stochastic Convergence.- 1 The Characterization of Convergence.- 2 Types of Convergence.- 3 Some Consequences.- 4 Convergence in rth Mean.- 17 Martingales.- 1 The Martingale Property.- 2 Kolmogorov's Inequality: the Law of Large Numbers.- 3 Martingale Convergence: Applications.- 4 The Optional Stopping Theorem.- 5 Examples of Stopped Martingales.- 18 Large-Deviation Theory.- 1 The Large-Deviation Property.- 2 Some Preliminaries.- 3 Cramer's Theorem.- 4 Some Special Cases.- 5 Circuit-Switched Networks and Boltzmarm Statistics.- 6 Multi-Class Traffic and Effective Bandwidth.- 7 Birth and Death Processes.- 19 Extension: Examples of the Infinite-Dimensional Case.- 1 Generalities on the Infinite-Dimensional Case.- 2 Fields and ?-Fields of Events.- 3 Extension on a Linear Lattice.- 4 Integrable Functions of a Scalar Random Variable.- 5 Expectations Derivable from the Characteristic Function: Weak Convergence324.- 20 Quantum Mechanics.- 1 The Static Case.- 2 The Dynamic Case.- References.

Controlling chaos in high dimensional systems.

Abstract: Recently formulated techniques for controlling chaotic dynamics face a fundamental problem when the system is high dimensional, and this problem is present even when the chaotic attractor is low dimensional. Here we introduce a procedure for controlling a chaotic time signal of an arbitrarily high dimensional system, without assuming any knowledge of the underlying dynamical equations. Specifically, we formulate a feedback control that requires modeling the local dynamics of only a single or a few of the possibly infinite number of phase-space variables.

Regenerative Stochastic Simulation

Abstract: Preface. Discrete-Event Simulations. Regenerative Stochastic Processes. Regenerative Simulation. Networks of Queues. Passage Times. Simulations With Simultaneous Events. Appendix A. Limit Theorems for Stochastic Processes. Appendix B. Random Number Generation.

Stationarity detection in the initial transient problem

Abstract: Let X = {X(t)}t ≥ 0 be a stochastic process with a stationary version X*. It is investigated when it is possible to generate by simulation a version X˜ of X with lower initial bias than X itself, in the sense that either X˜ is strictly stationary (has the same distribution as X*) or the distribution of X˜ is close to the distribution of X*. Particular attention is given to regenerative processes and Markov processes with a finite, countable, or general state space. The results are both positive and negative, and indicate that the tail of the distribution of the cycle length t plays a critical role. The negative results essentially state that without some information on this tail, no a priori computable bias reduction is possible; in particular, this is the case for the class of all Markov processes with a countably infinite state space. On the contrary, the positive results give algorithms for simulating X˜ for various classes of processes with some special structure on t. In particular, one can generate X˜ as strictly stationary for finite state Markov chains, Markov chains satisfying a Doeblin-type minorization, and regenerative processes with the cycle length t bounded or having a stationary age distribution that can be generated by simulation.

A deterministic approach to nonlinear systems analysis

Abstract: The electrophysiological investigation of the visual system poses two major problems for systems analysis and identification. This chapter aims to illuminate the old subject of white-noise systems analysis from a different angle in the hope of generating some new insights. It also aims to communicate methods that have proven extremely useful in visual electrophysiology. The chapter makes a distinction between stochastic and deterministic techniques based on the way the response properties are extracted. In a stochastic approach the system is stimulated by means of a random process, and the extraction of the response characteristics is based on the stochastic properties of the stimulus. In a deterministic analysis, on the other hand, one uses the exact input signal. Because the deterministic approach makes it possible to keep track of the number of occurrences of each configuration, it can also be used with nonwhite inputs.

The Asymptotic Validity of Sequential Stopping Rules for Stochastic Simulations

Abstract: : We establish general conditions for the asymptotic validity of sequential stopping rules to achieve fixed-volume confidence sets for simulation estimators of vector-valued parameters. The asymptotic validity occurs as the prescribed volume of the confidences set approaches zero. There are two requirements: a functional central limit theorem for the estimation process and strong consistency (with-probability-one convergence) for the variance or scaling matrix estimator. Applications are given for: sample means of i.i.d. random variables and random vectors, nonlinear functions of such sample means, jackknifing, Kiefer-Wolfowitz and Robbins-Monro stochastic approximation, and both regenerative and non-regenerative steady-state simulation. Keywords: Stochastic simulation, Variance estimators.

Density estimation by stochastic complexity

Abstract: The results by P. Hall and E.J. Hannan (1988) on optimization of histogram density estimators with equal bin widths by minimization of the stochastic complexity are extended and sharpened in two separate ways. As the first contribution, two generalized histogram estimators are constructed. The first has unequal bin widths which, together with the number of the bins, are determined by minimization of the stochastic complexity using dynamic programming. The other estimator consists of a mixture of equal bin width estimators, each of which is defined by the associated stochastic complexity. As the main contribution in the present work, two theorems are proved, which together extend the universal coding theorems to a large class of data generating densities. The first gives an asymptotic upper bound for the code redundancy in the order of magnitude, achieved with a special predictive type of histogram estimator, which sharpens a related bound. The second theorem states that this bound cannot be improved upon by any code whatsoever. >

Stochastic Processes with Applications

Abstract: Preface to the Classics Edition Preface Sample course outline 1. Random walk and Brownian motion 2, Discrete-parameter Markov chains 3. Birth-death Markov chains 4. Continuous-parameter Markov chains 5. Brownian motion and diffusions 6. Dynamic programming and stochastic optimization 7. An introduction to stochastic differential equations 8. A probability and measure theory overview Author index Subject index Errata.

Recursive structure of noncausal Gauss-Markov random fields

Abstract: An approach is developed for noncausal Gauss-Markov random fields (GMRFs) that enables the use of recursive procedures while retaining the noncausality of the field. Recursive representations are established that are equivalent to the original field. This is achieved by first presenting a canonical representation for GMRFs that is based on the inverse of the covariance matrix, which is called the potential matrix. It is this matrix rather than the field covariance that reflects in a natural way the MRF structure. From its properties, two equivalent one-sided representations are derived, each of which is obtained as the successive iterates of a Riccati-type equation. For homogeneous fields, these unilateral descriptions are symmetrized versions of each other, the study of only one Riccati equation being required. It is proven that this Riccati equation converges at a geometric rate, therefore the one-sided representations are asymptotically invariant. These unilateral representations make it possible to process the fields with well-known recursive techniques such as Kalman-Bucy filters and two-point smoothers. >

Stochastic Motion—Motion Under the Influence of Wind

Abstract: Stochastic approaches are very effective for modelling natural phenomena. This paper presents a motion model based on a stochastic process as well as physics, and proposes motion synthesis techniques for stochastic motion—motion under the influence of wind. The motion synthesis process is modelled by a cascade system of three components: wind model, dynamic model, and deformation model. Wind models produce spatio-temporal wind velocity fields using the power spectrum and auto-correlation of wind, just like fractal geometry. Dynamic models describe the dynamic response of the systems, using equation systems or response functions. Deformation models produce deformed shapes of objects according to the geometric models of the objects and the results of the dynamic systems. The biggest advantage of the model is its generality and consistency. The model is applicable to most of the existing trees and grass models, including structural models, particle systems, impressionist models, and 3D texture. It is demonstrated that the coupling of stochastic approaches and physically-based approaches can synthesize realistic motion of trees, grass and snow with modest computational cost.

Fractional diffusion equation on fractals: one-dimensional case and asymptotic behaviour

Abstract: A fractional differential equation is studied and its application for describing diffusion on random fractal structures is considered. It represents the simplest generalization of the fractional diffusion equation valid in Euclidean systems. The solution of the fractional equation in one dimension is discussed, and compared with exact results for the fractional Brownian motion and the one-dimensional version of the 'standard' diffusion equation on fractals. In higher dimensions, it correctly describes the asymptotic scaling behaviour of the probability density function on random fractals, as obtained recently by using scaling arguments and exact enumeration calculations for the infinite percolation cluster at criticality.