Showing papers on "Continuous-time stochastic process published in 1979"

Random variables and stochastic processes

Abstract: An electromagnetic pulse counter having successively operable, contact-operating armatures. The armatures are movable to a rest position, an intermediate position and an active position between the main pole and the secondary pole of a magnetic circuit.

Multiplicative stochastic processes in statistical physics

Abstract: For a large class of nonlinear stochastic processes with pure multiplicative fluctuations the corresponding time-dependent Fokker-Planck-equation is solved exactly by analytic methods. A universal eigenvalue spectrum and the corresponding set of eigenfunctions are obtained in closed form. The eigenvalue spectrum consists of a discrete as well as a continuous part. To emphasize the significance of the model proposed for the description of more-general stochastic processes the authors investigate its stability with respect to the inclusion of weak additive fluctuations. A discussion of the differences in the static as well as the dynamic behavior of multiplicative and additive stochastic processes is given in detail. It is shown explicitly how internal as well as externally imposed fluctuations can lead to multiplicative stochastic processes. The applications of the results to various fields such as nonlinear optics---subharmonic generation, parametric three-wave mixing, Raman scattering---electronic devices, autocatalytic chemical reactions, and population dynamics are given. In particular, a comparison with recent experiments by S. Kabashima et al., who investigated the statistical properties of electronic parametric oscillators driven by external noise, is carried out.

On the Stochastic Realization Problem

Abstract: Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density $\Phi $ such that $\Phi (\infty )$ is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations) require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman–Busy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algori...

Dynamic Programming Approach to Stochastic Evolution Equations

Abstract: In this paper stochastic regulator problems and optimal stationary control as well as stability are studied for infinite dimensional systems with state and control dependent noise. The stochastic model is described by a semigroup and Wiener processes in Hilbert space and Wonham’s approach using differential generators and dynamic programming is extended to infinite dimensions.

A stochastic realization approach to the smoothing problem

Abstract: The purpose of this paper is to develop a theory of smoothing for finite dimensional linear stochastic systems in the context of stochastic realization theory. The basic idea is to embed the given stochastic system in a class of similar systems all having the same output process and the same Kalman-Bucy filter. This class has a lattice structure with a smallest and a largest element; these two elements completely determine the smoothing estimates. This approach enables us to obtain stochastic interpretations of many important smoothing formulas and to explain the relationship between them.

The Carrying Dimension of a Stochastic Measure Diffusion

Abstract: A multiplicative stochastic measure diffusion process in $R^d$ is the continuous analogue of an infinite particle branching Markov process in which the particles move in $R^d$ according to a symmetric stable process of index $\alpha 0 \alpha$. The latter result is also proved by a different approach in the case $d = \alpha$.

Some averaging and stability results for random differential equations

Abstract: This paper concerns differential equations which contain strongly mixing random processes (processes for which the “past” and the “future” are asymptotically independent). When the “rate” of mixing is rapid relative to the rate of change of the solution process, information about the behavior of the solution is obtained. Roughly, the results fall into three categories:1. Quite generally, the solution process is well approximated by a deterministic trajectory, over a finite time interval. 2. For more restricted systems, this approximation extends to the infinite interval $[ {0,\infty } ),3$. Conditions for the asymptotic stability of $\dot x = AX$, where A is an $n \times n$ matrix-valued random process, are obtained.

A Stochastic Realization Approach to the Smoothing Problem. Revised.

Abstract: : The purpose of this paper is to develop a theory of smoothing for finite dimensional linear stochastic systems in the context of stochastic realization theory. The basic idea is to embed the given stochastic system in a class of similar systems all having the same output process and the same Kalman-Bucy filter. This class has a lattice structure with a smallest and a largest element; these two elements completely determine the smoothing estimates. This approach enables us to obtain stochastic interpretations of many important smoothing formulas and to explain the relationship between them. (Author)

Minimum contrast estimation in diffusion processes

Abstract: This paper is concerned with the asymptotic theory of estimates of an unknown parameter in continuous-time Markov processes, which are described by non-linear stochastic differential equations. The maximum likelihood estimate and the minimum contrast estimate are investigated. For these estimates strong consistency and asymptotic normality are proved. The unknown parameter is assumed to take its values either in an open or in a compact set of real numbers.

Time series recursions and stochastic approximation

Abstract: The thesis is divided into two parts. In Part I after an introductory chapter, a general approach is offered for the construction of recursions from a Prediction Error or Gaussian Likelihood viewpoint. This approach also allows other methods (for example, Model Reference) to be seen in context. Next, an heuristic argument is used to intuit the asymptotic properties of a general class of recursions (including the Prediction Error Recursions) and Time Series Stochastic Approximation schemes. The third chapter of Part I contains simulations illustrating some of the above ideas.

Dynamics under uncertainty

Abstract: THIS PAPER IS a preliminary investigation of dynamics under uncertainty. We attempt to develop a general approach to the continuous time stochastic processes that arise in dynamic economics from the maximizing behavior of agents. The analysis builds on recent results of Bismut [2, 3] concerning the characterization of the extrema of stochastic variational problems over a finite horizon and on our own investigations [6, 7, 20, 21] of the stability properties of the equations of dynamic economics.2 We consider a class of discounted infinite horizon maximum problems. While it is convenient to pose the basic economic problem as a stochastic control problem, to obtain the full benefit of Bismut's elegant characterization of a maximizing process it is convenient to transform this problem into an equivalent stochastic variational problem along the lines indicated by Rockafellar [27] in the deterministic case and generalized by Bismut [2] to the stochastic case. Within this framework we show that the idea of a competitive path introduced in the continuous time deterministic case in [21] generalizes in a natural way in the case of uncertainty to a competitive process. We show, under a concavity assumption on the basic integrand of the problem, that a competitive process which satisfies a transversality condition is optimal under a discounted catching up criterion (Section 2). In Section 3 we examine the sample path properties of a competitive process. If for almost every realization of a competitive process the associated dual price process generates a path of subgradients for the value function, we call the process McKenzie competitive, since it was McKenzie [22] who first recognized the importance of this property in the deterministic case. We show that two McKenzie competitive processes starting from distinct nonrandom initial conditions converge almost surely if the processes are bounded almost surely and if a certain curvature condition is satisfied by the Hamiltonian of the system. The earlier convergence result extensively studied in the deterministic case thus continues to hold in the stochastic case. The problem of finding sufficient conditions for the existence of a McKenzie competitive process remains an open problem. Section 4 examines the long-run behavior of the probability measure associated with a competitive process. We give conditions under which a McKenzie competitive process is a Markov process with an invariant probability measure and

Exact solutions of some multiplicative stochastic processes

Abstract: The theory of multiplicative stochastic processes with completely and quasicompletely random Gaussian statistics is discussed. Operator valued equations with stochastic coefficients are solved exactly for various types of statistics using the path integral technique. Generalizations of previous results for such stochastic processes are obtained.

Comparing Stochastic Systems Using Regenerative Simulation with Common Random Numbers.

Abstract: : Suppose two alternative designs for a stochastic system are to be compared. These two systems can be simulated independently or dependently. This paper presents a method for comparing two regenerative stochastic processes in a dependent fashion using common random numbers. A set of sufficient conditions is given that guarantees that the dependent simulations will produce a variance reduction over independent simulations. Numerical examples for a variety of simple stochastic models are included which illustrate the variance reduction achieved. (Author)

Asymptotic distribution of the log-likelihood function for stochastic processes

Abstract: Let X 0, X 1,⋯, X nbe r.v.'s coming from a stochastic process whose finite dimensional distributions are of known functional form except that they involve a k-dimensional parameter. From the viewpoint of statistical inference, it is of interest to obtain the asymptotic distributions of the log-likelihood function and also of certain other r.v.'s closely associated with the likelihood function. The probability measures employed for this purpose depend, in general, on the sample size n. These problems are resolved provided the process satisfies some quite general regularity conditions. The results presented herein generalize previously obtained results for the case of Markovian processes, and also for i.n.n.i.d. r.v.'s. The concept of contiguity plays a key role in the various derivations.

A Stochastic Representation for the Principal Eigenvalue of a Second-Order Differential Equation.

Abstract: : Using ideas from stochastic control a stochastic representation for the smallest eigenvalue of a second-order differential equation has been devised. As a side-result an associated stationary control problem in a very general setting has been solved.

Relationship between stochastic and differential models of compartmental systems

Abstract: This paper shows that the differential-equation model for compartmental systems is consistent with a stochastic description. Consequently, we may employ either a differential-equation or a stochastic formulation, either for parameter identification or for physical interpretation, as best suits the purpose. The differential-equation parameters, the so-called fractional transfer coefficients, may be determined from the corresponding set of stochastic parameters and vice versa.

Application of the Wigner distribution to harmonic analysis of generalized stochastic processes

Abstract: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.

Stochastic processes and estimation theory with applications

Abstract: A concise presentation which includes material on the application of time-sampling and spectra to a number of areas. It also includes detailed discussions of both Wierner-Kolmogorov theory and Kalman-Bucy theory. In addition, it applies recursive estimation to image enhancement. Appendices provide information on Dirac delta function, vector spaces and matrices, Fourier and bilateral Laplace transforms and their inversions, a special vector space, and state variables. Step-by-step, logical format.

Continuous Stochastic Models

Abstract: This chapter concentrates on continuous models for the dynamics of superionic conductors as opposed to jump models and lattice gas models, which are considered in Chap.4. By continuous models we mean that the diffusion of an ion is not represented by instantaneous jumps from an equilibrium site to another one, but by a continuous motion in between.

Stochastic approximation algorithm for the identification of linear multivariable systems

Abstract: A stochastic approximation algorithm is presented for on-line identification of linear, multivariable, discrete-time systems from noisy data without prior knowledge of the statistics of measurement noise. The algorithm uses a normalized mean-square error criterion which improves the initial convergence of the identification scheme. The results of a simulated example are given which indicate that the proposed algorithm provides good estimates even for large noise-to-signal ratios.

Robust estimation using the Robbins-Monro stochastic approximation algorithm

Abstract: The problem of minmax estimation of a location parameter introduced by Huber is considered. It is shown that under general conditions there exists a solution which is a form of the Robbins-Monro stochastic approximation algorithm. This generalizes earlier work by Martin and Masreliez who have given stochastic approximation (SA)-estimate solutions for two particular cases. As with the M -estimate solutions given by Huber, the SA solutions are completely determined by the probability distribution function with least Fisher information in the distribution set used to model the observation errors.

A dynamic stochastic version of the pitcher‐hamblin‐miller model of “collective violence"*

Abstract: The central equation of the deterministic diffusion model of Pitcher, Hamblin, and Miller (1978) is formulated as a time‐inhomogeneous stochastic process. It will be shown that the stochastic process leads to a negative binomial distribution. The deterministic diffusion function can be derived from the stochastic model and is identical to the expected value as a function of time. Therefore the deterministic model is supported in terms of the underlying stochastic process. Moreover the stochastic model allows the prediction of the distribution for any point in time and the construction of prediction intervals.