On the Stochastic Realization Problem. Revision.

We describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this covariance extension equation (CEE) is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem, answering a conjecture due to Georgiou (1987) in the affirmative. We also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the CEE, yielding some insights into the structure of solutions to the minimal partial stochastic realization problem. By combining this parameterization with some of the classical approaches in partial realization theory, we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations. As a corollary to these results, we note that, in sharp contrast with the deterministic case, there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length.

Previous work on the stochastic realization and approximation problem has cast this problem in the framework of theRV-coefficient, a measure of correlation recently introduced in the multivariate statistical literature. This allowed the introduction of a common measure for the “goodness of fit” for the different realization algorithms. This paper explores the deeper geometrical and logical foundation for this common measure in a unified theory for the data-driven and the exact covariance approaches.

This paper uses noncommutative dilation theory to develop a theory of stochastic bilinear systems. A certain stochastic bilinear realization and partial realization problem is solved. A new Caratheodory interpolation problem is presented and solved.

We analyse the properties of a fluid generating a spinning cosmic string spacetime with flat limiting cases corresponding to a constant angular momentum in the infinite past and static configuration in the infinite future. The spontaneous loss of angular momentum of a spinning cosmic string due to particle emission is discussed. The rate of particle production between the spinning cosmic string spacetime (t-) and a non-spinning cosmic string spacetime (t+) is calculated.

Purpose: Development of the apparatus of the stochastic processes econometric modeling. Discussion: The authors identify risk component in the dynamics of stochastic processes in the economy. Theoretical justification of the alternative and proportional expectations is used to make probabilistic nature of the risk. Results: The authors suggest stochastic process decomposition based on econometric approach to allocate a probability space of risks, and to identify shocks realizations that lie beyond the boundary of this space. Proportional expectations hypothesis distinguished two types of the event influence on the stochastic process realization: continuous (risk) and discrete (shock). The authors suggest model errors and residuals as the main source of information for the identification of the probability space of risks. The technique of econometric modeling of the price and return processes on stock market under the conditions of the proposed hypotheses is considered in the empirical part of the study. F-test results have not disproved the statement that the model residuals provide additional information about the simulated rate in the case of lack of relevant factors.

The problem of obtaining all stochastic realizations of a discrete-time non stalionary stochastic process defined on the set of integers is considered. The structure of the set of all covariances of the states of such realizations is studied. Special attention is paid to the minimum and maximum elements of this set. Finally, a non-Riccati-type algorithm—which does not require solution of a matrix Riccati equation—to generate (wide-sense) stochastic realizations is presented.

Regularity of a general equilibrium in a model with infinite past and future

We describe a special class of ballistic geodesics in Schwarzschild space-time, extending to the horizon in the infinite past and future of observer time, which are characterized by the property that they are in 1–1 correspondence, and completely degenerate in energy and angular momentum, with stable circular orbits. We derive analytic expressions for the source terms in the Regge–Wheeler and Zerilli–Moncrief equations for a point-particle moving on such a ballistic orbit, and compute the gravitational waves emitted during the infall in an extreme mass ratio black-hole binary coalescence. In this way a geodesic approximation to the plunge phase of compact binaries is obtained.

Efficient Spectral-Chaos Methods for Uncertainty Quantification in Long-Time Response of Stochastic Dynamical Systems

The brain integrates streams of sensory input and builds accurate predictions, while arriving at stable percepts under disparate time scales. This stochastic process bears different unfolding dynamics for different people, yet statistical learning (SL) currently averages out, as noise, individual fluctuations in data streams registered from the brain as the person learns. We here adopt a new analytical approach that instead of averaging out fluctuations in continuous electroencephalographic (EEG)-based data streams, takes these gross data as the important signals. Our new approach reassesses how individuals dynamically learn predictive information in stable and unstable environments. We find neural correlates for two types of learners in a visuomotor task: narrow-variance learners, who retain explicit knowledge of the regularity embedded in the stimuli. They seem to use an error-correction strategy steadily present in both stable and unstable environments. This strategy can be captured by current optimization-based computational frameworks. In contrast, broad-variance learners emerge only in the unstable environment. Local analyses of the moment-by-moment fluctuations, naïve to the overall outcome, reveal an initial period of memoryless learning, well characterized by a continuous gamma process starting out exponentially distributed whereby all future events are equally probable, with high signal (mean) to noise (variance) ratio. The empirically derived continuous Gamma process smoothly converges to predictive Gaussian signatures comparable to those observed for the error-corrective mode that is captured by current optimization-driven computational models. We coin this initially seemingly purposeless stage exploratory. Globally, we examine a posteriori the fluctuations in distributions' shapes over the empirically estimated stochastic signatures. We then confirm that the exploratory mode of those learners, free of expectation, random and memoryless, but with high signal, precedes the acquisition of the error-correction mode boasting smooth transition from exponential to symmetric distributions' shapes. This early naïve phase of the learning process has been overlooked by current models driven by expected, predictive information and error-based learning. Our work demonstrates that (statistical) learning is a highly dynamic and stochastic process, unfolding at different time scales, and evolving distinct learning strategies on demand.

A completeness proof for adapted probability logic

: Calculating the amount of information about a random function contained in another random function has important uses in communication theory. An expression for the mutual information for continuous time random processes has been given by Gelfand and Yaglom, Chiang, and Perez by generalizing Shannon's result in a natural way. Under a condition of absolute continuity of measures the continuous time expression has the same form as Shannon's result. For two Gaussian processes Gelfand and Yaglom express the mutual information in terms of a mean square estimation error. We generalize this result to diffusion processes and express the solution in a different form which is more naturally related to a corresponding filtering problem. We also use these results to calculate some information rates.

A great deal of data in economics, finance, engineering, and the natural sciences occur in the form of time series where observations are dependent and where the nature of this dependence is of interest. A model which describes the probability structure of time series observations \(X_t\), \(t=1, 2,\ldots , n\), is called a stochastic process. An \(X_t\) might be the value of a stock return at time t, the water level in a river at time t, and so on. In this chapter, we will introduce a variety of stochastic processes, e.g., stationary processes, linear processes, heavy-tailed processes, etc., and explain some of their elements, e.g., stationarity, spectral analysis, prediction, causality, etc. The primary purpose of this book is to provide statistical inference, Whittle estimation, model diagnostics, portmanteau tests, nonstandard testing problems for the boundary parameter, and causality tests for heavy-tailed stochastic processes. This relies on the asymptotic theory for higher order asymptotics, Bartlett adjustment, empirical likelihood method, asymptotic efficiency, robustness, etc. For these problems, some useful fundamentals and tools will be given. Throughout this book, we denote by \(\mathbb {N}\), \(\mathbb {Z}\), and \({\mathbb {R}}^m\), the set of all positive integers, the set of all integers, and the m-dimensional Euclidean space, respectively. Stochastic processes were born as a mathematical model describing random quantities which vary together with time. For each time \(t\in \mathbb {Z}\), there exists a random variable \(X_t\) defined on a probability space \((\Omega , \mathscr {A}, P)\), then the family of random variables \(\{X_t: t\in \mathbb {Z}\}\) is called a stochastic process. From the definition, \(\{X_t\}\) may be a family of any random variables. But, if we want to do mathematical or statistical analysis, a sort of regularity or invariance for \(\{X_t\}\) is needed. The most fundamental one is stationarity.

Hyperfinite models of adapted probability logic

In this paper, we study a parametric class of stochastic processes to model both fast and slow anomalous diffusions. This class, called generalized grey Brownian motion (ggBm), is made up of self-similar with stationary increments processes (H-sssi) and depends on two real parameters α ∊ (0, 2) and β ∊ (0, 1]. It includes fractional Brownian motion when α ∊ (0, 2) and β = 1, and time-fractional diffusion stochastic processes when α = β ∊ (0, 1). The latter have a marginal probability density function governed by time-fractional diffusion equations of order β. The ggBm is defined through the explicit construction of the underlying probability space. However, in this paper we show that it is possible to define it in an unspecified probability space. For this purpose, we write down explicitly all the finite-dimensional probability density functions. Moreover, we provide different ggBm characterizations. The role of the M-Wright function, which is related to the fundamental solution of the time-fractional diffusion equation, emerges as a natural generalization of the Gaussian distribution. Furthermore, we show that the ggBm can be represented in terms of the product of a random variable, which is related to the M-Wright function, and an independent fractional Brownian motion. This representation highlights the H-sssi nature of the ggBm and provides a way to study and simulate the trajectories. For this purpose, we developed a random walk model based on a finite difference approximation of a partial integro-differential equation of a fractional type.