Stable Stochastic Processes Research Articles

Збурення ізотропного $\alpha$-стійкого випадкового процесу псевдоградієнтом з узагальненим коефіцієнтом

The article is devoted to the perturbation of an isotropic $\alpha$-stable stochastic process in a finite-dimensional Euclidean space by a pseudo-gradient operator multiplied by a delta-function on a hypersurface. This is analogous to the construction of some membrane in the phase space. Semigroup of operators on the space of continuous bounded functions is constructed. It has the infinitesimal generator (in some generalized sense) $c\Delta_\alpha+(q\delta_S\nu,\nabla_\beta)$, where $c$ is some positive constant, $\Delta_\alpha$ is the fractional Laplacian of the order $\alpha$, $\delta_S$ is the delta-function on the hypersurface $S$, which has a normal vector $\nu$, $q$ is some continuous bounded function, $\nabla_\beta$ is a fractional gradient (pseudo-gradient), that is the pseudo-differentional operator defined by the symbol $i\lambda|\lambda|^{\beta-1}$. The order of the pseudo-gradient is less than $\alpha-1$. Some properties of the obtained semigroup are investigated. This semigroup defines a pseudo-process.

On single-layer potentials, pseudo-gradients and a jump theorem for an isotropic $$\alpha $$-stable stochastic process

On single-layer potentials, pseudo-gradients and a jump theorem for an isotropic $$\alpha $$-stable stochastic process

Bilateral estimates of some pseudo-derivatives of the transition probability density of an isotropic $\alpha$-stable stochastic process

In the paper, the transition probability density of an isotropic $\alpha$-stable stochastic process in a finite dimensional Euclidean space is considered. The results of applying pseudo-differential operators with respect spatial variables to this function are estimated from the both side: above and below. Operators in the consideration are defined by the symbols $|\lambda|^\varkappa$ and $\lambda|\lambda|^{\varkappa-1}$, where $\varkappa$ is some constant. The first operator with negative sign is fractional Laplacian and the second one multiplied by imaginary unit is fractional gradient.

Deep stable neural networks: Large-width asymptotics and convergence rates

In modern deep learning, there is a recent and growing literature on the interplay between large-width asymptotics for deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed weights, and classes of Gaussian stochastic processes (SPs). Such an interplay has proved to be critical in several contexts of practical interest, e.g. Bayesian inference under Gaussian SP priors, kernel regression for infinite-wide deep NNs trained via gradient descent, and information propagation within infinite-wide NNs. Motivated by empirical analysis, showing the potential of replacing Gaussian distributions with Stable distributions for the NN's weights, in this paper we investigate large-width asymptotics for (fully connected) feed-forward deep Stable NNs, i.e. deep NNs with Stable-distributed weights. First, we show that as the width goes to infinity jointly over the NN's layers, a suitable rescaled deep Stable NN converges weakly to a Stable SP whose distribution is characterized recursively through the NN's layers. Because of the non-triangular NN's structure, this is a non-standard asymptotic problem, to which we propose a novel and self-contained inductive approach, which may be of independent interest. Then, we establish sup-norm convergence rates of a deep Stable NN to a Stable SP, quantifying the critical difference between the settings of ``joint growth and ``sequential growth of the width over the NN's layers. Our work extends recent results on infinite-wide limits for deep Gaussian NNs to the more general deep Stable NNs, providing the first result on convergence rates for infinite-wide deep NNs.

Overreaction in Expectations: Evidence and Theory

Abstract We investigate biases in expectations across different settings through a large-scale randomized experiment where participants forecast stable stochastic processes. The experiment allows us to control forecasters’ information sets as well as the data-generating process, so we can cleanly measure biases in beliefs. We report three facts. First, forecasts display significant overreaction to the most recent observation. Second, overreaction is stronger for less persistent processes. Third, overreaction is also stronger for longer forecast horizons. We develop a tractable model of expectations formation with costly processing of past information, which closely fits the empirical facts. We also perform additional experiments to test the mechanism of the model.

Properties of single layer potentials for a pseudo- differential equation related to a linear transformation of a rotationally invariant stable stochastic process

This article is aimed at determining existence conditions of single layer potentials for pseudo-differential equations related to some linear transformations of a rotationally invariant stable stochastic process in a multidimensional Euclidean space and investigating their properties as well. The carrier surface of the potential is smooth enough. In this article, we consider two main cases: the first, when this surface is bounded and closed; the second, when it is unbounded, but could be presented by an explicit equation in some coordinate system. The density of this potential is a continuous function. It is bounded with respect to the spatial variable and, probably, has an integrable singularity with respect to the time variable at zero. Classic properties of this potential, including a jump theorem of the action result of some operator (an analog of the co-normal differential) at its surface points, considered.
 A rotationally invariant $\alpha$-stable stochastic process in $\mathbb{R}^d$ is a L\'{e}vy process with the characte\-ristic function of its value in the moment of time $t>0$ defined by the expression $\exp\{-tc|\xi|^\alpha\}$, $\xi\in\mathbb{R}^d$, where $\alpha\in(0,2]$, $c>0$ are some constants. If $\alpha=2$ and $c=1/2$, we get Brownian motion and classic theory of potential. There are many different results in this case. The situation of $\alpha\in(1,2)$ is considered in this paper. We study constant and invertible linear transformations of the rotationally invariant $\alpha$-stable stochastic process. The related pseudo-differential equation is the parabolic equation of the order $\alpha$ of the ``heat'' type in which the operator with respect to the spatial variable is the process generator. The single layer potential is constructed in the same way as the single layer potential for the heat equation in the classical theory of potentials. That is, we use the fundamental solution of the equation, which is the transition probability density of the related process. In our theory, the role of the gradient operator is performed by some vector pseudo-differential operator of the order $\alpha-1$. We have already studied the following main properties of the single layer potentials: the single layer potential is a solution of the relating equation outside of the carrier surface and the jump theorem is held. These properties can be useful to solving initial boundary value problems for the considered equations.

On multivariate fractional random fields: Tempering and operator-stable laws

In this paper, we define a new and broad family of vector-valued random fields called tempered operator fractional operator-stable random fields (TRF, for short). TRF is typically non-Gaussian and generalizes tempered fractional stable stochastic processes. TRF comprises moving average and harmonizable-type subclasses that are constructed by tempering (matrix-) homogeneous, matrix-valued kernels in time- and Fourier-domain stochastic integrals with respect to vector-valued, strictly operator-stable random measures. We establish the existence and fundamental properties of TRF. Assuming both Gaussianity and isotropy, we show the equivalence between certain moving average and harmonizable subclasses of TRF. In addition, we establish sample path properties in the scalar-valued case for several Gaussian instances.

Lower bound for local oscillations of Hermite processes

The most known example of a class of non-Gaussian stochastic processes which belongs to the homogeneous Wiener chaos of an arbitrary order N>1 is probably Hermite processes of rank N. They generalize fractional Brownian motion (fBm) and Rosenblatt process in a natural way. They were introduced several decades ago. Yet, in contrast with fBm and many other Gaussian and stable stochastic processes and fields related to it, few results on path behavior of Hermite processes are available in the literature. For instance the natural issue of whether or not their paths are nowhere differentiable functions has not yet been solved even in the most simple case of the Rosenblatt process. The goal of our article is to derive a quasi-optimal lower bound for the asymptotic behavior of local oscillations of paths of Hermite processes of any rank N, which, among other things, shows that these paths are nowhere differentiable functions.

Overreaction in Expectations: Evidence and Theory

We investigate biases in expectations across different settings through a large-scale randomized experiment where participants forecast stable stochastic processes. The experiment allows us to control forecasters’ information sets as well as the data generating process, so we can cleanly measure biases in beliefs. We find that forecasts display significant overreaction to the most recent observation. Moreover, overreaction is especially pronounced for less persistent processes and longer forecast horizons. We also find that commonly-used expectations models do not easily account for these variations in the degree of overreaction across different settings. To explain the observed patterns of overreaction, we develop a tractable model of expectations formation with costly information processing. Our model closely fits the empirical findings and generates additional predictions that we confirm in the data.

On single-layer potentials for a class of pseudo-differential equations related to linear transformations of a symmetric $\alpha$-stable stochastic process

In this article an arbitrary invertible linear transformations of a symmetric $\alpha$-stable stochastic process in $d$-dimensional Euclidean space $\mathbb{R}^d$ are investigated. The result of such transformation is a Markov process in $\mathbb{R}^d$ whose generator is the pseudo-differential operator defined by its symbol $(-(Q\xi,\xi)^{\alpha/2})_{\xi\in\mathbb{R}^d}$ with some symmetric positive definite $d\times d$-matrix $Q$ and fixed exponent $\alpha\in(1,2)$. The transition probability density of this process is the fundamental solution of some parabolic pseudo-differential equation. The notion of a single-layer potential for that equation is introduced and its properties are investigated. In particular, an operator is constructed whose role in our consideration is analogous to that the gradient in the classical theory. An analogy to the classical theorem on the jump of the co-normal derivative of the single-layer potential is proved. This result can be applied for solving some boundary-value problems for the parabolic pseudo-differential equations under consideration. For $\alpha=2$, the process under consideration is a linear transformation of Brownian motion, and all the investigated properties of the single-layer potential are well known.

On the third initial-boundary value problem for some class of pseudo-differential equations related to a symmetric $$\alpha $$ α -stable process

A fundamental solution of the so-called third initial-boundary value problem for one class of pseudo-differential equations is constructed. Those equations are related to a symmetric \(\alpha \)-stable stochastic process and our constructions are inspired by some probabilistic ideas. However, we expound our results in a way completely independent of any probabilistic notion. Only the last section of the paper is based on the notion of a stochastic process and also a pseudo-process and it gives some interpretation of our results in terms of stochastic analysis.

On Simple-Layer Potentials for One Class of Pseudodifferential Equations

We construct single-layer potentials for a class of pseudodifferential equations connected with symmetric stable stochastic processes. An operator similar to the operator of gradient in the classical potential theory is selected and an analog of the classical theorem on the jump of (co)normal derivative of single-layer potential is established. This result allows us to construct solutions of some initial-boundary-value problems for pseudodifferential equations of the indicated kind.

On the choice of GARCH parameters for efficient modelling of real stock price dynamics

We propose two different methods for optimal choice of GARCH(1,1) parameters for the efficient modelling of stock prices by using a particular return series. Using (as an example) stock return data for Intel Corporation, we vary parameters to fit the average volatility as well as fourth (linked to kurtosis of data) and eighth statistical moments and observe pure convergence of our simulated eighth moment to the stock data. Results indicate that fitting higher-order moments of a return series might not be an optimal approach for choosing GARCH parameters. In contrast, the simulated exponent of the Fourier spectrum decay is much less noisy and can easily fit the corresponding decay of the empirical Fourier spectrum of the used return series of Intel stock, allowing us to efficiently define all GARCH parameters. We compare the estimates of GARCH parameters obtained by fitting price data Fourier spectra with the ones obtained from standard software packages and conclude that the obtained estimates here are deeper in the stability region of parameters. Thus, the proposed method of using Fourier spectra of stock data to estimate GARCH parameters results in a more robust and stable stochastic process but with a shorter characteristic autocovariance time.

Dilatively stable stochastic processes and aggregate similarity

Dilatively stable processes generalize the class of infinitely divisible self-similar processes. We reformulate and extend the definition of dilative stability introduced by Igloi (Dilative stability, Ph.D. Thesis, University of Debrecen, Faculty of Informatics, http://www.inf.unideb.hu/valseg/dolgozok/igloi/dissertation.pdf (2008)) using characteristic functions. We also generalize the concept of aggregate similarity introduced by Kaj (Fractals in Engineering, New Trends in Theory and Applications, pp 199–218 (2005)). It turns out that these two notions are essentially the same for infinitely divisible processes. Examples of dilatively stable generalized fractional Levy processes are given and we point out that certain limit processes in aggregation models are dilatively stable.

Estimating the generalized autoregression model parameters for unknown noise distribution

We solve the problem of estimating the autoregressive parameters of a nonlinear stable stochastic process with discrete time of the AR(p)/ARCH(p) type with unknown ARCH(p) process parameters. For the AR(1)/ARCH(1) model, we solve the estimation problem for all unknown process parameters, i.e., the autoregression parameter and two parameters of the noise process ARCH(1). We assume that the noise distributions are unknown. We show that the least square estimates are strongly consistent.

Stochastic analysis of external and parametric dynamical systems under sub-Gaussian Levy white-noise

In this study stochastic analysis of non-linear dynamical systems under ?-stable, multiplicative white noise has been conducted. The analysis has dealt with a special class of ?-stable stochastic processes namely sub-Gaussian white noises. In this setting the governing equation either of the probability density function or of the characteristic function of the dynamical response may be obtained considering the dynamical system forced by a Gaussian white noise with an uncertain factor with ?/2-stable distribution. This consideration yields the probability density function or the characteristic function of the response by means of a simple integral involving the probability density function of the system under Gaussian white noise and the probability density function of the ?/2-stable random parameter. Some numerical applications have been reported assessing the reliability of the proposed formulation. Moreover a proper way to perform digital simulation of the sub-Gaussian ?-stable random process preventing dynamical systems from numerical overflows has been reported and discussed in detail.

Phase-modulated resonances modeled as self-similar processes with application to turbulent sounds

In this paper, we propose a nonlinear stochastic model for time-varying resonances where the instantaneous phase (and frequency) of a sinusoidal oscillation is allowed to vary proportionally to an /spl alpha/-stable self-similar stochastic processes. The main motivation of our work stems from previous experimental and theoretical evidence that speech resonances in fricative sounds can be modeled phenomenologically as AM-FM signals with randomly varying instantaneous frequencies and that several signal classes related to turbulent phenomena are self-similar 1/f processes. Our general approach is to model the instantaneous phase of an AM-FM resonance as a self-similar /spl alpha/-stable process. As a special case, this random phase model includes the class of random fractal signals known as fractional Brownian motion. We theoretically explore this random modulation model and analytically derive its autocorrelation and power spectrum. We also propose an algorithm to fit this model to arbitrary resonances with random phase modulation. Further, we apply the above ideas to real speech data and demonstrate that this model is suitable for resonances of fricative sounds.

Extremes of totally skewed α-stable processes11Research supported by `Naturvetenskapliga Forskningsrådet Contract F-AA/MA 9207-307, and by Office of Naval Research Grant N00014-93-1-0043.

We give upper and lower bounds for the probability for a local extrema of a totally skewed α -stable stochastic process. Often these bounds are sharp and coincide. The Gaussian case α =2 is not excluded, and there our results slightly improve existing general bounds. Applications focus on moving averages and fractional α -stable motions.

The Nature of Expectations in the Foreign Exchange Market: A Test of Competing Theories

In a 1982-83 article, Paul Davidson argues that rational expectations are a poor guide to real-world economic behavior. He supports this contention with a number of points, relying especially on the argument that entrepreneurs focus on exactly the sort of activity least likely to fit the RE mold: crucial decisions. Crucial decisions, once made, may forever change the circumstances surrounding future choices. As a consequence, ergodicity is violated and rational expectations cannot hold. A related question arising from this debate is whether or not economic agents passively forecast events or actually cause them. According to Davidson, rational expectations assume the former, though a realistic analysis would suggest the latter. This last point is the focus of the current paper. The Post Keynesian approach implies that expectations guide actions, and that those actions create the real world. This dynamic process tends to change the parameters affecting future decisions in the manner discussed above. Expectations are therefore causal. The rational expectations approach posits a world in which economic agents simply react, making forecasts of events that are driven by some independent and relatively stable stochastic process. In this context, expectations do not affect the objective variable. It is the purpose of this paper to perform an empirical test of these rival views. In particular, I examine the relationship between expectations and foreign exchange rates (using the forecast surveys published by Money Market Services). In the rational expectations model, where predictions and outcomes are expected to be correlated but not causal,

Rational belief structures and rational belief equilibria

The paper introduces some simplifying tools and methods for studying Rational Beliefs and for proving existence of Rational Belief Equilibria. We identify a set of stable non-stationary stochastic processes, named SIDS processes. Furthermore we introduce the concept of a Rational Belief Structure, which formulates the Rational Beliefs of the agents as beliefs about the distribution of exogenous variables and the beliefs of other agents. The use of the developed apparatus is demonstrated by showing existence of a set of Rational Belief Equilibria in an Overlapping Generations Model with money and one commodity.