Dimensional Stochastic Process Research Articles

Some algebraic structures in KPZ universality

We review some algebraic and combinatorial structures that underlie models in the KPZ universality class. Emphasis is placed on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov. We present how these combinatorial constructions are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions, Young tableaux and Gelfand-Tsetlin patterns. We also present how fundamental representation theoretic concepts, such as the Cauchy identity, the Pieri rule and the branching rule, can be used, alongside RSK correspondences, and can be combined with probabilistic ideas, in order to construct integrable stochastic dynamics on two dimensional arrays of Gelfand-Tsetlin type, in ways that couple different one dimensional stochastic processes. For example, interacting particle systems, on the one hand, and processes related to eigenvalues of random matrices, on the other, thus illuminating the emergence of random matrix distributions in interacting stochastic processes. The goal of the notes is to expose some of the overarching principles, which have driven a significant number of developments in the field.

Mathematical model of biological tissue surface based on one dimensional stochastic process of fractional Brownian motion

In this work the model for the profiling of biological tissue surfaces is introduced. The surface was considered as the realization of the stochastic process of the fractional Brownian motion (fBm) with the scale parameter σ and Hurst index H. The contour of the epidermal surface of the normal cut banana peel was studied. The magnified digital photos of the investigated specimens were made using the microscope (spatial resolution 1 μm) with the built-in camera. The function that linearly interpolated the surface contour was derived. The dispersion law of the differences between the values of the interpolating function in the adjacent knots was in good agreement with that one of the fBm stochastic process. The values of σ=0.1 and H=0.806 were obtained.

Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions

In this article we establish exponential moment bounds, moment bounds in fractional order smoothness spaces, a uniform H\"older continuity in time, and strong convergence rates for a class of fully discrete exponential Euler-type numerical approximations of infinite dimensional stochastic convolution processes. The considered approximations involve specific taming and truncation terms and are therefore well suited to be used in the context of SPDEs with non-globally Lipschitz continuous nonlinearities.

Optimal decision policy for real options under general Markovian dynamics

The Least-Squares Monte Carlo Method (LSM) has become the standard tool to solve real options modeled as an optimal switching problem. The method has been shown to deliver accurate valuation results under complex and high dimensional stochastic processes; however, the accuracy of the underlying decision policy is not guaranteed. For instance, an inappropriate choice of regression functions can lead to noisy estimates of the optimal switching boundaries or even continuation/switching regions that are not clearly separated. As an alternative to estimate these boundaries, we formulate a simulation-based method that starts from an initial guess of them and then iterates until reaching optimality. The algorithm is applied to a classical mine under a wide variety of underlying dynamics for the commodity price process. The method is first validated under a one-dimensional geometric Brownian motion and then extended to general Markovian processes. We consider two general specifications: a two-factor model with stochastic variance and a rich jump structure, and a four-factor model with stochastic cost-of-carry and stochastic volatility. The method is shown to be robust, stable, and easy-to-implement, converging to a more profitable strategy than the one obtained with LSM.

First passage time of nonlinear diffusion processes with singular boundary behavior

New theorems for moments of the first passage time of one dimensional nonlinear stochastic processes with an entrance boundary xe are formulated. This important class of one dimensional stochastic processes results among others from approximations of the energy or amplitude of second order nonlinear stochastic differential equations. Since the diffusion of a stochastic process vanishes at an entrance boundary, xe is called a singular point of the stochastic process. The theorems for the moments of the first passage times are validated based on existing analytical results. In addition, the first passage times of a forced and damped Mathieu oscillator, as well as a nonlinear stochastic differential equation, which is important for the determination of dangerous ship roll dynamics, are calculated. The proposed analytical expressions for the moments of the first passage times can be calculated very fast using standard quadrature formulas.

Cached Files Updating Revisited: The Distribution of Popularity-Weighted Average File Age

Low-latency information transmission is required in 5G communication networks, where the cache is commonly used to speed up the data access. Meanwhile, a large number of IoT applications need real-time information exchange, resulting in the information freshness more important. In this article, under the discrete-time model, we consider the average age of all files in a cached-files-updating system. To keep the cached files fresh, in each time slot the server refreshes files with certain probabilities. The age of one file or its age of information (AoI) is defined as the time the file stays in cache since it was sent to cache last time. Assume that each file in cache has its own request popularity, we determine the probability distribution of the popularity-weighted average file age. For the random age of single file, both the mean and its probability distribution can be obtained by establishing a simple Markov chain. Following the same line of thinking, we show that an $N$ -dimensional stochastic process can be constituted to characterize the changes of $N$ file ages simultaneously. By solving the steady state of the resulting process, we obtain the explicit expression of stationary probability for an arbitrary state vector. Then, the distribution of the popularity-weighted average file age can be found by mergering a proper set of stationary probabilities, which gives the complete description of the average age. We also provide the numerical results of the distribution for some simple cases. The age-distribution can be utilized to compute certain stochastic performance indices, such as the AoI violation probability of some status updating systems.

A $(2+1)$-dimensional growth process with explicit stationary measures

We introduce a class of (2+1)-dimensional stochastic growth processes, that can be seen as irreversible random dynamics of discrete interfaces. "Irreversible" means that the interface has an average non-zero drift. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. The model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting Hammersley processes. When the dynamical asymmetry parameter $(p-q)$ equals zero, the infinite-volume Gibbs measures $\pi_\rho$ (with given slope $\rho$) are stationary and reversible. When $p\ne q$, $\pi_\rho$ are not reversible any more but, remarkably, they are still stationary. In such stationary states, we find that the average height function at any given point $x$ grows linearly with time $t$ with a non-zero speed: $\mathbb E Q_x(t):=\mathbb E(h_x(t)-h_x(0))= V(\rho) t$ while the typical fluctuations of $Q_x(t)$ are smaller than any power of $t$ as $t\to\infty$. In the totally asymmetric case of $p=0,q=1$ and on the hexagonal lattice, the dynamics coincides with the "anisotropic KPZ growth model" introduced by A. Borodin and P. L. Ferrari. For a suitably chosen, "integrable", initial condition (that is very far from the stationary state), they were able to determine the hydrodynamic limit and a CLT for interface fluctuations on scale $\sqrt{\log t}$, exploiting the fact that in that case certain space-time height correlations can be computed exactly.

Scaling limits of a model for selection at two scales

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval [0,1] with dependence on a single parameter, λ. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on λ and the behavior of the initial data around 1. The second scaling leads to a measure-valued Fleming–Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.

High Dimensional Nonstationary Time Series Modelling with Generalized Dynamic Semiparametric Factor Model

(High dimensional) time series which reveal nonstationary and possibly periodic behavior occur frequently in many fields of science. In this article, we separate the modeling of high dimensional time series to time propagation of low dimensional time series and high dimensional time invariant functions via functional factor analysis. We propose a two-step estimation procedure. At the first step, we detect the deterministic trends of the time series by incorporating time basis selected by the group Lasso-type technique and choose the space basis based on smoothed functional principal component analysis. We show properties of this estimator under various situations extending current variable selection studies. At the second step, we obtain the detrended low dimensional stochastic process, but it also poses an important question: is it justified, from an inferential point of view, to base further statistical inference on the estimated stochastic time series? We show that the difference of the inference based on the estimated time series and true unobserved time series is asymptotically negligible, which finally allows one to study the dynamics of the whole high-dimensional system with a low dimensional representation together with the deterministic trend. We apply the method to our motivating empirical problems: studies of the dynamic behavior of temperatures (further used for pricing weather derivatives), implied volatilities and risk patterns and correlated brain activities (neuro-economics related) using fMRI data, where a panel version model is also presented.

Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence

We consider an individual-based spatially structured population for Darwinian evolution in an asexual population. The individuals move randomly on a bounded continuous space according to a reflected brownian motion. The dynamics involves also a birth rate, a density-dependent logistic death rate and a probability of mutation at each birth event. We study the convergence of the microscopic process in a long time scale when the population size grows to $$+\infty $$ and the mutation probability decreases to 0. We prove the convergence towards a jump process that jumps in the infinite dimensional space containing the monomorphic stable spatial distributions. The proof requires specific studies of the microscopic model. First, we study the extinction time of the branching diffusion processes that approximate small size populations. Then, we examine the upper bound of large deviation principle around the deterministic large population limit of the microscopic process. Finally, we find a lower bound on the exit time of a neighborhood of a stationary spatial distribution.

Bayesian estimation of Karhunen–Loève expansions; A random subspace approach

One of the most widely-used procedures for dimensionality reduction of high dimensional data is Principal Component Analysis (PCA). More broadly, low-dimensional stochastic representation of random fields with finite variance is provided via the well known Karhunen–Loève expansion (KLE). The KLE is analogous to a Fourier series expansion for a random process, where the goal is to find an orthogonal transformation for the data such that the projection of the data onto this orthogonal subspace is optimal in the L2 sense, i.e., which minimizes the mean square error. In practice, this orthogonal transformation is determined by performing an SVD (Singular Value Decomposition) on the sample covariance matrix or on the data matrix itself. Sampling error is typically ignored when quantifying the principal components, or, equivalently, basis functions of the KLE. Furthermore, it is exacerbated when the sample size is much smaller than the dimension of the random field. In this paper, we introduce a Bayesian KLE procedure, allowing one to obtain a probabilistic model on the principal components, which can account for inaccuracies due to limited sample size. The probabilistic model is built via Bayesian inference, from which the posterior becomes the matrix Bingham density over the space of orthonormal matrices. We use a modified Gibbs sampling procedure to sample on this space and then build probabilistic Karhunen–Loève expansions over random subspaces to obtain a set of low-dimensional surrogates of the stochastic process. We illustrate this probabilistic procedure with a finite dimensional stochastic process inspired by Brownian motion.

A class of infinite dimensional stochastic processes with unbounded diffusion

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron–Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.

A stochastic pseudospectral and T-matrix algorithm for acoustic scattering by a class of multiple particle configurations

We consider absorption and scattering of acoustic waves from uncertain configurations comprising multiple two dimensional bodies with various material properties (sound-soft, sound-hard, absorbing and penetrable) and develop tools to address the problem of quantifying uncertainties in the acoustic cross sections of the configurations. The uncertainty arises because the locations and orientations of the particles in the configurations are described through random variables, and statistical moments of the far-fields induced by the stochastic configurations facilitate quantification of the uncertainty. We develop an efficient algorithm, based on a hybrid of the stochastic pseudospectral discretization (to truncate the infinite dimensional stochastic process) and an efficient stable truncated version of Waterman's T-matrix approach (for cost effective realization at each multiple particle configuration corresponding to the pseudospectral quadrature points) to simulate the statistical properties of the stochastic model. We demonstrate the efficiency of the algorithm for configurations with non-smooth and non-convex bodies with distinct material properties, and random locations and orientations with normal and log-normal distributions.

Markov Process for Service Facility systems with perishable inventory and analysis of a single server queue with reneging Stochastic Model

ABSTRACT In this paper, we develop a supply network model for a service facility system with perishable inventory (on hand) by considering a two dimensional stochastic process of the form (L, X) =   0 )); ( ), ( (  t t X t L , where L (t) is the level of the on hand inventory and X (t) is the number of customers at time t. The inter-arrival time to the service station is assumed to be exponentially distributed with mean 1/λ. The service time for each customer is exponentially distributed with mean 1/ μ. The maximum inventory level is S and the maximum capacity of the waiting space is N. The replenishment process is assumed to be (S-1, S) with a replenishment of only one unit at any level of the inventory. Lead time is exponentially distributed with parameter β. The items are replenished at a rate of β whose mean replenishment time is 1/β. Item in inventory is perishable when it’s utility drops to zero or the inventory item become worthless while in storage. Perishable of any item occurs at a rate of γ. Once entered a queue, the customer may choose to leave the queue at a rate of α if they have not been served after a certain time (reneging). The steady state probability distributions for the system states are obtained. A numerical example is provided to illustrate the method described in the model.

Non-Fundamentalness in Structural Econometric Models: A Review

Economic theory typically assumes the existence of few unobserved unpredictable stochastic disturbances, called structural shocks, driving the whole economy. Would the economy be representable as a very high dimensional stochastic vector process, those shocks would be the reduced rank innovation of that process. In practice, however, only a few components of that process are observable, the innovation of which, in general, does not coincide with the structural shocks. As a result, a MA representation of the observed process in terms of the structural shocks will be a noninvertible one. In other words, the structural shocks driving the economy do not belong to the past of the observed series, but also involve their future, i.e. they are nonfundamental for the observed process. It follows that the present values of those structural shocks cannot be recovered from the observations; in particular, fitting causal VARs to the observed series can be extremely misleading. We review economic literature on VARs and we provide many examples of small size economic models that imply nonfundamentalness. If fundamental shocks nevertheless are to be recovered, the only solution consists in enlarging the space of observations, that is, in considering a very large panel of related time series. Among several alternatives, we consider the dynamic factor model methodology, which requires very little additional assumptions. We review first their economic interpretation and then we show how fundamentalness of the shocks is always guaranteed in this framework. Finally, by means of structural dynamic factor models we provide new empirical evidence on the relation between hours worked and technology shocks and we compare it with traditional VAR results.

High Dimensional Nonstationary Time Series Modelling with Generalized Dynamic Semiparametric Factor Model

(High dimensional) time series which reveal nonstationary and possibly periodic behavior occur frequently in many scientic elds. In this article, we separate the modeling of high dimensional time series to time propagation of low dimensional time series and high dimensional time invariant functions via functional factor analysis. We propose a two-step estimation procedure. At the rst step, we detect the deterministic trends of the time series by incorporating time basis selected by the group Lasso-type technique and choose the space basis based on smoothed functional principal component analysis. We show properties of this estimator under various situations extending current variable selection studies. At the second step, we obtain the detrended low dimensional stochastic process. However an important issue arises: is it justied, from an inferential point of view, to base further statistical inference on the estimated stochastic time series? We show that the dierence between the inference based on the estimated time series and true unobserved time series is asymptotically negligible, which ultimately allows one to study the dynamics of the whole high-dimensional system with a low dimensional representation together with the deterministic trend. We apply the method to our motivating empirical problems: studies of the dynamic behavior of temperatures (further used for pricing weather derivatives), implied volatilities and risk patterns and correlated brain activities (neuro-economics related) using fMRI data, where a panel version model is also presented.

Parametric estimation for discretely observed stochastic processes with jumps

We consider a two dimensional stochastic process (X,Y), which may have jump components and is not necessarily ergodic. There is an unknown parameter θ within the coefficients of (X,Y). The aim of this paper is to estimate θ from a regularly spaced sample of the process (X,Y). When the dynamic of X is known, an estimator is constructed by using a moment-based method. We show that our estimators will work if the Blumenthal-Getoor index of the jump part of Y is less than 2. What is perhaps the most interesting is the rate at which the estimators converge: it is $1/{\sqrt{n}}$ (as when the underlying processes are not contaminated by jumps) when that index is not greater than 1. When the dynamic of X is unknown, we introduce a spot volatility estimator-based approach to estimate θ. This approach can work even if the sample is contaminated by microstructure noise.

A stochastic process associated with the weighted white noise differentiation

In this paper, we give a relationship between the weighted white noise dierentiation and the Levy Laplacian introducing an operator changing the white noise ˙ B(t) by ˙ B(t) 2 . In addition, we give an infinite dimensional stochastic process generated by the weighted white noise dierentiation and a relationship between the stochastic process and the Levy Laplacian.

A SIMILARITY BETWEEN THE GROSS LAPLACIAN AND THE LÉVY LAPLACIAN

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

DIAGONALIZATION OF THE LÉVY LAPLACIAN AND RELATED STABLE PROCESSES

Eigenfunctions of the Lévy Laplacian with an arbitrary real number as an eigenvalue are constructed by means of a coordinate change of white noise distributions. The Lévy Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions and the corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional stochastic process related to the Lévy Laplacian is constructed from a one-dimensional stable process.