Vector-valued Stochastic Processes Research Articles

Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and which depend on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up of realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier–Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. The last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.

PERIODOGRAM ANALYSIS WITH MISSED OBSERVATION BETWEEN TWO VECTOR VALUED STOCHASTIC PROCESS.

10Nov 2017 PERIODOGRAM ANALYSIS WITH MISSED OBSERVATION BETWEEN TWO VECTOR VALUED STOCHASTIC PROCESS. M. A. Ghazal , A. I. El-desokey and A. M. Ben Aros. Department of Mathematics, Faculty of Science, University of Damietta, A. R. Egypt. Lecturer in Higher Future Institute for Specialized Technological Studies, Egypt. Department of Mathematics, Faculty of Science, University of Damietta, Egypt.

Fourth order Taylor–Kármán structured covariance tensor for gravity gradient predictions by means of the Hankel transformation

Vector-valued stochastic processes have been used for data processing, prediction, filtering, collocation, even network design and analysis in Geodesy, Physical Geodesy, and navigation etc. Recently, LEO satellite missions, such as CHAMP, GRACE and GOCE, provide a number of measurements to study the gravitational field of the Earth. When the gravity gradients, i.e. the second derivatives of the gravitational potential, are used for prediction and filtering by Kolmogorov–Wiener/Gauss–Markov concept, we will need the fourth-order covariance/correlation matrices of the gravity gradient signals. With the assumptions of homogeneous and isotropic field and the random functions of “potential type”, the paper aims at the development of the fourth order tensor-valued Taylor–Karman structured covariance/correlation matrices. The characteristic functions of these tensor-valued covariance/correlation matrices, namely the lateral and longitudinal components, will be derived for n-dimensional spaces, here specified for \(n=3\) dimensions in the paper. A special part is devoted to the Hankel transformation for gravity gradients and their variance-covariance in order to guarantee consistency well-known from problems in using Fourier transformations. We use the variance-covariance function of type (i) isotropic, (ii) homogeneous and (iii) potential as prior information for fitting the discrete data of variances and covariances estimated from observations.

Latent variables and space-time models for environmental problems

During the conference of the Italian Statistical Society on ‘‘Advances in Latent Variables’’ held in Brescia, Italy, June 19–21, 2013, the working group on environmental statistics, named GRASPA (http://www.graspa.org), organized the special track on ‘‘Space and space-time models: methods and environmental applications’’. This scientific activity was intended to celebrate the new condition as a permanent group of the Italian statistical society and renew GRASPA’s long life, which dates back to 1989. The idea of a special issue on the track main theme arose considering not only the number and scientific quality of the track contributions, but also the general interest of the topic, which exceeds the conference coverage. The result, raising from both track attendance and space-time statistical scientific community, is an interesting blend of various topics, which amounts at fourteen papers covering various environmental problems. To a large extent climate problems are considered, including upper atmosphere monitoring, rain simulation, air and water temperature and sea currents. Moreover, applications considered water body ecology, seismology and radon emissions. From the methodological point of view, areas covered by the special issue are scalar and vector-valued stochastic processes with continuous index over space or over spacetime. Moreover, functional data, point processes and time series methods are taken into account. Latent variables enter in a natural way in many of these papers either as spatial or temporal components. This methodological key is loosely used in the rest of this editorial to briefly review the special issue contributions. As for spatial processes methods, the work by RuizMedina and Porcu (2014) puts the theoretical basis for understanding the equivalence of Gaussian measures that index multivariate Gaussian fields. Their work will open the lines for research in estimation of Gaussian fields through tapering techniques as well as for assessing the properties of estimators of spatial dependence under infill asymptotics. For binary spatial data, which are built on the basis of a truncated latent Gaussian model, Bevilacqua et al. (2014) explore the possibility of building Euclidean likelihoods, obtaining high computational benefits and ensuring a reasonable level of statistical efficiency. In the same framework of spatial data, Verdin et al. (2014) introduce a stochastic weather generator for the variables of minimum and maximum temperature, as well as precipitation occurrence. In particular, temperature variables are modeled in a vector autoregressive framework, conditional on precipitation occurrence, whilst this last arises via a probit model. Both temperature and occurrence are spatially correlated using spatial Gaussian processes. Fontanella et al. (2014) consider a generalized latent-spatialquantile regression (GLSQR) model for the understanding of indoor radon gas monitoring in central Italy. Vallejo et al. (2014) consider a method for image landscape classification based on the assumption that the vector of image bands is a spatial multivariate process. They build such classification using the divergence of a modified Mahalanobis distance, given by the codispersion matrix. Methods for space-time processes are faced with great detail in this issue. If spatial design represents a critical issue, the paper by Stehlik et al. (2014) gives a clear picture A. Fasso (&) University of Bergamo, Viale Marconi 5, 24044 Dalmine BG, Italy e-mail: alessandro.fasso@unibg.it

Representation of Itô integrals by Lebesgue/Bochner integrals

In [Yong 2004], it was proved that as long as the integrand has certain properties, the corresponding It\^o integral can be written as a (parameterized) Lebesgue integral (or a Bochner integral). In this paper, we show that such a question can be answered in a more positive and refined way. To do this, we need to characterize the dual of the Banach space of some vector-valued stochastic processes having different integrability with respect to the time variable and the probability measure. The later can be regarded as a variant of the classical Riesz Representation Theorem, and therefore it will be useful in studying other problems. Some remarkable consequences are presented as well, including a reasonable definition of exact controllability for stochastic differential equations and a condition which implies a Black–Scholes market to be complete.

Discounting axioms imply risk neutrality

Although most applications of discounting occur in risky settings, the best-known axiomatic justifications are deterministic. This paper provides an axiomatic rationale for discounting in a stochastic framework. Consider a representation of time and risk preferences with a binary relation on a real vector space of vector-valued discrete-time stochastic processes on a probability space. Four axioms imply that there are unique discount factors such that preferences among stochastic processes correspond to preferences among present value random vectors. The familiar axioms are weak ordering, continuity and nontriviality. The fourth axiom, decomposition, is non-standard and key. These axioms and the converse of decomposition are assumed in previous axiomatic justifications for discounting with nonlinear intraperiod utility functions in deterministic frameworks. Thus, the results here provide the weakest known sufficient conditions for discounting in deterministic or stochastic settings. In addition to the four axioms, if there exists a von Neumann-Morgenstern utility function corresponding to the binary relation, then that function is risk neutral (i.e., affine). In this sense, discounting axioms imply risk neutrality.

MARM Processes Part I: General Theory

This paper introduces a new class of real vector-valued stochastic processes, called MARM (Multivariate Autoregressive Modular) processes, which generalizes the class of (univariate) ARM (Autoregressive Modular) processes. Like ARM processes, the key advantage of MARM processes is their ability to fit a strong statistical signature consisting of first-order and second-order statistics. More precisely, MARM processes exactly fit an arbitrary multi-dimensional marginal distribution and approximately fit a set of leading autocorrelations and cross-correlations. This capability appears to render the MARM modeling methodology unique in its ability to fit a multivariate model to such a class of strong statistical signatures. The paper describes the construction of two flavors of MARM processes, MARM+ and MARM−, studies the statistics of MARM processes (transition structure and second order statistics), and devises MARM-based fitting and forecasting algorithms providing point estimators and confidence intervals. The efficacy of the MARM fitting and forecasting methodology will be illustrated on real-life data in a companion paper.

Discounting and Risk Neutrality

Consider a binary relation on a real vector space of vector-valued discrete-time stochastic processes. If the binary relation satisfies four axioms, then there are unique discount factors such that preferences regarding stochastic processes induce prefences among present value random vectors. Three of the axioms are familiar: weak ordering, continuity, and non-triviality. The fourth axiom is decomposition. Also, if preferences satisfy the four axioms then the following properties are equivalent: the converse of decomposition, the existence of a felicity function on the set of random vectors, and risk neutrality. In this limited sense, discounting implies risk neutrality.

Approximations for the maximum of a vector-valued stochastic process with drift

Giving a generalization of Berkes and Horvath (2003), we consider the Euclidean norm of vector-valued stochastic processes, which can be approximated with a vector-valued Wiener process having a linear drift. The suprema of the Euclidean norm of the processes are not far away from the norm of the processes at the right most point. We also obtain an approximation for the supremum of the weighted Euclidean norm with a Wiener process.

A strong law of large numbers for nonexpansive vector-valued stochastic processes

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, $${\mathcal{F}}_0 \subset {\mathcal{F}}_1 \subset ... \subset {\mathcal{F}}_n $$ an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (xn) of strongly $${\mathcal{F}}_n $$ -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if $$E(x_{n + 1} \left| {{\mathcal{F}}_n ) = T(x_n )} \right.$$ . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (xn) be aT-martingale taking on values inH. If $$\sum\limits_{n = 1}^\infty {n^{ - 2} } E\left\| {x_{n + 1} - Tx_n } \right\|^2< \infty ,$$ then x n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (xn) be aT-martingale (taking on values inX). If $$\sum n ^{ - p} E\left( {\left\| {x_n - Tx_{n - 1} } \right\|^p } \right)< \infty ,$$ then there exists a continuous linear functionalf∈X * of norm 1 such that $$\mathop {\lim }\limits_{n \to \infty } f(x_n )/n = \mathop {\lim }\limits_{n \to \infty } \left\| {x_n } \right\|/n = inf\left\{ {\left\| {Tx - x} \right\|:x \in X} \right\} a.e.$$ If, in addition, the spaceX is strictly convex, x n /n converges weakly; and if the norm ofX * is Frechet differentiable (away from zero), x n /n converges strongly.

Equivalent martingale measures and no-arbitrage in stochastic securities market models

We characterize those vector-valued stochastic processes (with a finite index set and defined on an arbitrarystochasic base) which can become a martingale under an equivalent change of measure.This question is important in a widely studied problem which arises in the theory of finite period securities markets with one riskless bond and a finite number of risky stocks. In this setting, our characterization gives a criterion for recognizing when a securities market model allows for no arbitrage opportunities (free lunches). Intuitively, this can be interpreted as saying if one cannot win betting on a process, then it must be a martingale under an equivalent measure, and provides a converse to the classical notion that one cannot win betting on a martingale.

Discussion of two procedures for expanding a vector-valued stochastic process in an orthonormal way

Discussion of two procedures for expanding a vector-valued stochastic process in an orthonormal way

Vector-Valued Stochastic Processes. V. Optional and Predictable Variation of Stochastic Measures and Stochastic Processes

Vector-Valued Stochastic Processes. V. Optional and Predictable Variation of Stochastic Measures and Stochastic Processes

Vector-valued stochastic processes. I. Vector measures and vector-valued stochastic processes with finite variation

In this paper we study the relationshipμ V (M)=E(∫1 M dV S ) between operatorvalued processesV with finite variation ⋎V⋎ and operator-valued stochastic measuresμ V with finite variation |μ V |. The variations satisfy the inequality |μ V |⩽μ|V|, which, under certain conditions, is an equality (for example, ifV is measurable).

Vector-Valued Stochastic Processes. II. A Radon-Nikodym Theorem for Vector-Valued Processes with Finite Variation

Vector-Valued Stochastic Processes. II. A Radon-Nikodym Theorem for Vector-Valued Processes with Finite Variation

Vector-valued stochastic processes. V. Optional and predictable variation of stochastic measures and stochastic processes

Let μ \mu be a stochastic measure, with values in a Banach space E E , with finite variation | μ | |\mu | . If μ \mu is optional (resp. predictable), then | μ | |\mu | is also optional (resp. predictable) provided E E is separable, or the dual of a separable space, or has the Radon-Nikodym property. Let A A be a right continuous stochastic process with values in E E , with finite variation | A | |A| . If A A is measurable (resp. optional, predictable), then | A | |A| , the continuous part | A | c |A{|^c} and the discrete part | A | d |A{|^d} have the same property.

Vector-valued stochastic processes. II. A Radon-Nikodým theorem for vector-valued processes with finite variation

Given a real-valued process A A with finite variation | A | \left | A \right | and a vector-valued process B B with finite variation | B | \left | B \right | such that for each ω \omega , the Stieltjes measure d B . ( ω ) dB{\mathbf {.}}(\omega ) is absolutely continuous with respect to d A . ( ω ) dA.(\omega ) , there exists a vector-valued process H H which, under certain separability conditions, satisfies B t = ∫ [ 0 , t ] H s d A s {B_t} = \int _{[0,t]} {{H_s}d{A_s}} and | B | t = ∫ [ 0 , t ] ‖ H s ‖ d | A | s {\left | B \right |_t} = \int _{[0,t]} {\left \| {{H_s}} \right \|d{{\left | A \right |}_s}} for every t ≥ 0 t \geq 0 . If, moreover, A A and B B are optional or predictable, then so is H H .

The Andersen-Jessen theorem revisited

AbstractFor stochastic processes which are induced by a signed measure, the Andersen-Jessen theorem asserts almost sure convergence and yields the identification of the limit. This result has been extended to real and vector-valued stochastic processes which are induced by a finitely additive set function or a set function process. In the present paper, we study the structure of such induced stochastic processes in order to locate the Andersen-Jessen theorem and its extensions in the family of convergence theorems for martingales and their generalizations. As an application of these results, we also show that the Andersen-Jessen theorem and its extensions can be deduced from the convergence theorems for conditional expectations and positive supermartingales.

Initial correlations of the multiplicative process, driven by random jumps

The authors consider a vector-valued stochastic process, which is multiplicatively driven by the Markov jump process. They obtain a closed expression for the average of the vector process by solving the Burshtein equation for the marginal average. It is shown that the solution for t>t0 requires knowledge of an initial correlation operator, due to the finite correlation time of the jump process. They derive the equation for the steady-state solution, which is applied to evaluate the stationary correlation functions. Then some specific limits of the jump process, are discussed, which arise if one takes the correlation time to be zero or infinite. For the special case L(x)=A+xB they derive from the Burshtein equation a recurrence relation between the moments of the vector process. This relation is solved and it is shown how all initial moments at t0 determine the moments for t>or=t0. The recurrence relation is applied to solve the two-state and the three-state process more explicitly. It is pointed out that the occurring initial correlations cannot be neglected in general.

Stationary symbol sequences from variable-length word sequences

Variable-length word sequences (VLWS) and symbol sequences obtained from VLWS are discussed. A VLWS is regarded as a vector-valued stochastic process in which the sequence of vector dimensions is itself a stochastic process. For a given VLWS, the symbol sequence is not uniquely defined because it depends on how the choice of the time origin is made. In particular, a deterministic choice leads to a nonstationary symbol sequence even if the originating VLWS is stationary. A suitable random choice must be made in order to get a stationary symbol sequence from a stationary VLWS. Attention is given to the spectral analysis of both the VLWS and the corresponding stationary symbol sequence. An application to a VLWS consisting of mutually independent words is given.