Stochastic realization of a Gaussian stochastic control system

The stochastic realization problem is considered of representing a stationary Gaussian process as the observation process of a Gaussian stochastic control system. The problem formulation includes that the lastm components of the observation process form the Gaussian white noise input process to the system. Identifiability of this class of systems motivates the problem. The results include a necessary and sufficient condition for the existence of a stochastic realization. A subclass of Gaussian stochastic control systems is defined that is almost a canonical form for this stochastic realization problem. For a structured Gaussian stochastic control system an equivalent condition for identifiability of the parametrization is stated.The research of this paper is supported in part by the Commission of the European Communities through the SCIENCE Program by the projectSystem Identification with contract number SC1-CT92-0779.     

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the non-linear stochastic heat equation in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.     

Stochastic evolution equations with general white noise disturbance

Existence and uniqueness theorems are proved for a general class of stochastic linear abstract evolution equations, with a general type of stochastic forcing term. The abstract evolution equation is modeled using an evolution operator (or 2-parameter semigroup) approach and this includes linear partial differential equations and linear differential delay equations. The stochastic forcing term is modeled by defining an Itô stochastic integral with respect to a Hilbert space-valued orthogonal increments process, which can be used to model both Gaussian and non-Gaussian white noise processes. The theory is illustrated by examples of stochastic partial differential equations and delay equations, which arise in filtering problems for distributed and delay systems.     

Statistical structuring theory in parametrically excitable dynamical systems with a Gaussian pump

Based on the idea of the statistical topography, we analyze the problem of emergence of stochastic structure formation in linear and quasilinear problems described by first-order partial differential equations. The appearance of a parametric excitation on the background of a Gaussian pump is a specific feature of these problems. We obtain equations for the probability density of the solutions of these equations, whence it follows that the stochastic structure formation emerges with probability one, i.e., for almost every realization of the random parameters of the medium.     

Estimation and filtering on a doubly stochastic poisson process

An explicit formula for the characteristic function of a doubly stochastic Poisson process is derived in this paper by means of the harmonic decomposition of its intensity function that we suppose to be Gaussian. The statistical moments are then obtained, as well as the sample function density of the process. These results are applied to estimate the parameters of several well-known processes. Finally, a linear filtering procedure for the intensity function is developed and the algorithm is implemented by computers.     

一类多维参数高斯过程的弱逼近

本文研究了一类多维参数高斯过程的弱极限问题.在一般情况下,利用泊松过程得到了此类过程的弱极限定理,此多维参数高斯过程可表示为确定的核函数关于维纳过程的随机积分,且包含多维参数的分数布朗运动.     

Existence of the linear prediction for Banach space valued Gaussian processes

The correspondence between Gaussian stochastic processes with values in a Banach space E and cylindrical processes which are related to them is studied. It is shown that the linear prediction of an E-valued Gaussian process is an E-valued random variable as well as the spectral measure of an E-valued Gaussian stationary process is a Gaussian random measure.     

A linear system‐free Gaussian RBF method for the Gross‐Pitaevskii equation on unbounded domains

Gaussian radial basis function (RBF) interpolation methods are theoretically spectrally accurate. However, in applications this accuracy is seldom realized due to the necessity of solving a very poorly conditioned linear system to evaluate the methods. Recently, by using approximate cardinal functions and restricting the method to a uniformly spaced grid (or a smooth mapping thereof), it has been shown that the Gaussian RBF method can be formulated in a matrix free framework that does not involve solving a linear system [ 1 ]. In this work, we differentiate the linear system‐free Gaussian (LSFG) method and use it to solve partial differential equations on unbounded domains that have solutions that decay rapidly and that are negligible at the ends of the grid. As an application, we use the LSFG collocation method to numerically simulate Bose‐Einstein condensates. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 389–401, 2012     

Stochastic Integrals and Evolution Equations with Gaussian Random Fields

The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of the chaos space of the random field, combined with the Wick product, leads to the Itô-Skorokhod integral, and provides an efficient tool to study the integral, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields. Also considered are the corresponding linear stochastic evolution equations.     

Gaussian and sparse processes are limits of generalized Poisson processes

The theory of sparse stochastic processes offers a broad class of statistical models to study signals, far beyond the more classical class of Gaussian processes. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by Lévy white noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Lévy process—from Gaussian to sparse—can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects.     

Wiener Chaos Approach to Optimal Prediction

The chaos expansion of a general non-linear function of a Gaussian stationary increment process conditioned on its past realizations is derived. This work combines the Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on a realization of its past. This is done by considering special bases for the Gaussian space 𝒢 generated by the process, which allows us to obtain an orthogonal basis for the Fock space of 𝒢 such that each basis element is either measurable or independent with respect to the given samples. This allows us to easily derive the chaos expansion of a random variable conditioned on part of the sample path. We provide a general method for the construction of such basis when the underlying process is Gaussian with stationary increment. We evaluate the basis elements in the case of the fractional Brownian motion, which leads to a prediction formula for this process.     

Two-parameter diffusions random fields

This paper presents an alternative method for calculating the diffusion, drift, and mixed coefficients of an example of biparameter Gaussian diffusion defined as a solution of a linear hyperbolic stochastic partial differential equation (Nualart & Sanz , 1979). To derive the expression of these coefficients, we part from an integral stochastic repre , sentation given by these authors for this class of biparameter diffusion processes arising from biparameter Gaussian random fields verifying a particular Markov property     

On stochastic processes with linear conditional moments

In the paper, we show that a stochastic process with linear regression and linear conditional variance must be either a Gaussian process or a Poisson-type process. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

Nonlinear Transformations of Smooth Measures on Infinite-Dimensional Spaces

We investigate the properties of the image of a differentiable measure on an infinitely-dimensional Banach space under nonlinear transformations of the space. We prove a general result concerning the absolute continuity of this image with respect to the initial measure and obtain a formula for density similar to the Ramer–Kusuoka formula for the transformations of the Gaussian measure. We prove the absolute continuity of the image for classes of transformations that possess additional structural properties, namely, for adapted and monotone transformations, as well as for transformations generated by a differential flow. The latter are used for the realization of the method of characteristics for the solution of infinite-dimensional first-order partial differential equations and linear equations with an extended stochastic integral with respect to the given measure.     

Operator differential-algebraic equations with noise arising in fluid dynamics

We study linear semi-explicit stochastic operator differential algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able to include Gaussian noise and stochastic convolution terms as perturbations in the differential as well as in the constraint equation. By the application of the polynomial chaos expansion method, we reduce the stochastic operator DAE to an infinite system of deterministic operator DAEs for the stochastic coefficients. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. This then allows to prove the existence and uniqueness of the solution of the initial stochastic operator DAE in a certain weighted space of stochastic processes.     

The pricing of Asian options under stochastic interest rates

The purpose of this paper is to analyse the effect of stochastic interest rates on the pricing of Asian options. It is shown that a stochastic, in contrast to a deterministic, development of the term structure of interest rates has a significant influence. The price of the underlying asset, e.g. a stock or oil, and the prices of bonds are assumed to follow correlated two-dimensional Itô processes. The averages considered in the Asian options are calculated on a discrete time grid, e.g. all closing prices on Wednesdays during the lifetime of the contract. The value of an Asian option will be obtained through the application of Monte Carlo simulation, and for this purpose the stochastic processes for the basic assets need not be severely restricted. However, to make comparison with published results originating from models with deterministic interest rates, we will stay within the setting of a Gaussian framework.     

Well posedness,large deviations and ergodicity of the stochastic 2D Oldroyd model of order one

In this work, we establish the unique global solvability of the stochastic two dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows perturbed by multiplicative Gaussian noise. A local monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. The Laplace principle for the strong solution of the stochastic system is established in a suitable Polish space using a weak convergence approach. The Wentzell–Freidlin large deviation principle is proved using the well known results of Varadhan and Bryc. The large deviations for shot time are also considered. We also establish the existence of a unique ergodic and strongly mixing invariant measure for the stochastic system with additive Gaussian noise, using the exponential stability of strong solutions.     

On a characterization of processes by the independence of linear forms in a triangular system

Necessary and sufficient conditions are given for a stochastic process to be either a Gaussian or an independent increments process. These conditions are based on the independence of some linear forms in a triangular system and the linearity of regression. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

Efficient transient noise analysis in circuit simulation

Stochastic differential algebraic equations (SDAEs) arise as a mathematical model for electrical network equations that are influenced by additional sources of Gaussian white noise. We discuss adaptive linear multi-step methods for their numerical integration, in particular stochastic analogues of the trapezoidal rule and the two-step backward differentiation formula, and we obtain conditions that ensure mean-square convergence of this methods. For the case of small noise we present a strategy for controlling the step-size in the numerical integration. It is based on estimating the mean-square local errors and leads to step-size sequences that are identical for all computed paths. Test results illustrate the performance of the presented methods. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A developed solution of the stochastic Milne problem using probabilistic transformations

This paper considers the solution of the Milne problem of radiative transfer with isotropic scattering in a continuous stochastic medium. Properties of the medium are assumed to be continuous random functions of the spatial dimensions. The available solutions - in literature - for this stochastic integro-differential equation (SIDE) are represented only by the ensemble average of the radiant energy density. In this paper, a developed algorithm, based on the implementation of the random variable transformation technique together with an integral transformation to the stochastic properties, is introduced. A complete stochastic solution represented by the probability-density function (p.d.f) of the radiant energy density is obtained. Using the closed form of the p.d.f, the nth moment of the stochastic solution is evaluated. In realization of this work, Exponential and Gaussian statistics for the medium properties are assumed. Results are physically acceptable and found to be compatible with those in the literature.