Some recent results in finitely additive white noise theory

We present a short survey of some very recent results on the finitely additive white noise theory. We discuss the Markov property of the solution of a stochastic differential equation driven directly by a white noise, study the Radon-Nikodym derivative of the measure induced by nonlinear transformation on a Hilbert space with respect to the canonical Gauss measure thereon and obtain a representation for nonlinear filter maps.     

Connections between three-dimensional inverse scattering and linear least-squares estimation of random fields

The three-dimensional Schrödinger equation inverse scattering problem with a nonspherically-symmetric potential is related to the filtering problem of computing the linear leastsquares estimate of the three-dimensional random field on the surface of a sphere from noisy observations inside the sphere. The relation consists of associating an estimation problem with the inverse scattering problem, and vice-versa. This association allows equations and quantities for one problem to be given interpretations in terms of the other problem. A new fast algorithm is obtained for the estimation of random fields using this association. The present work is an extension of the connections between estimation and inverse scattering already known to exist for stationary random processes and one-dimensional scattering potentials, and isotropic random fields and radial scattering protentials.     

Stochastic resonance in a linear static system driven by correlated multiplicative and additive noises

In this work, we investigate the signal transmission in a linear static system driven by correlated multiplicative and additive noises. When the input signal is periodic, we depict the stochastic resonance (SR) phenomenon by employing the signal-to-noise ratio (SNR) theory; while the input signal is aperiodic, we describe the SR phenomenon by using the input–output cross correlation theory. And the exact analytic expressions of the output SNR and the normalized time averaged cross covariance between input and output are obtained. The results show: under the condition of negative correlated noises, SR arises; while with positive correlated or uncorrelated noises, there is no SR. This result may extend the SR theory to a common linear static system.     

New filter design for linear time-delay systems

This paper develops new robust delay-dependent filter design for a class of linear systems with time-varying delays and convex-bounded parameter uncertainties. The design procedure hinges upon the constructive use of an appropriate Lyapunov functional plus a free-weighting matrices in order to exhibit the delay-dependent dynamics. The developed approach utilizes smaller number of LMI decision variables thereby leading to less conservative solutions to the delay-dependent stability and filtering problems. Subsequently, linear matrix inequalities (LMIs)-based conditions are characterized such that the linear delay system is robustly asymptotically stable with an γ-level L₂-gain. All the developed results are tested on representative examples.     

On the abstract theory of additive and multiplicative Schwarz algorithms

Summary. In recent years, it has been shown that many modern iterative algorithms (multigrid schemes, multilevel preconditioners, domain decomposition methods etc.) for solving problems resulting from the discretization of PDEs can be interpreted as additive (Jacobi-like) or multiplicative (Gauss-Seidel-like) subspace correction methods. The key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space into a sum of subspaces and the splitting of the variational problem on into auxiliary problems on these subspaces. In this paper, we propose a modification of the abstract convergence theory of the additive and multiplicative Schwarz methods, that makes the relation to traditional iteration methods more explicit. The analysis of the additive and multiplicative Schwarz iterations can be carried out in almost the same spirit as in the traditional block-matrix situation, making convergence proofs of multilevel and domain decomposition methods clearer, or, at least, more classical. In addition, we present a new bound for the convergence rate of the appropriately scaled multiplicative Schwarz method directly in terms of the condition number of the corresponding additive Schwarz operator. These results may be viewed as an appendix to the recent surveys [X], [Ys]. Received February 1, 1994 / Revised version received August 1, 1994     

Simplified order 4.0 weak Taylor schemes for additive noise

Following Kloeden and Platen [P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992] Taylor schemes are considered here as the starting point to obtain simplified Taylor schemes replacing the multiple integrals by simpler variables. The conditions that a group of variables has to fulfill so that the new scheme reaches weak-order 4.0 in the additive noise case are given explicitly, as well as the way to find such groups. For a particular selection, a pair of stochastic schemes with order 4.0 in the weak sense, correcting the one proposed in [P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992], is obtained.     

Suprathreshold stochastic resonance in multilevel threshold system driven by multiplicative and additive noises

The suprathreshold stochastic resonance in multithreshold neuronal networks system driven by multiplicative Gaussian noise and additive Gaussian noise is studied. The expression of the mutual information is derived, and the effects of the noise intensity and system parameter on mutual information are discussed. It is found that adjusting the additive noise intensity is more effective than adjusting the multiplicative noise intensity to enhance information transmission, and the more the number of devices, the more apparent the phenomenon of suprathreshold stochastic resonance. Moreover, we also found that the selection of threshold is very important in the process of information transmission.     

Dynamic Markov bridges motivated by models of insider trading

Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own filtration and satisfies X₁=Z₁. We call X a dynamic bridge, because its terminal value Z₁ is not known in advance. We compute its semimartingale decomposition explicitly under both its own filtration F^X and the filtration F^X,Z jointly generated by X and Z. Our construction is heavily based on parabolic partial differential equations and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading that can be viewed as a non-Gaussian generalization of the model of Back and Pedersen (1998) [3], where the insider’s additional information evolves over time.     

Forgetting the initial distribution for Hidden Markov Models

The forgetting of the initial distribution for discrete Hidden Markov Models (HMM) is addressed: a new set of conditions is proposed, to establish the forgetting property of the filter, at a polynomial and geometric rate. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using different HMM of interest: the dynamic tobit model, the nonlinear state space model and the stochastic volatility model.     

Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes

Multiplicative programming problems (MPPs) are global optimization problems known to be NP-hard. In this paper, we employ algorithms developed to compute the entire set of nondominated points of multi-objective linear programmes (MOLPs) to solve linear MPPs. First, we improve our own objective space cut and bound algorithm for convex MPPs in the special case of linear MPPs by only solving one linear programme in each iteration, instead of two as the previous version indicates. We call this algorithm, which is based on Benson’s outer approximation algorithm for MOLPs, the primal objective space algorithm. Then, based on the dual variant of Benson’s algorithm, we propose a dual objective space algorithm for solving linear MPPs. The dual algorithm also requires solving only one linear programme in each iteration. We prove the correctness of the dual algorithm and use computational experiments comparing our algorithms to a recent global optimization algorithm for linear MPPs from the literature as well as two general global optimization solvers to demonstrate the superiority of the new algorithms in terms of computation time. Thus, we demonstrate that the use of multi-objective optimization techniques can be beneficial to solve difficult single objective global optimization problems.     

Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise

A class ofimplicit Runge-Kutta schemes for stochastic differential equations affected bymultiplicative Gaussian white noise is shown to be optimal with respect to global order of convergence in quadratic mean. A test equation is proposed in order to investigate the stability of discretization methods for systems of this kind. Herestability is intended in a truly probabilistic sense, as opposed to the recently introduced extension of A-stability to the stochastic context, given for systems with additive noise. Stability regions for the optimal class are also given.Partially supported by the Italian Consiglio Nazionale delle Ricerche.     

Mean first-passage time in a bistable system driven by multiplicative and additive colored noises with colored cross-correlation

The effect of any system parameter (include p, q, λ, τ₁, τ₂, τ₃) on the mean first-passage time is investigated in a symmetric bistable system driven by multiplicative and additive colored noise with colored cross-correlation. Expression of the mean first-passage time (MFPT) has been obtained by applying the steepest-descent approximation. The effects of the additive noise intensity and the multiplicative noise intensity on the MFPT are mostly different only with bigger positive values of the strength of correlations between multiplicative and additive noise; and the MFPT has a maximum with the increase of the multiplicative noise intensity. The effects of the noise self-correlation times on the MFPT are quite different from that of the correlation time between the noises as the strength of correlations between multiplicative and additive noise is positive.     

Algorithms of filtration and extrapolation for discrete-time dynamical systems

The bases of theory and the recursive filtration algorithms ensuring the guaranteed precision of estimate for an extrapolated state of a dynamic system are described. A determined precision is ensured by corresponding choice of algorithm parameters.The different algorithms of filtration and extrapolation are investigated. These algorithms may be used in constructing tracking systems, organizing of corresponding measurements and estimation of parameters in information systems.     

Descent Kummer theory via Weil restriction of multiplicative groups

We prove a Kummer duality for certain fields without roots of unity by using the Weil restriction of the multiplicative groups. This is a natural generalization of the classical Kummer theory.     

Stability radii of positive linear Volterra-Stieltjes equations

We study stability radii of linear Volterra-Stieltjes equations under multi-perturbations and affine perturbations. A lower and upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer functions. Then, the class of positive linear Volterra-Stieltjes equations is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices. As direct consequences of the obtained results, we get some results on robust stability of positive linear integro-differential equations and of positive linear functional differential equations. To the best of our knowledge, most of the results of this paper are new.     

Multidimensional constant linear systems

A continuous resp. discrete r-dimensional (r1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).Partially supported by US Air Force Grant AFOSR-87-0249 and by Office of Naval Research Grant N 00014-86-K-0538 through the Center for Mathematical System Theory, University of Florida, Gainesville, Florida, U.S.A.     

Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems

We provide a short and elementary proof for the recently proved result by G. da Prato and H. Frankowska that - under minimal assumptions - a closed set is invariant with respect to a stochastic control system if and only if it is invariant with respect to the (associated) deterministic control system.     

Multilevel additive and multiplicative methods for orthogonal spline collocation problems

Summary. Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are presented. Received March 1, 1994 / Revised version received January 23, 1996