Stochastic optimization theory in Hilbert spaces—1

We study a class of stochastic optimization problems in which the state as well as the observation spaces are permitted to be (Hilbert spaces) of non-finite dimension. Although there have been previous attempts in the Hilbert space setting, our results, techniques, as well as applications, are totally different. We initiate the use of Gauss measure on a Hilbert space even though it is only finitely additive; and an associated theory of white noise, in contrast to the Wiener process theory, which is novel even in the finite dimensional case. We only treat time-invariant systems, but no strong ellipticity or coercivity conditions are used; we exploit the theory of semigroups of operators in contrast to the Lions-Magenes theory. A key result involves a far-reaching generalization of the Factorization theorem of Krein. We apply the results to the problem of boundary observation and control for partial differential equations. By the creation of a special state space, we can apply the theory to problems in which the state equations are finitedimensional but the noise does not have a rational spectrum. In a final section, we present a stochastic theory for inverse problems (System Identification) in the Hilbert space setting. The basic theoretical problem is the calculation of R-N derivatives for finitely additive measures. A fundamental result concerns Identifiability; in particular the identifiability of diffusion coefficients from boundary data is treated here for the first time.     

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.     

A self-dual variational approach to stochastic partial differential equations

Unlike many of their deterministic counterparts, stochastic partial differential equations are not amenable to the methods of calculus of variations à la Euler–Lagrange. In this paper, we show how self-dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. We construct solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We show how a stochastic version of Bolza's duality leads to solutions for equations with additive noise. We then use a Hamiltonian formulation to construct solutions for non-linear equations with non-additive noise such as the stochastic Navier–Stokes equations in dimension two.     

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.     

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise.     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.     

On the splitting-up method for rough (partial) differential equations

This article introduces the splitting method to systems driven by rough paths. The focus is on (nonlinear) partial differential equations with rough noise but we also cover rough differential equations. Applications to stochastic partial differential equations arising in control theory and nonlinear filtering are given.     

Particle Approximations for a Class of Stochastic Partial Differential Equations

The paper presents a particle approximation for a class of nonlinear stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The new results permit the treatment of filtering problems where the signal noise is no longer independent of the observation noise.     

Robust Parameter Estimation for Stochastic Differential Equations

We consider the estimation of parameters in stochastic differential equations (SDEs). The problem is treated in the setting of nonlinear filtering theory with a degenerate diffusion matrix. A robust stochastic Feynman–Kac representation for solutions of SDEs of Zakai-type is derived. It is verified that these solutions are conditional densities for the conditional measures defined by degenerate filtering problems. We show that the corresponding estimator for the parameters is robust in the following sense: It depends continuously on both the measurement path and on the intensity of the measurement noise. An algorithm based on a Monte-Carlo approach is given for the practical application of the estimator, and numerical results are reported. Mathematics Subject Classifications (2000) Primary: 62M05, 62M20; secondary: 62F15.     

Predictor-corrector pseudospectral methods for stochastic partial differential equations with additive white noise

Commonly used finite-difference numerical schemes show some deficiencies in the integration of certain types of stochastic partial differential equations with additive white noise. In this paper efficient predictor-corrector spectral schemes to integrate these equations are discussed. They are all based on the discretization of the system in Fourier space. The nonlinear terms are treated using a pseudospectral approach so as to speed up the computations without a significant loss of accuracy. The proposed schemes are applied to solve, both in one and two spatial dimensions, two paradigmatic continuum models arising in the context of nonequilibrium dynamics of growing interfaces: the Kardar-Parisi-Zhang and Lai-Das Sarma-Villain equations. Numerical results about the Lai-Das Sarma-Villain equation in two spatial dimensions have not been previously reported in the literature.     

A New Approach for Solving a Class of Delay Fractional Partial Differential Equations

In this article, a new numerical approach has been proposed for solving a class of delay time-fractional partial differential equations. The approximate solutions of these equations are considered as linear combinations of Müntz–Legendre polynomials with unknown coefficients. Operational matrix of fractional differentiation is provided to accelerate computations of the proposed method. Using Padé approximation and two-sided Laplace transformations, the mentioned delay fractional partial differential equations will be transformed to a sequence of fractional partial differential equations without delay. The localization process is based on the space-time collocation in some appropriate points to reduce the fractional partial differential equations into the associated system of algebraic equations which can be solved by some robust iterative solvers. Some numerical examples are also given to confirm the accuracy of the presented numerical scheme. Our results approved decisive preference of the Müntz–Legendre polynomials with respect to the Legendre polynomials.     

Stochastic Evolution Equations Driven by a Fractional White Noise

Abstract

In this paper we study stochastic evolution equations driven by a fractional white noise with arbitrary Hurst parameter in infinite dimension. We establish the existence and uniqueness of a mild solution for a nonlinear equation with multiplicative noise under Lipschitz condition by using a fixed point argument in an appropriate inductive limit space. In the linear case with additive noise, a strong solution is obtained. Those results are applied to stochastic parabolic partial differential equations perturbed by a fractional white noise.     

Integral Kernel Operators on Fock Space – Generalizations and Applications to Quantum Dynamics

A general theory of operators on Boson Fock space is discussed in terms of the white noise distribution theory on Gaussian space (white noise calculus). An integral kernel operator is generalized from two aspects: (i) The use of an operator-valued distribution as an integral kernel leads us to the Fubini type theorem which allows an iterated integration in an integral kernel operator. As an application a white noise approach to quantum stochastic integrals is discussed and a quantum Hitsuda–Skorokhod integral is introduced. (ii) The use of pointwise derivatives of annihilation and creation operators assures the partial integration in an integral kernel operator. In particular, the particle flux density becomes a distribution with values in continuous operators on white noise functions and yields a representation of a Lie algebra of vector fields by means of such operators.     

Partial differential equations driven by rough paths

We study a class of linear first and second order partial differential equations driven by weak geometric p-rough paths, and prove the existence of a unique solution for these equations. This solution depends continuously on the driving rough path. This allows a robust approach to stochastic partial differential equations. In particular, we may replace Brownian motion by more general Gaussian and Markovian noise. Support theorems and large deviation statements all become easy corollaries of the corresponding statements of the driving process. In the case of first order equations with Gaussian noise, we discuss the existence of a density with respect to the Lebesgue measure for the solution.     

On abstract stochastic bilinear equations with white noise inputs

We study the response of a class of stochastic bilinear hyperbolic (oscillatory) systems to multiplicative noise inputs. We formulate them as bilinear abstract Cauchy problems in the framework of (finitely additive) white noise theory. We show that the solution can be expressed in terms of the integral of the noise, which enables us to study relevant stochastic properties of the solution.     

Additive noise driven phase transitions in a predator-prey system

We explore the impact of additive noise on phase transitions in a predator-prey system, which is formulated by stochastic partial differential equations (SPDEs). The system is observed to experience twice phase transitions under certain level of additive noise. We extend the multiple scaling approach from single SPDE to multiple SPDEs and find a necessary and sufficient condition to excite the occurrence of the first transition from a spatially homogeneous state to a spatially regular spiral wave. Numerical experiments show the second phase transition from the regular spatial pattern to spiral turbulence. We conclude that additive noise has a destabilising effect on population dynamics by triggering the onset of Hopf bifurcation.     

Extended and unscented filtering algorithms using one-step randomly delayed observations

In this paper, the least squares filtering problem is investigated for a class of nonlinear discrete-time stochastic systems using observations with stochastic delays contaminated by additive white noise. The delay is considered to be random and modelled by a binary white noise with values of zero or one; these values indicate that the measurement arrives on time or that it is delayed by one sampling time. Using two different approximations of the first and second-order statistics of a nonlinear transformation of a random vector, we propose two filtering algorithms; the first is based on linear approximations of the system equations and the second on approximations using the scaled unscented transformation. These algorithms generalize the extended and unscented Kalman filters to the case in which the arrival of measurements can be one-step delayed and, hence, the measurement available to estimate the state may not be up-to-date. The accuracy of the different approximations is also analyzed and the performance of the proposed algorithms is compared in a numerical simulation example.     

A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The approach presented here can be used in a first course on differential equations for science and engineering majors.     

A White-Noise Approach to Stochastic Calculus

During the past 15 years a new technique, called the stochastic limit of quantum theory, has been applied to deduce new, unexpected results in a variety of traditional problems of quantum physics, such as quantum electrodynamics, bosonization in higher dimensions, the emergence of the noncrossing diagrams in the Anderson model, and in the large-N-limit in QCD, interacting commutation relations, new photon statistics in strong magnetic fields, etc. These achievements required the development of a new approach to classical and quantum stochastic calculus based on white noise which has suggested a natural nonlinear extension of this calculus. The natural theoretical framework of this new approach is the white-noise calculus initiated by T. Hida as a theory of infinite-dimensional generalized functions. In this paper, we describe the main ideas of the white-noise approach to stochastic calculus and we show that, even if we limit ourselves to the first-order case (i.e. neglecting the recent developments concerning higher powers of white noise and renormalization), some nontrivial extensions of known results in classical and quantum stochastic calculus can be obtained.