Wiener Chaos and Nonlinear Filtering

The paper discusses two algorithms for solving the Zakai equation in the time-homogeneous diffusion filtering model with possible correlation between the state process and the observation noise. Both algorithms rely on the Cameron-Martin version of the Wiener chaos expansion, so that the approximate filter is a finite linear combination of the chaos elements generated by the observation process. The coefficients in the expansion depend only on the deterministic dynamics of the state and observation processes. For real-time applications, computing the coefficients in advance improves the performance of the algorithms in comparison with most other existing methods of nonlinear filtering. The paper summarizes the main existing results about these Wiener chaos algorithms and resolves some open questions concerning the convergence of the algorithms in the noise-correlated setting. The presentation includes the necessary background on the Wiener chaos and optimal nonlinear filtering.     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

   Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

Stochastic system for generalized polytropic filtration

In this paper, we study a certain class of stochastic quasilinear parabolic equations describing a generalized polytropic elastic filtration in the framework of variable exponents Lebesgue and Sobolev spaces. We establish an existence result in the infinite dimensional framework of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions, and the governing equations are subjected to cylindrical Wiener processes. We use a Galerkin method, derive crucial a priori estimates for the approximate solutions, and combine profound analytic and probabilistic compactness results in order to pass to the limit. Several difficulties arise in obtaining these uniform bounds and passing to the limit since the nonlinear elliptic part of the leading operator admits nonstandard growth. Apart from adapting the above essential tools, we extend classical methods of monotonicity to the present situation.     

Nonlinear Filtering for Diffusions in Random Environments

Suppose that the signal X to be estimated is a diffusion process in a random medium W and the signal is correlated with the observation noise. We study the historical filtering problem concerned with estimating the signal path up until the current time based upon the back observations. Using Dirichlet form theory, we introduce a filtering model for general rough signal X ^W and establish a multiple Wiener integrals representation for the unnormalized pathspace filtering process. Then, we construct a precise nonlinear filtering model for the process X itself and give the corresponding Wiener chaos decomposition.     

Wave Equation Driven by Fractional Generalized Stochastic Processes

Fractional derivatives of generalized stochastic processes have the global properties and keep the memory of their own. They are applicable for processes with memory. We employ them in solving equations driven by fractional derivatives of singular noises and singular initial data. We work on the perturbation of the wave equation by fractional time and space derivatives of generalized processes, in particular with Wiener process and a nonlinear term. The Wiener process is used to represent the integral of a Gaussian white noise process, and so is useful as a model of noise in electronics engineering, instrument errors in filtering theory and unknown forces in control theory.     

State estimation for partially observed jump processes

In this paper partially observed jump processes are considered and optimal filtering equations are given for the conditional expectation of a functional on the past of the process.Rudemo [6] derived filtering equations for a partially observed jump Markov process. Snyder [3] gives equations for the conditional characteristic function of a jump process. Segall et al. [2] discuss filtering for processes with counting observations. Their work carries over to processes with counting observations the martingale methods that Fujisaki et al. [1] had used to derive nonlinear filtering equations for processes governed by Ito equations. Many further references to filtering for processes with discrete state measurements are given in the references cited.The objective of this paper is to show that by making use of the concept of a representation of a functional the idea of Rudemo's proof of [6, pp. 595–599] can be carried over to jump processes. The author feels that this is a very interesting proof because of its simplicity. It involves only calculations with conditional expectations and the rule for differentiation of a quotient.     

Optimal speed of detection in generalized Wiener disorder problems

We define a general Wiener disorder problem in which a sudden change in a time profile of unknown size has to be detected in white noise of small intensity. Since both the time of the change and its size are unknown, this problem is considerably harder than standard Wiener disorder problems where the size of the change is assumed to be known a priori. We formulate the problem within the Bayesian framework of nonlinear filtering theory, and use Strassen's functional law of the iterated logarithm to bound stochastic measures which arise in the nonlinear filtering equations. This leads to explicit expressions for the detection delay in the optimal statistics for small noise intensities, and we indicate how these can be used to analyse the detection delays of recursive suboptimal detection algorithms for this problem.     

Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs

In this paper we study mathematically and computationally optimal control problems for stochastic elliptic partial differential equations. The control objective is to minimize the expectation of a tracking cost functional, and the control is of the deterministic, distributed type. The main analytical tool is the Wiener-Itô chaos or the Karhunen-Loève expansion. Mathematically, we prove the existence of an optimal solution; we establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations; we represent the input data in their Wiener-Itô chaos expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the optimality system through the discretizations of the probability space and the spatial space by the finite element method; we also derive error estimates in terms of both types of discretizations.     

Numerical analysis of the stochastic Stokes equations of Wick type

We propose a finite element method for the numerical solution of the stochastic Stokes equations of the Wick type. We give existence and uniqueness results for the continuous problem and its approximation. Optimal error estimates are derived and algorithmic aspects of the method are discussed. Our method will reduce the problem of solving stochastic Stokes equations to solving a set of deterministic ones. Moreover, one can reconstruct particular realizations of the solution directly from Wiener chaos expansions once the coefficients are available. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Numerical schemes for random ODEs with affine noise

Numerical schemes for random ordinary differential equations, abbreviated RODEs, with an affine structure can be derived in a similar way as for affine control systems using Taylor expansions that resemble stochastic Taylor expansions for Stratonovich stochastic differential equations. The driving noise processes can be quite general, such as Wiener processes or fractional Brownian motions with continuous sample paths or compound Poisson processes with piecewise constant sample paths, and even more general noises. Such affine-Taylor schemes of arbitrarily high order are constructed here. It is shown how their structure simplifies when the noise terms are additive or commutative. A derivative free counterpart is given and multi-step schemes are derived too. Numerical comparisons are provided for various explicit one-step and multi-step schemes in the context of a toggle switch model from systems biology.     

Interface cracks in anisotropic composites

The linear model equations of elasticity often give rise to oscillatory solutions in some vicinity of interface crack fronts. In this paper we apply the Wiener–Hopf method which yields the asymptotic behaviour of the elastic fields and, in addition, criteria to prevent oscillatory solutions. The exponents of the asymptotic expansions are found as eigenvalues of the symbol of corresponding boundary pseudodifferential equations. The method works for three‐dimensional anisotropic bodies and we demonstrate it for the example of two anisotropic bodies, one of which is bounded and the other one is its exterior complement. The common boundary is a smooth surface. On one part of this surface, called the interface, the bodies are bonded, while on the complementary part there is a crack. By applying the potential method, the problem is reduced to an equivalent system of Boundary Pseudodifferential Equations (BPE) on the interface with the stress vector as the unknown. The BPEs are defined via Poincaré–Steklov operators. We prove the unique solvability of these BPEs and obtain the full asymptotic expansion of the solution near the crack front. As a special case we consider the interface crack between two different isotropic materials and derive an explicit criterion which prevents oscillatory solutions. Copyright © 1999 John Wiley & Sons, Ltd.     

Asymptotic expansions for the variance of stopping times in nonlinear renewal theory

We treat the problem of finding asymptotic expansions for the variance of stopping times for Wiener processes with positive drift (continuous time case) as well as sums of i.i.d. random variables with positive mean (discrete time case). Carrying over the setting of nonlinear renewal theory to Wiener processes, we obtain an asymptotic expansion up to vanishing terms in the continuous time case. Applying the same methods to sums of i.i.d. random variables, we also provide an expansion in the discrete time case up to terms of order o(b^1/2) where the leading term is of order O(b), as b → ∞. The possibly unbounded term is the covariance of nonlinear excess and stopping time.     

Correspondence between frame shrinkage and high-order nonlinear diffusion

Nonlinear diffusion filtering and wavelet/frame shrinkage are two popular methods for signal and image denoising. The relationship between these two methods has been studied recently. In this paper we investigate the correspondence between frame shrinkage and nonlinear diffusion.We show that the frame shrinkage of Ron-Shen?s continuous-linear-spline-based tight frame is associated with a fourth-order nonlinear diffusion equation. We derive high-order nonlinear diffusion equations associated with general tight frame shrinkages. These high-order nonlinear diffusion equations are different from the high-order diffusion equations studied in the literature. We also construct two sets of tight frame filter banks which result in the sixth- and eighth-order nonlinear diffusion equations.The correspondence between frame shrinkage and diffusion filtering is useful to design diffusion-inspired shrinkage functions with competitive performance. On the other hand, the study of such a correspondence leads to a new type of diffusion equations and helps to design frame-inspired diffusivity functions. The denoising results with diffusion-inspired shrinkages provided in this paper are promising.     

Filtering with Marked Point Process Observations via Poisson Chaos Expansion

We study a general filtering problem with marked point process observations. The motivation comes from modeling financial ultra-high frequency data. First, we rigorously derive the unnormalized filtering equation with marked point process observations under mild assumptions, especially relaxing the bounded condition of stochastic intensity. Then, we derive the Poisson chaos expansion for the unnormalized filter. Based on the chaos expansion, we establish the uniqueness of solutions of the unnormalized filtering equation. Moreover, we derive the Poisson chaos expansion for the unnormalized filter density under additional conditions. To explore the computational advantage, we further construct a new consistent recursive numerical scheme based on the truncation of the chaos density expansion for a simple case. The new algorithm divides the computations into those containing solely system coefficients and those including the observations, and assign the former off-line.     

The wavelet transform for Wiener functionals and some applications

The wavelet transform is defined for Wiener functionals. We characterize global and local regularities of Wiener functionals and we give a criterion for the existence and regularity of densities. Such a criterion is applied to diffusion processes and to the solutions to backward stochastic differential equations.     

Second order probabilistic parametrix method for unbiased simulation of stochastic differential equations

In this article, following the paradigm of bias–variance trade-off philosophy, we derive parametrix expansions of order two, based on the Euler–Maruyama scheme with random partitions, for the purpose of constructing an unbiased simulation method for multidimensional stochastic differential equations. These formulas lead to Monte Carlo simulation methods which can be easily parallelized. The second order method proposed here requires further regularity of coefficients in comparison with the first order method but achieves finite moments even when Poisson sampling is used for the partitions, in contrast to Andersson and Kohatsu-Higa (2017). Moreover, using an exponential scaling technique one achieves an unbiased simulation method which resembles a space importance sampling technique which significantly improves the efficiency of the proposed method. A hint of how to derive higher order expansions is also presented.     

The Skorohod integral in conuclear spaces

We define an anticipative stochastic integral with respect to a nonhomogeneous Wiener process in a dual of a nuclear space and investigate its basic properties. The theory is developed without the use of chaos expansions.     

Uncertainty investigations in nonlinear aeroelastic systems

In this paper, the stochastic collocation method (SCM) is applied to investigate the nonlinear behavior of an aeroelastic system with uncertainties in the system parameter and the initial condition. Numerical case studies for problems with uncertainties are carried out. In particular, the performance of the SCM is compared with solutions based on other computational techniques such as Monte Carlo simulation, Wiener chaos expansion and wavelet chaos expansion. From the computational results, we conclude that the SCM is an effective tool to study a nonlinear aeroelastic system with random parameters.     

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of non-autonomous linear parabolic initial- and final-boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary dimension. Under certain conditions regarding their joint endpoint distributions, we also prove that such processes become reversible Markov diffusions. Furthermore we show that those diffusions satisfy two Itô equations for some suitably constructed Wiener processes, and from that analysis derive Feynman–Kac representations for the solutions to the given equations. We then illustrate some of our results by considering the heat equation with Neumann boundary conditions both in a one-dimensional bounded interval and in a two-dimensional disk.