Wiener Noise Research Articles

Stability, uniqueness and existence of solutions to McKean–Vlasov SDEs: a multidimensional Yamada–Watanabe approach

We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean–Vlasov equations with random non-Lipschitz continuous coefficients. In the deterministic case, we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada–Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise exponential stability. Furthermore, Lyapunov exponents are computed explicitly.

Partial Smoothing of the Stochastic Wave Equation and Regularization by Noise Phenomena

We establish partial smoothing properties of the transition semigroup (Pt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(P_t)$$\\end{document} associated to the linear stochastic wave equation driven by a cylindrical Wiener noise on a separable Hilbert space. These new results allow the study of related vector-valued infinite-dimensional PDEs in spaces of functions which are Hölder continuous along special directions. As an application we prove strong uniqueness for semilinear stochastic wave equations involving nonlinearities of Hölder type. We stress that we are able to prove well-posedness although the Markov semigroup (Pt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(P_t)$$\\end{document} is not strong Feller.

Optimal control of stochastic Markovian jump systems with Wiener and Poisson noises: Two reinforcement learning approaches

Optimal control of stochastic Markovian jump systems with Wiener and Poisson noises: Two reinforcement learning approaches

Analytical equivalent transformation method for nonlinear stochastic dynamics with multiple noises in high dimensions

In many practical applications, stochastic problems are high dimensionality and multiple noises. The study focuses on developing an analytical approach for processing multi-noise systems in high dimensions, which overcomes the challenge solving stochastic dynamics problems. Applying an analytical equivalent transformation and introducing a new partial differentiation, the author demonstrate a convenient method for obtaining the Fokker–Planck equation corresponding to Langevin dynamics equations for high-dimensional and multi-noise systems. This methodology is applied to analyze the cubic laser model subjected to white noises and the saturation laser model subjected to bounded sine Wiener noise. The results verify the correction, efficiency, and superiority of the method of processing multiple noises in high dimensions. This approach provides a novel framework for solving other practical dynamics problems for high-dimensional and multi-noise systems.

Stability and moment estimates for the stochastic singular Φ-Laplace equation

We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a homogeneous diffusivity. Our results cover the singular stochastic p-Laplace equations and, more generally, singular stochastic Φ-Laplace equations with zero Dirichlet boundary conditions. We obtain improved moment estimates and quantitative convergence rates of the ergodic semigroup to the unique invariant measure, classified in a systematic way according to the degree of local degeneracy of the potential at the origin. We obtain new concentration results for the invariant measure and establish maximal dissipativity of the associated Kolmogorov operator. In particular, we recover the results for the curve shortening flow in the plane by Es-Sarhir et al. (2012) [22], and improve the results by Liu and Tölle (2011) [41].

Stationary local random countable sets over the Wiener noise

The times of Brownian local minima, maxima and their union are three distinct examples of local, stationary, dense, random countable sets associated with classical Wiener noise. Being local means, roughly, determined by the local behavior of the sample paths of the Brownian motion, and stationary means invariant relative to the Lévy shifts of the sample paths. We answer to the affirmative Tsirelson’s question, whether or not there are any others, and develop some general theory for such sets. An extra ingredient to their structure, that of an honest indexation, leads to a splitting result that is akin to the Wiener–Hopf factorization of the Brownian motion at the minimum (or maximum) and has the latter as a special case. Sets admitting an honest indexation are moreover shown to have the property that no stopping time belongs to them with positive probability. They are also minimal: they do not have any non-empty proper local stationary subsets. Random sets, of the kind studied in this paper, honestly indexed or otherwise, give rise to nonclassical one-dimensional noises, generalizing the noise of splitting. Some properties of these noises and the inter-relations between them are investigated. In particular, subsets are connected to subnoises.

Talagrand’s transportation inequality for SPDEs with locally monotone drifts

The purpose of this paper is twofold. Firstly, we prove transportation inequalities T2(C) on the space of continuous paths with respect to the uniform metric for the law of the solution to a class of non-linear monotone stochastic partial differential equations (SPDEs) driven by the Wiener noise. Furthermore, we also establish the T1(C) property for such SPDEs but with merely locally monotone coefficients, including the stochastic Burgers type equation and stochastic 2-D Navier–Stokes equation.

Polynomial Propagation of Moments in Stochastic Differential Equations

We address the problem of approximating the moments of the solution, , of an Itô stochastic differential equation (SDE) with drift and diffusion terms over a time grid . In particular, we assume an explicit numerical scheme for the generation of sample paths and then obtain recursive equations that yield any desired noncentral moment of as a function of the initial condition . The core of the methodology is the decomposition of the numerical solution into a “central part” and an “effective noise” term. The central term is computed deterministically from the ordinary differential equation (ODE) that results from eliminating the diffusion term in the SDE, while the effective noise accounts for the stochastic deviation from the numerical solution of the ODE. For simplicity, we describe the proposed methodology based on an Euler–Maruyama integrator, but other explicit numerical schemes can be exploited in the same way. We also apply the moment approximations to construct estimates of the 1-dimensional marginal probability density functions of based on a Gram–Charlier expansion. Both for the approximation of moments and 1-dimensional densities, we describe how to handle the cases in which the initial condition is fixed (i.e., for some deterministic and known ) or random. In the latter case, we resort to polynomial chaos expansion (PCE) schemes in order to approximate the target moments. The methodology has been inspired by the PCE and differential algebra methods used for uncertainty propagation in astrodynamics problems. Hence, we illustrate its application for the quantification of uncertainty in a 2-dimensional Keplerian orbit perturbed by a Wiener noise process.

A BSDEs approach to pathwise uniqueness for stochastic evolution equations

We prove strong well-posedness for a class of stochastic evolution equations in Hilbert spaces H when the drift term is Hölder continuous. This class includes examples of semilinear stochastic Euler-Bernoulli beam equations which describe elastic systems with structural damping, and semilinear stochastic 3D heat equations. In the deterministic case, there are examples of non-uniqueness in our framework. Strong (or pathwise) uniqueness is restored by means of a suitable additive Wiener noise. The proof of uniqueness relies on the study of related systems of infinite dimensional forward-backward SDEs (FBSDEs). This is a different approach with respect to the well-known method based on the Itô formula and the associated Kolmogorov equation (the so-called Zvonkin transformation or Itô-Tanaka trick). We deal with approximating FBSDEs in which the linear part generates a group of bounded linear operators in H; such approximations depend on the type of SPDEs we are considering. We also prove Lipschitz dependence of solutions from their initial conditions.

Orders of strong and weak averaging principle for multi-scale SPDEs driven by α-stable process

In this paper, the averaging principle is studied for a class of multi-scale stochastic partial differential equations driven by α-stable process, where α∈(1,2). Using the technique of Poisson equation, we prove that the orders of strong and weak convergence are 1−1/α and 1−r for any r∈(0,1) respectively. The main contributions extend Wiener noise considered by Bréhier in [6] to α-stable process, and the finite dimensional case considered by Sun et al. in [36] to the infinite dimensional case.

Nonuniqueness in law for stochastic hypodissipative Navier–Stokes equations

We study the incompressible hypodissipative Navier–Stokes equations with dissipation exponent 0<α<12 on the three-dimensional torus perturbed by an additive Wiener noise term and prove the existence of an initial condition for which distinct probabilistic weak solutions exist. To this end, we employ convex integration methods to construct a pathwise probabilistically strong solution, which violates a pathwise energy inequality up to a suitable stopping time. This paper seems to be the first in which such solutions are constructed via Beltrami waves instead of intermittent jets or flows in a stochastic setting.

Large deviation principle for spatial economic growth model on networks

In this paper we study a spatially structured economic growth model on a finite network in the presence of a Wiener noise acting on the system. We consider an extension of the Solow’s model under the assumption of Lipschitz type for the production function and uniform boundedness of the productivity operator. Our interest is mainly set in studying the small noise asymptotics of the system. In our model, we obtain bounds on the probability that the logarithm of the capital stock will differ from its deterministic steady state level by a given amount. We show that this probability decays exponentially with the intensity of the noise.

Singular perturbations and asymptotic expansions for SPDEs with an application to term structure models

We study the dependence of mild solutions to linear stochastic evolution equations on Hilbert space driven by Wiener noise, with drift having linear part of the type A+εG, on the parameter ε. In particular, we study the limit and the asymptotic expansions in powers of ε of these solutions, as well as of functionals thereof, as ε→0, with good control on the remainder. These convergence and series expansion results are then applied to a parabolic perturbation of the Musiela SPDE of mathematical finance modeling the dynamics of forward rates.

Identification of Continuous-Discrete Hidden Markov Models with Multiplicative Observation Noise

The paper aims to identify hidden Markov model parameters. The unobservable state represents a finite-state Markov jump process. The observations contain Wiener noise with state-dependent intensity. The identified parameters include the transition intensity matrix of the system state, conditional drift and diffusion coefficients in the observations. We propose an iterative identification algorithm based on the fixed-interval smoothing of the Markov state. Using the calculated state estimates, we restore all required system parameters. The paper contains a detailed description of the numerical schemes of state estimation and parameter identification. The comprehensive numerical study confirms the high precision of the proposed identification estimates.

Wong-Zakai approximation and support theorem for 2D and 3D stochastic convective Brinkman-Forchheimer equations

In this work, we demonstrate the Wong-Zakai approximation results for two- and three-dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations forced by Hilbert space valued Wiener noise on bounded domains. Even though the existence and uniqueness of a pathwise strong solution to SCBF equations is known, the existence of a unique solution to the approximating system is not immediate from the solvability results of SCBF equations, and we prove it by using the Faedo-Galerkin approximation technique and monotonicity arguments. Moreover, as an application of the Wong-Zakai approximation, we obtain the support of the distribution of solutions to SCBF equations.

Optimal Stabilization of Linear Stochastic System with Statistically Uncertain Piecewise Constant Drift

The paper presents an optimal control problem for the partially observable stochastic differential system driven by an external Markov jump process. The available controlled observations are indirect and corrupted by some Wiener noise. The goal is to optimize a linear function of the state (output) given a general quadratic criterion. The separation principle, verified for the system at hand, allows examination of the control problem apart from the filter optimization. The solution to the latter problem is provided by the Wonham filter. The solution to the former control problem is obtained by formulating an equivalent control problem with a linear drift/nonlinear diffusion stochastic process and with complete information. This problem, in turn, is immediately solved by the application of the dynamic programming method. The applicability of the obtained theoretical results is illustrated by a numerical example, where an optimal amplification/stabilization problem is solved for an unstable externally controlled step-wise mechanical actuator.

A computational framework for mean square responses of bidirectional nonlinear systems under correlated stochastic excitation

A mathematical framework using Ito–Taylor expansion based stochastic integration scheme for planar structural systems subjected to correlated bidirectional excitation at the base is presented. The bidirectional excitation is modeled using correlated Wiener processes. Instead of simple lumped mass representation of structural systems, a more realistic planar multidirectional model is adopted here. Such assumption adequately captures the behavior of systems under real ground excitation which are often multidimensional in nature. The bidirectional excitation model is developed using a mathematically rigorous framework of correlated Wiener derivatives, exploiting the relationship between Wiener and white noise processes, something that is rarely adopted in literature. An Ito–Taylor expansion based Taylor 1.5 integration scheme is developed first for generating system responses, and, the same Ito–Taylor based scheme is utilized to create an automated framework for the evolution of mean-square responses, a manually difficult exercise otherwise. The proposed framework facilitates the development and simultaneous evaluation of n+nC2 (where, n is the number of system states) mean square equations after obtaining the mean square terms using Kronecker product. Extensive simulation studies are carried out for a torsionally coupled linear bidirectional frame, stochastic bidirectional duffing system and a Bouc–Wen hysteretic system at the base. The results for each of the case studies show that non-accounting of correlation in the bidirectional excitation leads to as much as 30% error in the estimation of displacement mean square responses.

Finite-time Annular Domain Stability and Stabilization of Itô-type Stochastic Time-Varying Systems with Wiener and Poisson noises

This paper investigates finite time annular domain (FTAD) stability and stabilization for Itô-type stochastic time-varying systems with continuous Wiener and discontinuous Poisson noises (STVSWPNs). First, using Itô-Levy formula and time-varying multiple quadratic Lyapunov functions, two less conservative FTAD-stability conditions based generalised differential Lyapunov equations (GDLEs) and differential linear matrix inequalities (DLMIs) are obtained. Second, the FTAD-stabilization is studied and some new sufficient conditions for the existence of state feedback and static output feedback controllers are presented by tractable differential linear matrix inequalities. Moreover, a new numerical algorithm is given. Finally, a numerical example and a real-world example are utilized to show the effectiveness of the proposed methods.

An equivalent analytical method to deal with cross-correlated exponential type noises in the nonlinear dynamic system

An equivalent analytical method to deal with cross-correlated exponential type noises in the nonlinear stochastic dynamical system is presented, which are equivalent to a linear combination of multiplicative white noises. This equivalent analytical method solves a recent conundrum which arose in the study of cross-correlated sine Wiener noises and provides a simple and effective method to deal with the related stochastic dynamical problem of exponential type noises. Applying this method, we investigate the FitzHugh-Nagumo dynamic system with a periodic signal and driven by cross-correlated sine-wiener noises, make steady-state analysis, and point out stochastic resonance phenomena induced by cross-correlated sine-wiener noises. We show the effects of the noise parameters on the stationary probability density and the Signal-to-Noise Ratio of the stochastic dynamical system. The cross-correlation time τ and the cross-correlation strength λ enhance the stochastic resonance excited by the additive sine Wiener noise η(t), in contrast, and inhibit the stochastic resonance excited by the multiplicative sine Wiener noise ε(t).

Finite-time annular domain stability and stabilization of Itô stochastic systems with Wiener noise and Poisson jumps-differential Gronwall inequality approach

This paper investigates the finite-time annular domain stability and stabilization of stochastic systems, described by an Itô-type differential equation, in which the systems are driven by both Wiener noises and Poisson jumps. First, a new inequality called reverse differential Gronwall inequality is established to achieve less conservative conditions for finite-time annular domain stability. Second, some sufficient conditions are derived to guarantee that the closed-loop system is finite-time annular domain stable by constructing a state feedback controller and an observer-based controller, respectively. All related conditions can be expressed in terms of matrix inequalities and corresponding algorithms are given. Finally, two numerical examples are presented to verify the validity of the derived results, and the effect of Poisson jump intensity on the boundary of the system states is illustrated.