Q-Wiener Process Research Articles

Correctness and regularization of stochastic problems

Abstract The paper is devoted to the regularization of ill-posed stochastic Cauchy problems in Hilbert spaces: (0.1) d ⁢ u ⁢ ( t ) = A ⁢ u ⁢ ( t ) ⁢ d ⁢ t + B ⁢ d ⁢ W ⁢ ( t ) , t > 0 , u ⁢ ( 0 ) = ξ . du(t)=Au(t)dt+BdW(t),\quad t>0,\qquad u(0)=\xi. The need for regularization is connected with the fact that in the general case the operator A is not supposed to generate a strongly continuous semigroup and with the divergence of the series defining the infinite-dimensional Wiener process { W ⁢ ( t ) : t ≥ 0 } {\{W(t):t\geq 0\}} . The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q-Wiener processes.

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)

This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

Stratonovich-Henstock integral for the operator-valued stochastic process

In this paper, we introduce the Stratonovich-Henstock integral of an operator-valued stochastic process with respect to a Q-Wiener process. We also formulate a version of Itô’s formula for this integral.

A numerical method for a nonlocal diffusion equation with additive noise

We consider a nonlocal evolution equation representing the continuum limit of a large ensemble of interacting particles on graphs forced by noise. The two principle ingredients of the continuum model are a nonlocal term and a Q-Wiener process describing the interactions among the particles in the network and stochastic forcing, respectively. The network connectivity is given by a square integrable function called a graphon. We prove that the initial value problem for the continuum model is well-posed. Further, we construct semidiscrete (discrete in space and continuous in time) and fully discrete schemes for the nonlocal model. The former is obtained by a discontinuous Galerkin method and the latter is based on further discretizing time using the Euler–Maruyama method. We prove convergence and estimate the rate of convergence in each case. For the semidiscrete scheme, the rate of convergence is expressed in terms of the regularity of the graphon, the Q-Wiener process, and the initial data. We work in generalized Lipschitz spaces, which allows us to treat models with data of lower regularity. This is important for applications as many interesting types of connectivity, including small-world and power-law, are expressed by graphons that are not smooth. The error analysis of the fully discrete scheme, on the other hand, reveals that for some models common in applied science, one has a higher speed of convergence than that predicted by the standard estimates for the Euler–Maruyama method. The rate of convergence analysis is supplemented with detailed numerical experiments, which are consistent with our analytical results. As a by-product, this work presents a rigorous justification for taking continuum limit for a large class of interacting dynamical systems on graphs subject to noise.

Stochastic optimal control in infinite dimensions with state constraints

We consider the state-constrained stochastic optimal control problem in infinite-dimensional separable Hilbert spaces, where the state process is driven by the Q-Wiener process and the (possibly unbounded) linear operator. By applying the stochastic target theory and the backward reachability approach, we show that the original (possibly discontinuous) value function can be represented by the zero-level set of the auxiliary (continuous) value function. The auxiliary value function is obtained from the penalized unconstrained stochastic control problem (in infinite dimensions) that includes an additional control variable as a consequence of the (infinite-dimensional) martingale representation theorem. We then prove that the auxiliary value function is a unique (continuous) viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation in infinite dimensions. Note that the viscosity analysis developed in our paper generalizes that presented in the existing literature, since the corresponding infinite-dimensional HJB equation includes an additional operator-valued control variable in the Hamiltonian maximization and depends on an additional initial state variable.

Stationary approximations of inertial manifolds for stochastic retarded semilinear parabolic equations

A class of retarded semilinear parabolic equation driven by an additive Q-Wiener process is studied. We construct a process via the Wiener shift to provide stationary approximations of the noise, and show that solutions of stationary approximations converge pointwisely to solutions of the stochastic retarded semilinear parabolic equation. We also prove the existence of Ck,γ smooth inertial manifolds for the stationary approximations, and show that such inertial manifolds converge to the those of the stochastically perturbed retarded semilinear parabolic equation.

A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier\u2013Stokes Equations

We analyze the control problem of the stochastic Navier–Stokes equations in multi-dimensional domains considered in Benner and Trautwein (Math Nachr 292(7):1444–1461, 2019) restricted to noise terms defined by a Q-Wiener process. The cost functional related to this control problem is nonconvex. Using a stochastic maximum principle, we derive a necessary optimality condition to obtain explicit formulas the optimal controls have to satisfy. Moreover, we show that the optimal controls satisfy a sufficient optimality condition. As a consequence, we are able to solve uniquely control problems constrained by the stochastic Navier–Stokes equations especially for two-dimensional as well as for three-dimensional domains.

Mathematical Modeling of Working Memory in the Presence of Random Disturbance using Neural Field Equations

In this paper, we describe a neural field model which explains how a population of cortical neurons may encode in its firing pattern simultaneously the nature and time of sequential stimulus events. Moreover, we investigate how noise-induced perturbations may affect the coding process. This is obtained by means of a two-dimensional neural field equation, where one dimension represents the nature of the event (for example, the color of a light signal) and the other represents the moment when the signal has occurred. The additive noise is represented by a Q-Wiener process. Some numerical experiments reported are carried out using a computational algorithm for two-dimensional stochastic neural field equations.

Fractional Stochastic Active Scalar Equations Generalizing the Multi-Dimensional Quasi-Geostrophic & 2D-Navier–Stokes Equations: The General Case

We introduce and study the well posedness: global existence, uniqueness and regularity of the solutions, of a class of d-dimensional fractional stochastic active scalar equations defined either on the torus $$ {\mathbb {T}}^d$$ or $$ {\mathbb {R}}^d$$ or $$ O\subsetneq {\mathbb {R}}^d$$ bounded, $$ d\ge 1$$ . This class includes, among others the stochastic; dD-quasi-geostrophic equation, fractional Burgers equation, fractional nonlocal transport equation and the 2D-fractional vorticity Navier–Stokes equation. We consider a locally Lipschitz diffusion term and a cylindrical Q-Wiener process with finite trace covariance Q. In particular, for $$ O={\mathbb {T}}^d$$ or $$ O\subsetneq {\mathbb {R}}^d$$ bounded, we prove the existence and the uniqueness of a global mild solution for the free divergence mode in the subcritical regime ( $$\alpha >\alpha _0(d)\ge 1$$ ), martingale solutions in the general regime ( $$\alpha \in (0, 2)$$ , supercritical, critical and subcritical) and free divergence mode and a local mild solution for the general mode and subcritical regime. Different kinds of regularity are also established for these solutions. For $$ O={\mathbb {R}}^d$$ , we studied the subcritical regime and we proved the existence of a global martingale solution for the free divergence mode and a local mild solution for the general mode.

Legendre spectral element method (LSEM) to simulate the two-dimensional system of nonlinear stochastic advection–reaction–diffusion models

In this work, we develop a Legendre spectral element method (LSEM) for solving the stochastic nonlinear system of advection–reaction–diffusion models. The used basis functions are based on a class of Legendre functions such that their mass and diffuse matrices are tridiagonal and diagonal, respectively. The temporal variable is discretized by a Crank–Nicolson finite-difference formulation. In the stochastic direction, we also employ a random variable W based on the Q-Wiener process. We inspect the rate of convergence and the unconditional stability for the achieved semi-discrete formulation. Then, the Legendre spectral element technique is used to obtain a full-discrete scheme. The error estimation of the proposed numerical scheme is substantiated based upon the energy method. The numerical results confirm the theoretical analysis.

Conjugate dynamics on center-manifolds for stochastic partial differential equations

In this paper, we prove that for a random differential equation with the driving noise constructed from a Q-Wiener process and the Wiener shift, there exists a local center, unstable, stable, center-unstable, center-stable manifold, and a local stable foliation, an unstable foliation on the center-unstable manifold, and a stable foliation on the center-stable manifold, the smoothness of which depend on the vector fields of the equation. Also we show that any two arbitrarily local center manifolds constructed as above are conjugate.

Spatial convergence for semi-linear backward stochastic differential equations in Hilbert space: a mild approach

In this paper, we present convergence results of a spatial semi-discrete approximation of a Hilbert space-valued backward stochastic differential equations with noise driven by a cylindrical Q-Wiener process. Both the solution and its space discretization are formulated in mild forms. Under suitable assumptions of the final value and the drift, a convergence rate is established.

Double Lusin condition and Vitali convergence theorem for the Itô–McShane Integral

In this paper, we formulate a version of Vitali convergence theorem for the Ito–McShane integral of an operator-valued stochastic process with respect to a Q-Wiener process. We also characterize the integral using double Lusin condition.

Travelling waves for reaction–diffusion equations forced by translation invariant noise

Inspired by applications, we consider reaction–diffusion equations on R that are stochastically forced by a small multiplicative noise term that is white in time, coloured in space and invariant under translations. We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in Hamster and Hupkes (2018,2019) for noise processes driven by a single Brownian motion. The main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed. We illustrate our approach by two fully worked-out examples, which highlight the predictive power of our expansions.

Iterated stochastic integrals in infinite dimensions: approximation and error estimates

Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden et al. (Stoch Anal Appl 10(4):431–441, 1992. https://doi.org/10.1080/07362999208809281) and by Wiktorsson (Ann Appl Probab 11(2):470–487, 2001. https://doi.org/10.1214/aoap/1015345301) for the approximation of two-times iterated stochastic integrals involved in numerical schemes for finite dimensional stochastic ordinary differential equations to an infinite dimensional setting. These methods clear the way for new types of approximation schemes for SPDEs without commutative noise. Precisely, we analyze two algorithms to approximate two-times iterated integrals with respect to an infinite dimensional Q-Wiener process in case of a trace class operator Q given the increments of the Q-Wiener process. Error estimates in the mean-square sense are derived and discussed for both methods. In contrast to the finite dimensional setting, which is contained as a special case, the optimal approximation algorithm cannot be uniquely determined but is dependent on the covariance operator Q. This difference arises as the stochastic process is of infinite dimension.

Finite element methods and their error analysis for SPDEs driven by Gaussian and non-Gaussian noises

In this paper, we investigate the mean square error of numerical methods for SPDEs driven by Gaussian and non-Gaussian noises. The Gaussian noise considered here is a Hilbert space valued Q-Wiener process and the non-Gaussian noise is defined through compensated Poisson random measure associated to a Lévy process. As the models consider the influences of Gaussian and non-Gaussian noises simultaneously, this makes the models more realistic when the models are also influenced by some randomly abrupt factors, but more complicated. As a consequence, the numerical analysis of the problems becomes more involved. We first study the regularity for the mild solution. Next, we propose a semidiscrete finite element scheme in space and a fully discrete linear implicit Euler scheme for the SPDEs, and rigorously obtain their error estimates. Both the regularity results of the mild solution and error estimates obtained in the paper are novel.

Numerical solutions of stochastic Fisher equation to study migration and population behavior in biological invasion

In this paper, numerical solutions of the stochastic Fisher equation have been obtained by using a semi-implicit finite difference scheme. The samples for the Wiener process have been obtained from cylindrical Wiener process and Q-Wiener process. Stability and convergence of the proposed finite difference scheme have been discussed scrupulously. The sample paths obtained from cylindrical Wiener process and Q-Wiener process have also been shown graphically.

A linear quadratic control problem for the stochastic heat equation driven by Q-Wiener processes

We consider a control problem for the stochastic heat equation with Neumann boundary condition, where controls and noise terms are defined inside the domain as well as on the boundary. The noise terms are given by independent Q-Wiener processes. Under some assumptions, we derive necessary and sufficient optimality conditions stochastic controls have to satisfy. Using these optimality conditions, we establish explicit formulas with the result that stochastic optimal controls are given by feedback controls. This is an important conclusion to ensure that the controls are adapted to a certain filtration. Therefore, the state is an adapted process as well.

Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations

In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schrödinger equations driven by a multiplicative Q-Wiener process. Beyond the uniform boundedness of moments for high order derivatives of the exact solution, the key requirement of our approach is the exponential integrability of both the exact and numerical solutions. By constructing and analyzing a Lyapunov functional and its discrete correspondence, we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution, which immediately yields the well-posedness of both the continuous and discrete problems. The latter exponential integrability is obtained through a variant of a criterion given by Cox, Hutzenthaler and Jentzen [arXiv:1309.5595]. As a by-product of this exponential integrability, we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation-type result on the dependence of the noise with first order strong convergence rate.

Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation

In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process {u(t)}t∈[0,T ] satisfying a stochastic linear evolution equation with a positive-type memory term driven by an additive Gaussian noise. The equation can be written in an abstract form as du+ (∫ t 0 b(t− s)Au(s) ds ) dt = dW , t ∈ (0, T ]; u(0) = u0 ∈ H, where W is a Q-Wiener process on H = L(D) and where the main example of b we consider is given by b(t) = tβ−1/Γ (β), 0 0 such that A−α has finite trace and that Q is bounded from H into D(A) for some real κ with α− 1 β+1 < κ ≤ α. The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter ∆t = T/n), and a standard continuous finite element method in space (parameter h). Let un,h be the discrete solution at T = n∆t. We show that ( E‖un,h − u(T )‖ )1/2 = O(h +∆t), for any γ < (1− (β + 1)(α− κ))/2 and ν ≤ 1 β+1 − α+ κ. M. Kovacs Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, 9054, New Zealand. E-mail: mkovacs@maths.otago.ac.nz J. Printems Laboratoire d’Analyse et de Mathematiques Appliquees CNRS UMR 8050, 61, avenue du General de Gaulle, Universite Paris–Est, 94010 Creteil, France. E-mail: printems@u-pec.fr 2 Mihaly Kovacs, Jacques Printems