Weak Convergence Rate Research Articles

Weak convergence rates for temporal numerical approximations of the semilinear stochastic wave equation with multiplicative noise

Weak convergence rates for temporal numerical approximations of the semilinear stochastic wave equation with multiplicative noise

Rate of Weak Convergence of Random Walk with a Generalized Reflecting Barrier

In this study, a random walk process with generalized reflecting barrier is considered and an inequality for rate of weak convergence of the stationary distribution of the process of interest is propounded. Under some conditions, the weak convergence theorem for the stationary distribution of the considered process is obtained in literature, however, the rate of the convergence is not studied sufficiently. Nonetheless, one of the most crucial issues in probability theory is the convergence rate in limit theorems, as it affects the precision and effectiveness of using these theorems in practice. Therefore, for the rate of convergence for the examined process, comparatively simple inequality is represented. The obtained inequality demonstrates that the rate of convergence is correlated with the tail of the distribution of ladder heights of the random walk.

On the weak rate of convergence for the Euler–Maruyama scheme with Hölder drift

We consider stochastic differential equations with bounded α-Hölder continuous drift, with α∈(0,1), driven by multiplicative noise. We moreover assume that the diffusion coefficient is uniformly elliptic and bounded, with bounded first and second order partial derivatives. We show that the weak rate of convergence for the Euler–Maruyama scheme is arbitrarily close to (1+α)/2.

Weak Convergence of the Rosenbrock Semi-implicit Method for Semilinear Parabolic SPDEs Driven by Additive Noise

Abstract This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.

Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients

Abstract We present an error analysis of weak convergence of one-step numerical schemes for stochastic differential equations (SDEs) with super-linearly growing coefficients. Following Milstein’s weak error analysis on the one-step approximation of SDEs, we prove a general result on weak convergence of the one-step discretization of the SDEs mentioned above. As applications, we show the weak convergence rates for several numerical schemes of half-order strong convergence, such as tamed and balanced schemes. Numerical examples are presented to verify our theoretical analysis.

On the limit distribution for stochastic differential equations driven by cylindrical non-symmetric α-stable Lévy processes

This paper deals with the limit distribution for a stochastic differential equation driven by a non-symmetric cylindrical [Formula: see text]-stable process. Under suitable conditions, it is proved that the solution of this equation converges weakly to that of a stochastic differential equation driven by a Brownian motion in the Skorohod space as [Formula: see text]. Also, the rate of weak convergence, which depends on [Formula: see text], for the solution towards the solution of the limit equation is obtained. For illustration, the results are applied to a simple one-dimensional stochastic differential equation, which implies the rate of weak convergence is optimal.

An application of the multiplicative Sewing Lemma to the high order weak approximation of stochastic differential equations

We introduce a variant of the multiplicative Sewing Lemma in [Gerasimovičs, Hocquet, Nilssen; J. Funct. Anal. 281 (2021)] which yields arbitrary high order weak approximations to stochastic differential equations, extending the cubature approximation on Wiener space introduced by Lyons and Victoir. Our analysis allows to derive stability estimates and explicit weak convergence rates. As a particular example, a cubature approximation for stochastic differential equations driven by continuous Gaussian martingales is given.

Weak convergence for weighted sums of a class of random variables with related statistical applications

In this paper, we study the weak convergence and convergence rate in the weak law of large numbers for weighted sums of a class of random variables satisfying the Rosenthal type inequality. The necessary and sufficient conditions for the convergence rates in the weak law of large numbers under some mild conditions are provided. Moreover, the main results that we established are applied to simple linear errors-in-variables regression models and nonparametric regression models based on a class of random errors. Finally, we present some numerical simulations to assess the finite sample performance of the theoretical results.

Error estimates for a finite volume scheme for advection–diffusion equations with rough coefficients

We study the implicit upwind finite volume scheme for numerically approximating the advection–diffusion equation with a vector field in the low regularity DiPerna–Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique solution of the continuous model is at least one. The numerical error is estimated in terms of logarithmic Kantorovich–Rubinstein distances and provides a bound on the rate of weak convergence.

On the rate of convergence of an exponential scheme for the non-linear stochastic Schrödinger equation with finite-dimensional state space

We address the numerical solution of the finite-dimensional non-linear stochastic Schrödinger equation, which is a locally Lipschitz stochastic differential equation modeling, for instance, quantum measurement processes. We study the rate of weak convergence of an exponential scheme that reproduces the norm of the desired solution by projecting onto the unit sphere. This justifies the use of the Talay-Tubaro extrapolation procedure in the numerical simulation of open quantum systems. In particular, we prove that an Euler-Exponential scheme converges with weak-order one, and we obtain the leading order term of its weak error expansion with respect to the step-size. Then, applying the Talay-Tubaro extrapolation procedure to the Euler-Exponential scheme under consideration we get a second-order method for computing the mean values of smooth functions of the solution of the non-linear stochastic Schrödinger equation. We also prove that the exponential scheme under study has order of strong convergence 1/2, which gives theoretical support to the use of the multilevel Monte Carlo method in simulating open quantum systems. We present a numerical experiment with a quantized electromagnetic field in interaction with a reservoir that illustrates the good performance of the weak second-order method, and the multilevel Monte Carlo method.

The Kolmogorov infinite dimensional equation in a Hilbert space via deep learning methods

We consider the nonlinear Kolmogorov equation posed in a Hilbert space H, not necessarily of finite dimension. This model was recently studied by Cox et al. [12] in the framework of weak convergence rates of stochastic wave models. Here, we propose a complementary approach by providing an infinite-dimensional Deep Learning method to approximate suitable solutions of this model. Based in the work by Hure, Pham and Warin [25] concerning the finite dimensional case, and our previous work [9] dealing with Lévy based processes, we generalize an Euler scheme and consistency results for the Forward Backward Stochastic Differential Equations to the infinite dimensional Hilbert valued case. Since our framework is general, we require the recently developed DeepOnets neural networks [10,29] to describe in detail the approximation procedure. Also, the framework developed by Fuhrman and Tessitore [17] to fully describe the stochastic approximations will be adapted to our setting.

Random walks on the circle and Diophantine approximation.

Random walks on the circle group whose elementary steps are lattice variables with span or taken mod exhibit delicate behavior. In the rational case, we have a random walk on the finite cyclic subgroup , and the central limit theorem and the law of the iterated logarithm follow from classical results on finite state space Markov chains. In this paper, we extend these results to random walks with irrational span , and explicitly describe the transition of these Markov chains from finite to general state space as along the sequence of best rational approximations. We also consider the rate of weak convergence to the stationary distribution in the Kolmogorov metric, and in the rational case observe a phase transition from polynomial to exponential decay after steps. This seems to be a new phenomenon in the theory of random walks on compact groups. In contrast, the rate of weak convergence to the stationary distribution in the total variation metric is purelyexponential.

Splitting integrators for stochastic Lie–Poisson systems

We study stochastic Poisson integrators for a class of stochastic Poisson systems driven by Stratonovich noise. Such geometric integrators preserve Casimir functions and the Poisson map property. For this purpose, we propose explicit stochastic Poisson integrators based on a splitting strategy, and analyse their qualitative and quantitative properties: preservation of Casimir functions, existence of almost sure or moment bounds, asymptotic preserving property, and strong and weak rates of convergence. The construction of the schemes and the theoretical results are illustrated through extensive numerical experiments for three examples of stochastic Lie–Poisson systems, namely: stochastically perturbed Maxwell–Bloch, rigid body and sine–Euler equations.

A posteriori error analysis and adaptivity for high-dimensional elliptic and parabolic boundary value problems

We derive a posteriori error estimates for the (stopped) weak Euler method to discretize SDE systems which emerge from the probabilistic reformulation of elliptic and parabolic (initial) boundary value problems. The a posteriori estimate exploits the use of a scaled random walk to represent noise, and distinguishes between realizations in the interior of the domain and those close to the boundary. We verify an optimal rate of (weak) convergence for the a posteriori error estimate on deterministic meshes. Based on this estimate, we then set up an adaptive method which automatically selects local deterministic mesh sizes, and prove its optimal convergence in terms of given tolerances. Provided with this theoretical backup, and since corresponding Monte-Carlo based realizations are simple to implement, these methods may serve to efficiently approximate solutions of high-dimensional (initial-)boundary value problems.

Weak convergence of the backward Euler method for stochastic Cahn–Hilliard equation with additive noise

We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn–Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen–Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn–Hilliard equation setting for the first time.

Nonparametric estimation of a smooth trend in the presence of a periodic sequence

We investigate the estimators obtained from reversing the three-step estimation procedure of Vogt and Linton (2014) for a regression model including a smooth trend function, a periodic component and a stochastic error term. The asymptotic properties of the estimators are given: (a) we provide the uniform weak convergence rates of the estimators of the trend function and periodic sequence; (b) we establish the asymptotic normality for the trend function estimator; and (c) we show that the period estimator is consistent. Plug-in type bandwidths are proposed for the trend estimator. A simulation study and an application to sea level data are employed in order to complement our theoretical results.

Globally Adaptive Longitudinal Quantile Regression with High Dimensional Compositional Covariates.

In this work, we propose a longitudinal quantile regression framework that enables a robust characterization of heterogeneous covariate-response associations in the presence of high-dimensional compositional covariates and repeated measurements of both response and covariates. We develop a globally adaptive penalization procedure, which can consistently identify covariate sparsity patterns across a continuum set of quantile levels. The proposed estimation procedure properly aggregates longitudinal observations over time, and ensures the satisfaction of the sum-zero coefficient constraint that is needed for proper interpretation of the effects of compositional covariates. We establish the oracle rate of uniform convergence and weak convergence of the resulting estimators, and further justify the proposed uniform selector of the tuning parameter in terms of achieving global model selection consistency. We derive an efficient algorithm by incorporating existing R packages to facilitate stable and fast computation. Our extensive simulation studies confirm the theoretical findings. We apply the proposed method to a longitudinal study of cystic fibrosis children where the association between gut microbiome and other diet-related biomarkers is of interest.

Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise

This paper aims to investigate the finite element weak convergence rate for semilinear parabolic stochastic partial differential equations(SPDEs) driven by additive noise. In contrast to many results in the current scientific literature, we investigate the more general case where the nonlinearity is allowed to be of Nemytskii-type and the linear operator is not necessarily self-adjoint, which is more challenging and models more realistic phenomena such as convection–reaction–diffusion processes. Using Malliavin calculus, Kolmogorov equations and by splitting the linear operator into a self-adjoint and non self-adjoint parts, we prove the convergence of the finite element approximation and obtain a weak convergence rate that is twice the strong convergence rate.

Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates

In this study, we present an adaptive multilevel Monte Carlo (AMLMC) algorithm for approximating deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient and geometric singularities in bounded domains of Rd. Our AMLMC algorithm is built on the results of the weak convergence rates in the work (Moon et al., 2006) for an adaptive algorithm using isoparametric d-linear quadrilateral finite element approximations and the dual weighted residual error representation in a deterministic setting. Designed to suit the geometric nature of the singularities in the solution, our AMLMC algorithm uses a sequence of deterministic, non-uniform auxiliary meshes as a building block. The above-mentioned deterministic adaptive algorithm generates these meshes, corresponding to a geometrically decreasing sequence of tolerances. In particular, for a given realization of the diffusivity coefficient and accuracy level, AMLMC constructs its approximate sample using the first mesh in the hierarchy that satisfies the corresponding bias accuracy constraint. This adaptive approach is particularly useful for the lognormal case treated here, which lacks uniform coercivity and thus produces functional outputs that vary over orders of magnitude when sampled. Furthermore, we discuss iterative solvers and compare their efficiency with direct ones. To reduce computational work, we propose a stopping criterion for the iterative solver with respect to the quantity of interest, the realization of the diffusivity coefficient, and the desired level of AMLMC approximation.From the numerical experiments, based on a Fourier expansion of the diffusivity coefficient field, we observe improvements in efficiency compared with both standard Monte Carlo (MC) and standard MLMC (SMLMC) for a problem with a singularity similar to that at the tip of a slit modeling a crack.

Strong and weak convergence rates of logarithmic transformed truncated EM methods for SDEs with positive solutions

To inherit numerically the positivity of stochastic differential equations (SDEs) with non-globally Lipschitz coefficients, we devise a novel explicit method, called logarithmic transformed truncated Euler–Maruyama method. There is however a price to be paid for the preserving positivity, namely that the logarithmic transformation would cause the coefficients of the transformed SDEs growing super-linearly or even exponentially, which makes the strong and weak convergence analysis more complicated. Based on the exponential integrability, truncation techniques and some other arguments, we show that the strong convergence rate of the underlying numerical method is 1/2, and the weak convergence rate can be arbitrarily close to 1. To the best of our knowledge, this is the first result establishing the weak convergence rate of numerical methods for the general SDEs with positive solutions. Numerical experiments are finally reported to confirm our theoretical results.