Strong Convergence Rate Research Articles

Strong convergence rates in averaging principle for slow-fast McKean-Vlasov SPDEs

In this paper, we aim to study the asymptotic behavior for a class of McKean-Vlasov stochastic partial differential equations with slow and fast time-scales. Using the variational approach and classical Khasminskii time discretization, we show that the slow component strongly converges to the solution of the associated averaged equation. In particular, the corresponding convergence rates are also obtained. The main results can be applied to demonstrate the averaging principle for various McKean-Vlasov nonlinear SPDEs such as stochastic porous media type equation, stochastic p-Laplace type equation and also some McKean-Vlasov stochastic differential equations.

Strong and weak convergence rates for slow–fast stochastic differential equations driven by α-stable process

In this paper, we study the averaging principle for a class of stochastic differential equations driven by α-stable processes with slow and fast time-scales, where α∈(1,2). We prove that the strong and weak convergence order are 1−1/α and 1 respectively. We show, by a simple example, that 1−1/α is the optimal strong convergence rate.

Strong rates of convergence of space-time discretization schemes for the 2D Navier–Stokes equations with additive noise

We consider the strong solution of the 2D Navier–Stokes equations in a torus subject to an additive noise. We implement a fully implicit time numerical scheme and a finite element method in space. We prove that the space-time rate of convergence is the “optimal” one, namely, [Formula: see text] in time and 1 in space. Let us mention that the coefficient [Formula: see text] is equal to the time regularity of the solution with values in [Formula: see text]. Our method relies on the existence of finite exponential moments for both the solution and its time approximation. Unlike previous results, our main new idea is the use of a discrete Grönwall lemma for the error estimate without any localization.

Rate of strong convergence to Markov-modulated Brownian motion

AbstractLatouche and Nguyen (2015b) constructed a sequence of stochastic fluid processes and showed that it converges weakly to a Markov-modulated Brownian motion (MMBM). Here, we construct a different sequence of stochastic fluid processes and show that it converges strongly to an MMBM. To the best of our knowledge, this is the first result on strong convergence to a Markov-modulated Brownian motion. Besides implying weak convergence, such a strong approximation constitutes a powerful tool for developing deep results for sophisticated models. Additionally, we prove that the rate of this almost sure convergence is $o(n^{-1/2} \log n)$ . When reduced to the special case of standard Brownian motion, our convergence rate is an improvement over that obtained by a different approximation in Gorostiza and Griego (1980), which is $o(n^{-1/2}(\log n)^{5/2})$ .

An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise

<abstract><p>In this paper, we investigate the stochastic evolution equations (SEEs) driven by a bounded $ \log $-Whittle-Mat$ \acute{{\mathrm{e}}} $rn (W-M) random diffusion coefficient field and $ Q $-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.</p></abstract>

Semiconductor full quantum hydrodynamic model with non-flat doping profile: (II) semi-classical limit

This is the other part of our series of studies concerning the full quantum hydrodynamic (FQHD) model for semiconductors with non-flat doping profile. In this paper, as a sequel to Hu H, Zhang K (Semiconductor full quantum hydrodynamic model with non-flat doping profile: (I) stability. 2021, submitted), we proceed to rigorously justify the semi-classical limit of both steady states and global-in-time solutions as Planck's constant ε tends to zero; the strong convergence rates are also obtained. Our main results are proved by using the elaborated energy estimates and compactness analysis.

Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise

Stochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such equations. In the case of approximation results for strong convergence rates, semilinear stochastic wave equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic wave equation is constant, that is, it is assumed that the considered wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the continuous version of the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation and the Hölder-inequality for Schatten norms.

Strong convergence rates of multiple change-point estimator for ρ-mixing sequence

We consider the detection of multiple change-points for ρ-mixing sequence. A CUSUM-type method is proposed to estimate the multiple mean change-points and the strong consistency of the multiple mean change-point estimation is proved when the number of change-point is known. Furthermore, we establish the strong convergence rate of this change-point estimation under mild conditions and simulations of multiple change-point detection are also presented.

Strong error estimates for a space-time discretization of the linear-quadratic control problem with the stochastic heat equation with linear noise

Abstract We propose a time-implicit, finite-element-based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear noise of type $[X(t)+\sigma (t)]\,\,\textrm{d}W(t)$ and prove optimal convergence w.r.t. both space and time discretization parameters. In particular, we employ the stochastic Riccati equation as a proper analytical tool to handle the linear noise, and thus extend the applicability of the earlier work by Prohl & Wang (2021, Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation. ESAIM Control Optim. Calc. Var., 27, 54), where the error analysis was restricted to additive noise.

The convergence of a numerical scheme for additive fractional stochastic delay equations with [formula omitted

In this paper, we investigate the strong convergence of the exponential Euler method to stochastic delay differential equations with fractional Brownian motion (FSDDEs) of Hurst parameter H∈(12,1). We establish the strong convergence rate H of the method for FSDDEs to the exact solution. Also we justify our theoretical results with some numerical examples of these equations alongside insignificant step size.

An adaptive Euler–Maruyama scheme for McKean–Vlasov SDEs with super-linear growth and application to the mean-field FitzHugh–Nagumo model

In this paper, we introduce fully implementable, adaptive Euler–Maruyama schemes for McKean–Vlasov stochastic differential equations (SDEs) assuming only a standard monotonicity condition on the drift and diffusion coefficients but no global Lipschitz continuity in the state variable for either, while global Lipschitz continuity is required for the measure component. We prove moment stability of the discretised processes and a strong convergence rate of 1/2. Several numerical examples, centred around a mean-field model for FitzHugh–Nagumo neurons, illustrate that the standard uniform scheme fails and that the adaptive approach shows in most cases superior performance to tamed approximation schemes. In addition, we introduce and analyse an adaptive Milstein scheme for a certain sub-class of McKean–Vlasov SDEs with linear measure-dependence of the drift.

Numerical methods for stochastic Volterra integral equations with weakly singular kernels

AbstractIn this paper we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations (SVIEs) with weakly singular kernels, with singularities $\alpha \in (0, 1)$ for the drift term and $\beta \in (0, 1/2)$ for the stochastic term. Subsequently, we propose a $\theta $-Euler–Maruyama scheme and a Milstein scheme to solve the equations numerically and obtain strong rates of convergence for both schemes in $L^{p}$ norm for any $p\geqslant 1$. For the $\theta $-Euler–Maruyama scheme the rate is $\min \big\{1-\alpha ,\frac{1}{2}-\beta \big\}~ $ and for the Milstein scheme is $\min \{1-\alpha ,1-2\beta \}$. These results on the rates of convergence are significantly different from those it is similar schemes for the SVIEs with regular kernels. The source of the difficulty is the lack of Itô formula for the equations. To get around this difficulty we use the Taylor formula subsequently carrying out a sophisticated analysis of the equation.

A note on the Euler–Maruyama scheme for 1-dimensional stochastic differential equations involving the local time of the unknown process

In this paper, we study both strong and weak rate of convergence of Euler-Maruyama scheme for one-dimensional stochastic differential equations involving the local time (SDELT) of the unknown process, whose the solution corresponds to divergence form operator with a discontinuous coefficients at point ξ ∈ ℝ. The paper is concerned with stochastic differential equation with local time of type, where is the local time of X in {ξ}. The main idea of this paper is to use a space transform in order to transform the original SDELT to a new auxiliary equation with discontinuous coefficients but without local time. Then, we apply the Euler-Maruyama scheme to this new equation. Finally, we obtain an approximation of the original SDELT by transforming the approximation of the auxiliary equation.

Statistical Inference for the Heteroscedastic Partially Linear Varying-Coefficient Errors-in-Variables Model with Missing Censoring Indicators

In this paper, we focus on heteroscedastic partially linear varying-coefficient errors-in-variables models under right-censored data with censoring indicators missing at random. Based on regression calibration, imputation, and inverse probability weighted methods, we define a class of modified profile least square estimators of the parameter and local linear estimators of the coefficient function, which are applied to constructing estimators of the error variance function. In order to improve the estimation accuracy and take into account the heteroscedastic error, reweighted estimators of the parameter and coefficient function are developed. At the same time, we apply the empirical likelihood method to construct confidence regions and maximum empirical likelihood estimators of the parameter. Under appropriate assumptions, the asymptotic normality of the proposed estimators is studied. The strong uniform convergence rate for the estimators of the error variance function is considered. Also, the asymptotic chi-squared distribution of the empirical log-likelihood ratio statistics is proved. A simulation study is conducted to evaluate the finite sample performance of the proposed estimators. Meanwhile, one real data example is provided to illustrate our methods.

Evaluation and Numerical Simulation of Music Education Informationization Based on the Local Linear Regression Method

The study of this theoretical problem enables sparse or dense functional data, including educational information evaluation data. The choice of different weights is subjected to principal component analysis. The evaluation of music education informatization level mainly evaluates the status quo of music education informatization development, provides a basis for formulating and adjusting music education informatization development policies, and provides support for educational decision-making, to promote the sustainable and balanced development of music education informatization. The evaluation of music education informatization has become the key promotion work of music education informatization at this stage. This paper studies the convergence rate of functional principal components based on the local linear method under general weighting conditions. First, we introduce the related research on the estimation of mean and covariance function under general weighting. Secondly, for principal functional components under general weighting, namely, eigenvalues and eigen functions, the text gives the corresponding estimated values and derives its strong uniform convergence rate. Finally, the convergence rate was verified by simulation research. The estimation methods and conclusions of this article enrich the research of functional linear regression models and will help analyze the complex and changeable problems encountered in the application of music education information.

A note on explicit Milstein-type scheme for stochastic differential equation with Markovian switching

An explicit Milstein-type scheme for stochastic differential equation with Markovian switching is derived and its strong convergence in L2-sense is established without using the Itô–Taylor expansion formula. The rate of strong convergence is shown to be equal to 1.0 under the assumptions that the coefficients satisfy mild regularity conditions. More precisely, the coefficients are assumed to be only once differentiable which are more relaxed conditions than those made in the existing literature.

Strong convergence rate of truncated Euler-Maruyama method for stochastic differential delay equations with Poisson jumps.

We study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.

Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators

The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time change given by an inverse subordinator. SDEs to be considered are unique in two different aspects: i) they contain two drift terms, one driven by the random time change and the other driven by a regular, non-random time variable; ii) the standard Lipschitz assumption is replaced by that with a time-varying Lipschitz bound. The difficulty imposed by the first aspect is overcome via an approach that is significantly different from a well-known method based on the so-called duality principle. On the other hand, the second aspect requires the establishment of a criterion for the existence of exponential moments of functions of the random time change.

Higher Order Approximation for Stochastic Space Fractional Wave Equation Forced by an Additive Space-Time Gaussian Noise

The infinitesimal generator (fractional Laplacian) of a process obtained by subordinating a killed Brownian motion catches the power-law attenuation of wave propagation. This paper studies the numerical schemes for the stochastic wave equation with fractional Laplacian as the space operator, the noise term of which is an infinite dimensional Brownian motion or fractional Brownian motion (fBm). Firstly, we establish the regularity of the mild solution of the stochastic fractional wave equation. Then a spectral Galerkin method is used for the approximation in space, and the space convergence rate is improved by postprocessing the infinite dimensional Gaussian noise. In the temporal direction, when the time derivative of the mild solution is bounded in the sense of mean-squared $L^p$-norm, we propose a modified stochastic trigonometric method, getting a higher strong convergence rate than the existing results, i.e., the time convergence rate is bigger than $1$. Particularly, for time discretization, the provided method can achieve an order of $2$ at the expenses of requiring some extra regularity to the mild solution. The theoretical error estimates are confirmed by numerical experiments.

Numerical methods for mean-field stochastic differential equations with jumps

In this paper, we are devoted to the numerical methods for mean-field stochastic differential equations with jumps (MSDEJs). By combining with the mean-field Itô formula (see Sun, Yang, and Zhao, Numer. Math. Theor. Meth. Appl., 10, pp. 798–828 (2017)), we first develop the Itô formula and further construct the Itô-Taylor expansion for MSDEJs. Then based on the Itô-Taylor expansions, we propose the strong order γ and the weak order η Itô-Taylor schemes for MSDEJs. We theoretically prove the strong convergence rate γ of the strong order γ Itô-Taylor scheme and the weak convergence rate η of the weak order η Itô-Taylor scheme, respectively. Some numerical tests are also presented to verify our theoretical conclusions.