Strong Convergence Rate Research Articles

Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations

Abstract In this paper, we are concerned with the limit theory of stochastic McKean–Vlasov equations. First, we prove the optimal L p {L^{p}} ( p ⩾ 2 {p\geqslant 2} ) strong convergence rate of the Wong–Zakai approximation for stochastic McKean–Vlasov equations. Then we show the support theorem for stochastic McKean–Vlasov equations.

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.

Optimal strong convergence rate for a class of McKean–Vlasov SDEs with fast oscillating perturbation

In this paper, we consider the averaging principle for a class of McKean–Vlasov stochastic differential equations perturbed by a fast oscillating term. By using the technique of Poisson equation, we prove the occurrence of the averaging principle, i.e., the solution Xɛ converges to the solution X̄ of the corresponding averaged equation in L2(Ω,C([0,T],Rn)) with the optimal convergence order 1/2. To the best of authors’ knowledge, this is the first result about the strong averaging principle when the coefficients in the slow equation depends on the law of the fast component.

PV Power Forecasting Based on Relevance Vector Machine with Sparrow Search Algorithm Considering Seasonal Distribution and Weather Type

Accurate photovoltaic (PV) power forecasting is indispensable to enhancing the stability of the power grid and expanding the absorptive photoelectric capacity of the power grid. As an excellent nonlinear regression model, the relevance vector machine (RVM) can be employed to forecast PV power. However, the optimization of the free parameters is still a key problem for improving the performance of the RVM. Taking advantage of the strong global search capability, good stability, and fast convergence rate of the sparrow search algorithm (SSA), this paper optimizes the parameters of the RVM by using the SSA to develop an excellent RVM (called SSA-RVM). Consequently, a novel hybrid PV power forecasting model via the SSA-RVM is proposed to perform ultra-short-term (4 h ahead) prediction. In addition, the effects of seasonal distribution and weather type on PV power are fully considered, and different seasonal prediction models are established separately to improve the prediction capability. The benchmark is used to verify the accuracy of the SSA-RVM-based forecasting model under various conditions, and the experiment results demonstrate that the proposed SSA-RMV method outperforms the traditional RVM and support vector machine models, and it even shows a better prediction effect than the RVM models with other optimization approaches.

On the strong convergence rate for the Euler–Maruyama scheme of one-dimensional SDEs with irregular diffusion coefficient and local time

The strong rate of convergence for the Euler–Maruyama scheme of stochastic differential equations (SDEs) driven by a Brownian motion with Hölder continuous diffusion coefficient or irregular drift coefficient have been widely studied. In the case of irregular diffusion coefficient, however, there are few studies. In this article, under Le Gall's condition on the diffusion coefficient, which leads to conclude the pathwise uniqueness for SDEs, we provide the same result on the strong rate of convergence as in the case of 1/2-Hölder continuous diffusion coefficient. The idea of the proof is to use a version of Avikainen's inequality. As an application, we introduce a numerical scheme for SDEs with local time.

Density function of numerical solution of splitting AVF scheme for stochastic Langevin equation

In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. We first show the existence of the density function of the numerical solution by proving its exponential integrability property, Malliavin differentiability and the almost surely non-degeneracy of the associated Malliavin covariance matrix. Then the smoothness of the density function is obtained through a lower bound estimate of the smallest eigenvalue of the corresponding Malliavin covariance matrix. Meanwhile, we derive the optimal strong convergence rate in every Malliavin–Sobolev norm of the numerical solution via Malliavin calculus. Combining the strong convergence result and the smoothness of the density functions, we prove that the convergence order of the density function of the numerical scheme coincides with its strong convergence order.

Quantile-Wavelet Nonparametric Estimates for Time-Varying Coefficient Models

The paper considers quantile-wavelet estimation for time-varying coefficients by embedding a wavelet kernel into quantile regression. Our methodology is quite general in the sense that we do not require the unknown time-varying coefficients to be smooth curves of a common degree or the errors to be independently distributed. Quantile-wavelet estimation is robust to outliers or heavy-tailed data. The model is a dynamic time-varying model of nonlinear time series. A strong Bahadur order O2mn3/4(logn)1/2 for the estimation is obtained under mild conditions. As applications, the rate of uniform strong convergence and the asymptotic normality are derived.

Strong convergence rate of the averaging principle for a class of slow–fast stochastic evolution equations

ABSTRACT We prove a strong convergence rate of the averaging principle for general two-time-scales stochastic evolution equations driven by cylindrical Wiener processes. In particular, our general result can be used to deal with a large class of quasi-linear stochastic partial differential equations, such as stochastic reaction–diffusion equations, stochastic p-Laplace equations, stochastic porous media equations, and so on.

Improvement of Split-Step Forward Milstein Schemes for SODEs Arising in Mathematical Physics

In the present investigation, new explicit approaches by the Milstein method and increment function of the Jacobian derivative of the drift coefficient are designed. Several numerical tests such as Cox–Ingersoll–Ross process, stochastic Brusselator, and Davis-Skodje system are presented to illustrate the accuracy and the efficiency of our schemes. Furthermore, we show that the strong convergence rate of our procedures is approximately one.

PD-like finite time controller with simple structure for satellite attitude control

A finite time controller with PD-like structure for satellite attitude control is proposed in this paper. The controller is constructed with simple structure based on standard PD controller. The fractional order term is designed hence system could both have strong robustness and finite time convergence rate, and the advantage of finite time control and PD control is combined in this paper. System convergence rate is discussed by Lyapunov method, and the constraint on control parameters is given by implementing the coupled term of angular velocity and attitude quaternion. Moreover, the accuracy at steady stage depending on control parameters is given hence system could converge to this field within finite time. System stability and performance is demonstrated by numerical simulation results.

The truncated Euler–Maruyama method for CIR model driven by fractional Brownian motion

Recently, Hong et al. (2020) established the strong convergence rate of the backward Euler scheme for the Cox–Ingersoll–Ross (CIR) model driven by fractional Brownian motion with Hurst parameter H∈(1/2,1), which may effect the efficiency of computation. Taking advantage of being explicit and easily implementable, a positivity preserving explicit scheme is proposed in this paper. For overcoming the difficulties caused by the unbounded diffusion coefficient, an auxiliary equation with a constant diffusion coefficient obtained by proper Lamperti transformation is used. By means of Malliavin calculus, we show that the truncated Euler–Maruyama scheme applied to this auxiliary equation not only ensures the positivity of the numerical solution, but also has the H-order rate of the root mean square error over a finite time interval. Furthermore, by transforming back, an explicit scheme for the original CIR model is obtained and has the same convergence order. Finally, some numerical experiments are provided to illustrate the theoretical results.

Galerkin Approximation for Stochastic Volterra Integral Equations with Doubly Singular Kernels

This paper is concerned with the more general nonlinear stochastic Volterra integral equations with doubly singular kernels, whose singular points include both s=t and s=0. We propose a Galerkin approximate scheme to solve the equation numerically, and we obtain the strong convergence rate for the Galerkin method in the mean square sense. The rate is min{2−2(α1+β1),1−2(α2+β2)} (where α1,α2,β1,β2 are positive numbers satisfying 0<α1+β1<1, 0<α2+β2<12), which improves the results of some numerical schemes for the stochastic Volterra integral equations with regular or weakly singular kernels. Moreover, numerical examples are given to support the theoretical result and explain the priority of the Galerkin method.

$$L^p$$-convergence rate of backward Euler schemes for monotone SDEs

We give a unified method to derive the strong convergence rate of the backward Euler scheme for monotone SDEs in $L^p(\Omega)$-norm, with general $p \ge 4$. The results are applied to the backward Euler scheme of SODEs with polynomial growth coefficients. We also generalize the argument to the Galerkin-based backward Euler scheme of SPDEs with polynomial growth coefficients driven by multiplicative trace-class noise.

An averaging principle for slow–fast fractional stochastic parabolic equations on unbounded domains

In this paper we shall prove an averaging principle for two-time-scale fractional stochastic parabolic equations on unbounded domains. First, the exponential ergodicity for invariant measures of the fractional stochastic parabolic equation on the unbounded domain Rn is showed. Then an averaging principle for two-time-scale fractional stochastic parabolic equations on unbounded domains is derived. As a byproduct, the rate of strong convergence for the slow component towards the solution of the fractional stochastic parabolic effective equation on the unbounded domain is presented. As far as we know, this is the first result on unbounded domains in this topic.

Strong $$L^2$$ convergence of time Euler schemes for stochastic 3D Brinkman–Forchheimer–Navier–Stokes equations

We prove that some time Euler schemes for the 3D Navier-Stokes equations modified by adding a Brinkman-Forchheimer term and a random perturbation converge in $L^2(\Omega)$. This extends previous results concerning the strong rate of convergence of some time discretization schemes for the 2D Navier Stokes equations. Unlike the 2D case, our proposed 3D model with the Brinkman-Forchheimer term allows for a strong rate of convergence of order almost 1/2, that is independent of the viscosity parameter.

Strong convergence rate of Euler-Maruyama approximations in temporal-spatial Hölder-norms

Classical approximation results for stochastic differential equations analyze the Lp-distance between the exact solution and its Euler-Maruyama approximations. In this article we measure the error with temporal-spatial Hölder-norms. Our motivation for this is multigrid approximations of the exact solution viewed as a function of the starting point. We establish the classical strong convergence rate 0.5 with respect to temporal-spatial Hölder-norms if the coefficient functions have bounded derivatives of first and second order.

Well-posedness and averaging principle of McKean-Vlasov SPDEs driven by cylindrical α-stable process

In this paper, we first study the well-posedness of a class of McKean-Vlasov stochastic partial differential equations driven by cylindrical α-stable process, where Then by the method of the Khasminskii’s time discretization, we prove the averaging principle of a class of multiscale McKean-Vlasov stochastic partial differential equations driven by cylindrical α-stable processes. Meanwhile, we obtain a specific strong convergence rate.

Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations

Strong convergence rates for fuly discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities, such as the stochastic Allen–Cahn equation with space-time white noise, are shown. The obtained strong rates of convergence are essentially sharp.

Bernstein polynomial of recursive regression estimation with censored data

In this paper, we deal with the problem of the regression estimation near the edges under censoring. For this purpose, we consider a new recursive estimator based on the stochastic approximation algorithm and Bernstein polynomials of the regression function when the response random variable is subject to random right censoring. We give the central limit theorem and the strong pointwise convergence rate for our proposed nonparametric recursive estimators under some mild conditions. Finally, we provide pointwise moderate deviation principles (MDP) for the proposed estimators. We corroborate these theoretical results through simulations as well as the analysis of a real data set.

Strong Rates of Convergence of a Splitting Scheme for Schrödinger Equations with Nonlocal Interaction Cubic Nonlinearity and White Noise Dispersion

Strong Rates of Convergence of a Splitting Scheme for Schrödinger Equations with Nonlocal Interaction Cubic Nonlinearity and White Noise Dispersion