Strong Convergence Rate Research Articles

On non parametric kernel estimation of the mode of the regression function in the strong mixing random design model with censored data

. This study delves into the conditional mode estimation of a randomly censored scalar response variable operating within the framework of strong mixing conditions. We introduce a kernel-based estimator for the conditional mode function. The principal contribution of this investigation lies in the derivation of the asymptotic distribution and the strong rate of convergence of the newly proposed estimators. These findings are established under a set of fairly comprehensive structural assumptions governing the underlying models. Additionally, we conduct a series of simulation studies to showcase the finite sample performance characteristics of the proposed estimator.

Convergence analysis of a fully discrete scheme for diffusion-wave equation forced by tempered fractional Brownian motion

In this paper, we investigate a fully discrete approximation of stochastic diffusion-wave equation driven by additive tempered fractional Gaussian noise. This additive noise exhibits semi-long range dependence. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative and a tempered fractional Brownian motion. The space discretization is achieved via the spectral Galerkin method. The Caputo fractional derivative of order α∈(1,2) is approximated by using the Grünwald–Letnikov formula. Combining Mittag-Leffler function, Laplace transform and z-transform, we establish the error estimates of the full discretization in the sense of mean-squared L2-norm. Numerical experiments are presented to confirm the strong convergence rates.

Strong Consistency and Rate of Convergence of Switched Least Squares System Identification for Autonomous Markov Jump Linear Systems

Strong Consistency and Rate of Convergence of Switched Least Squares System Identification for Autonomous Markov Jump Linear Systems

The rates of strong consistency for estimators in heteroscedastic partially linear errors-in-variables model for widely orthant dependent samples

In this article, some general results on the rates of strong consistency for the least squares estimators and weighted least squares estimators in heteroscedastic partially linear errors-in-variables model based on widely orthant dependent random errors are presented. The results can deduce strong consistency and convergence rates under some mild conditions. Some numerical simulations are also provided to support the theoretical results.

A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients

Abstract The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($\text{Tr} (Q) < \infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +\tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d \le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.

Partial Derivatives Estimation of Multivariate Variance Function in Heteroscedastic Model via Wavelet Method

For derivative function estimation, conventional research only focuses on the derivative estimation of one-dimensional functions. This paper considers partial derivatives estimation of a multivariate variance function in a heteroscedastic model. A wavelet estimator of partial derivatives of a multivariate variance function is proposed. The convergence rates of a wavelet estimator under different estimation errors are discussed. It turns out that the strong convergence rate of the wavelet estimator is the same as the optimal uniform almost sure convergence rate of nonparametric function problems.

Strong consistency properties of the variance change point estimator based on strong-mixing samples

<p>In this paper, our primary attention was centered on the issue of detecting the variance change point for strong-mixing samples. We delved into the cumulative sum (CUSUM) estimator of variance change model and established the strong convergence rate of the variance change point estimation. Furthermore, to corroborate the effectiveness of the CUSUM based methodology, we have conducted a series of simulations, the outcomes of which underscored its validity.</p>

Strong convergence rates of a fully discrete scheme for the stochastic Cahn-Hilliard equation with additive noise

Strong convergence rates of a fully discrete scheme for the stochastic Cahn-Hilliard equation with additive noise

Strong convergence rate of an exponentially integrable scheme for stochastic nonlinear wave equation

Strong convergence rate of an exponentially integrable scheme for stochastic nonlinear wave equation

Convergence of a exponential tamed method for a general interest rate model

We prove mean-square convergence of a exponential tamed method, for a generalized Ait-Sahalia interest rate model. The method is based on a Lamperti transform, splitting and applying a tamed numerical method for the nonlinearity. The main difficulty in the analysis is caused by the non-globally Lipschitz drift coefficients of the model. We consider the existence, uniqueness of the solution and boundedness of moments for the transformed SDE before proving bounded moments and inverse moment bounds for the numerical approximation. The exponential tamed method is a hybrid method in the sense that a backstop method is invoked to prevent solutions from overshooting zero and becoming negative. We successfully recover the strong convergence rate of order one for the exponential tamed method. In addition we prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. In our numerical experiments we compare to other numerical methods in the literature for realistic parameter values.

Density convergence of a fully discrete finite difference method for stochastic Cahn–Hilliard equation

This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn–Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in L 1 ( R ) L^1(\mathbb {R}) of the numerical solution. Our results partially answer positively to the open problem posed by J. Cui and J. Hong [J. Differential Equations 269 (2020), pp. 10143–10180] on computing the density of the exact solution numerically.

Relative regularity conditions and linear regularity properties for split feasibility problems in normed linear spaces

The bounded linear regularity property plays a key role in the study of the strong convergence and/or convergence rate of the CQ algorithm for solving split feasibility problems. To establish some sufficient conditions ensuring the bounded linear regularity property for split feasibility problems in normed linear spaces, we introduce the notion of a relative regularity condition and its associated relative regularity constant in spirit of the regularity condition used in Burke and Ferris (1995) [12]. Based on convex analysis techniques, we explore equivalent characterizations of the relative regularity condition, which in particular extend the classical results in Burke and Ferris (1995) [12] from the Euclidean space to general normed linear spaces, and then establish some important and useful properties in terms of the related relative regularity constant. Consequently, we develop a new technique to establish some sufficient conditions ensuring the bounded linear regularity property for split feasibility problems in normed linear spaces. The sufficient conditions presented in this paper are in terms of the relative regularity constant, which seem completely new. Applied to the case of Hilbert spaces, our results extend and improve the corresponding ones in Wang et al. (2017) [35] by relaxing the relevant assumptions.

Best practices for portfolio optimization by quantum computing, experimented on real quantum devices

In finance, portfolio optimization aims at finding optimal investments maximizing a trade-off between return and risks, given some constraints. Classical formulations of this quadratic optimization problem have exact or heuristic solutions, but the complexity scales up as the market dimension increases. Recently, researchers are evaluating the possibility of facing the complexity scaling issue by employing quantum computing. In this paper, the problem is solved using the Variational Quantum Eigensolver (VQE), which in principle is very efficient. The main outcome of this work consists of the definition of the best hyperparameters to set, in order to perform Portfolio Optimization by VQE on real quantum computers. In particular, a quite general formulation of the constrained quadratic problem is considered, which is translated into Quadratic Unconstrained Binary Optimization by the binary encoding of variables and by including constraints in the objective function. This is converted into a set of quantum operators (Ising Hamiltonian), whose minimum eigenvalue is found by VQE and corresponds to the optimal solution. In this work, different hyperparameters of the procedure are analyzed, including different ansatzes and optimization methods by means of experiments on both simulators and real quantum computers. Experiments show that there is a strong dependence of solutions quality on the sufficiently sized quantum computer and correct hyperparameters, and with the best choices, the quantum algorithm run on real quantum devices reaches solutions very close to the exact one, with a strong convergence rate towards the classical solution, even without error-mitigation techniques. Moreover, results obtained on different real quantum devices, for a small-sized example, show the relation between the quality of the solution and the dimension of the quantum processor. Evidences allow concluding which are the best ways to solve real Portfolio Optimization problems by VQE on quantum devices, and confirm the possibility to solve them with higher efficiency, with respect to existing methods, as soon as the size of quantum hardware will be sufficiently high.

Higher-order adaptive methods for exit times of Itô diffusions

Abstract We construct a higher-order adaptive method for strong approximations of exit times of Itô stochastic differential equations (SDEs). The method employs a strong Itô–Taylor scheme for simulating SDE paths, and adaptively decreases the step size in the numerical integration as the solution approaches the boundary of the domain. These techniques complement each other nicely: adaptive timestepping improves the accuracy of the exit time by reducing the magnitude of the overshoot of the numerical solution when it exits the domain, and higher-order schemes improve the approximation of the state of the diffusion process. We present two versions of the higher-order adaptive method. The first one uses the Milstein scheme as the numerical integrator and two step sizes for adaptive timestepping: $h$ when far away from the boundary and $h^2$ when close to the boundary. The second method is an extension of the first one using the strong Itô–Taylor scheme of order 1.5 as the numerical integrator and three step sizes for adaptive timestepping. Under some regularity assumptions, we show that for any $\xi>0$, the strong error is ${\mathcal{O}}(h^{1-\xi })$ and ${\mathcal{O}}(h^{3/2-\xi })$ for the first and second method, respectively. Provided quite restrictive commutativity conditions hold for the diffusion coefficient, we further show that the expected computational cost for both methods is ${\mathcal{O}}(h^{-1} \log (h^{-1}))$. This results in a near doubling/trebling of the strong error rate compared to the standard Euler–Maruyama-based approach, while the computational cost rate is kept close to order one. Numerical examples that support the theoretical results are provided, and we discuss the potential for extensions that would further improve the strong convergence rate of the method.

Random periodic solutions of SDEs: Existence, uniqueness and numerical issues

The aim of this paper is to discuss existence and uniqueness of random periodic solutions to stochastic differential equations (SDEs) with multiplicative noise under a one-sided Lipschitz condition, as well as on their numerical approximation via two classes of stochastic θ-methods, i.e., θ-Maruyama methods with θ∈[1/2,1] and θ-Milstein ones with θ∈[0,1]. The existence of the random periodic solutions as the limit of the pull-back flows of the discretized SDEs and the strong convergence rate of the aforementioned methods are also investigated. Selected numerical experiments confirming the theoretical analysis are also given.

Convergence Rates for the Truncated Euler-Maruyama Method for Nonlinear Stochastic Differential Equations

In this paper, our main aim is to investigate the strong convergence rate of the truncated Euler-Maruyama approximations for stochastic differential equations with superlinearly growing drift coefficients. When the diffusion coefficient is polynomially growing or linearly growing, the strong convergence rate of arbitrarily close to one half is established at a single time T or over a time interval [0,T], respectively. In both situations, the common one-sided Lipschitz and polynomial growth conditions for the drift coefficients are not required. Two examples are provided to illustrate the theory.

Wong-Zakai approximations for stochastic differential equations with path-dependent coefficients

In this paper, we shall prove an L p ( p > 2 ) L^{p}(p>2) strong convergence rate of the non-adapted Wong-Zakai approximation for stochastic differential equations with path-dependent coefficients under d ∞ d_{\infty } -Lipschitz conditions.

Strong convergence rates of an explicit scheme for stochastic Cahn–Hilliard equation with additive noise

Strong convergence rates of an explicit scheme for stochastic Cahn–Hilliard equation with additive noise

A Strong Convergence Rate of the Averaging Principle for Two-Time-Scale Forward-Backward Stochastic Differential Equations

A Strong Convergence Rate of the Averaging Principle for Two-Time-Scale Forward-Backward Stochastic Differential Equations

Strong convergence for an explicit fully‐discrete finite element approximation of the Cahn‐Hillard‐Cook equation with additive noise

AbstractIn this paper, we consider an explicit fully‐discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity‐tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform moment bounds for the numerical approximations. To the best of our knowledge, the main contribution of this work is the first result in the literature which establishes strong convergence for an explicit fully‐discrete finite element approximation of the CHC equation. Finally, numerical results are finally reported to confirm the previous theoretical findings.