arXiv subjects

Xinghu Jin

Publications and source records attributed to Xinghu Jin.

7 recordsLinked to original sources

arXiv2025

Optimal Rates for Ergodic SDEs Driven by Multiplicative $α$-Stable Processes in Wasserstein-1 distance

This paper establishes the quantitative stability of invariant measures $μ_α$ for $\mathbb{R}^d$-valued ergodic stochastic differential equations driven by rotationally invariant multiplicative $α$-stable processes with $α\in(1,2]$. Under structural assumptions on the coefficients with a fixed parameter vector $\bmθ$, we derive optimal convergence rates in the Wasserstein-$1$ ($\cW_{1}$) distance between the invariant measures introduced above, namely, \item[(i)] For any interval $[α_0, \vartheta_0] \subset (1,2)$, there exists $C_1 = C(α_0, \vartheta_0,\bmθ,d) > 0$ such that \cW_{1}(μ_α, μ_\vartheta) \leq C_1 |α- \vartheta|, \quad \forall α, \vartheta \in [α_0, \vartheta_0]. \item[(ii)] For any $α_0\in (1,2)$, there exists $C_2 = C(α_0, \bmθ) > 0$ such that \begin{align*} \cW_{1}(μ_α, μ_2) \leq C_2\, d(2 - α), \quad \forall α\in [α_0, 2). The optimality of these rates is rigorously verified by explicit calculations for the Ornstein-Uhlenbeck systems in \cite{Deng2023Optimal}. It is worth emphasizing that \cite{Deng2023Optimal} addressed only case (ii) under additive noise, whereas our analysis establishes results for both cases (i) and (ii) under multiplicative $α$-stable noise, employing fundamentally different analytical methods.

math.PR↗

arXiv2024

Unbiased approximation of the ergodic measure for piecewise $α$-stable Ornstein-Uhlenbeck processes arising in queueing networks

Piecewise $α$-stable Ornstein-Uhlenbeck (OU) processes arising in queue networks usually do not have an explicit dissipation, which makes the related numerical methods such as Euler-Maruyama (EM) scheme more difficult to analyze. We develop an EM scheme with decreasing step size $Λ=(η_n)_{n\in \mathbb{N}}$ to approximate their ergodic measures. This approximation does not have a bias and has a rate $η^{1/α}_n$ in Wasserstein-1 distance. We show by the classical OU process that our convergence rate is optimal. We further prove the central limit theorem (CLT) and moderate derivation principle (MDP) for the empirical measure of these piecewise $α$-stable Ornstein-Uhlenbeck processes. In addition, we use the Sinkhorn--Knopp algorithm to compute the Wasserstein-1 distance and conduct simulations for several concrete examples.

math.PR↗

arXiv2023

Approximation of the invariant measure for stable SDE by the Euler-Maruyama scheme with decreasing step-sizes

Let $(X_t)_{t \ge 0}$ be the solution of the stochastic differential equation $$dX_t = b(X_t) dt+A dZ_t, \quad X_{0}=x,$$ where $b: \mathbb{R}^d \rightarrow \mathbb R^d$ is a Lipschitz function, $A \in \mathbb R^{d \times d}$ is a positive definite matrix, $(Z_t)_{t\geq 0}$ is a $d$-dimensional rotationally invariant $α$-stable Lévy process with $α\in (1,2)$ and $x\in\mathbb{R}^{d}$. We use two Euler-Maruyama schemes with decreasing step sizes $Γ= (γ_n)_{n\in \mathbb{N}}$ to approximate the invariant measure of $(X_t)_{t \ge 0}$: one with i.i.d. $α$-stable distributed random variables as its innovations and the other with i.i.d. Pareto distributed random variables as its innovations. We study the convergence rate of these two approximation schemes in the Wasserstein-1 distance. For the first scheme, when the function $b$ is Lipschitz and satisfies a certain dissipation condition, we show that the convergence rate is $γ^{1/α}_n$. Under an additional assumption on the second order directional derivatives of $b$, this convergence rate can be improved to $γ^{1+\frac 1 α-\frac{1}κ}_n$ for any $κ\in [1,α)$. For the second scheme, when the function $b$ is twice continuously differentiable, we obtain a convergence rate of $γ^{\frac{2-α}α}_n$. We show that the rate $γ^{\frac{2-α}α}_n$ is optimal for the one dimensional stable Ornstein-Uhlenbeck process. Our theorems indicate that the recent remarkable result about the unadjusted Langevin algorithm with additive innovations can be extended to the SDEs driven by an $α$-stable Lévy process and the corresponding convergence rate has a similar behaviour. Compared with the previous result, we have relaxed the second order differentiability condition to the Lipschitz condition for the first scheme.

math.PR↗

arXiv2023

Approximation of the ergodic measure of SDEs with singular drift by Euler-Maruyama scheme

We study the approximation of the ergodic measure of the following stochastic differential equation (SDE) on $\mathbb{R}^d$: \begin{eqnarray}\label{e:SDEE} d X_t &=& (b_1(X_t)+b_2(X_t)) d t+σ(X_t) d W_t, \end{eqnarray} where $W_t$ is a $d$-dimensional standard Brownian motion, and $b_1: \mathbb{R}^d \mapsto \mathbb{R}^d$, $b_2: \mathbb{R}^d \mapsto \mathbb{R}^d$ and $σ: \mathbb{R}^d \mapsto \mathbb{R}^{d\times d}$ are the functions to be specified in Assumption 2.1 below. In particular, $b_1$ satisfies $b_1\in \mathbb{L}^\infty(\mathbb{R}^d)\cap \mathbb{L}^1(\mathbb{R}^d)$ or $b_1 \in \mathcal{C}_b^α(\mathbb{R}^d)$ with $α\in (0,1)$, which makes the standard numerical schemes not work or fail to give a good convergence rate. In order to overcome these two difficulties, we first apply a Zvonkin's transform to SDE and obtain a new SDE which has coefficients with nice properties and admits a unique ergodic measure $\widehat μ$, then discretize the new equation by Euler-Maruyama scheme to approximate $\widehat μ$, and finally use the inverse Zvonkin's transform to get an approximation of the ergodic measure of SDE, denoted by $μ$. Our approximation method is inspired by Xie and Zhang [22]. The proof of our main result is based on the method of introducing a stationary Markov chain, a key ingredient in this method is establishing the regularity of a Poisson equation, which is done by combining the classical PDE local regularity and a nice extension trick introduced by Gurvich [10].

math.PR↗

arXiv2022

An approximation to the invariant measure of the limiting diffusion of G/Ph/n+GI queues in the Halfin-Whitt regime and related asymptotics

In this paper, we develop a stochastic algorithm based on the Euler--Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of $G/Ph/n+GI$ queues in the Halfin-Whitt regime. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein's method for multi-dimensional diffusions, in which the regularity of Stein's equation developed by Gurvich (2014, 2022) plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of $G/Ph/n+GI$ queues and its Euler-Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein's equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role.

math.PR↗

arXiv2021

An approximation to steady-state of M/Ph/n+M queue

In this paper, we develop a stochastic algorithm based on Euler-Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of the $M/Ph/n+M$ queue. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion of the queueing model. Our result also provides an approximation to the steady-state of the diffusion-scaled queueing processes in the Halfin-Whitt regime given the well established interchange of limits property. To establish the error bound, we employ the recently developed Stein's method for multi-dimensional diffusions, in which the regularity of Stein's equation developed by Gurvich \cite{Gur1} plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of the $M/Ph/n+M$ queue and its Euler-Maruyama scheme. In particular, the variance of the CLT of the limiting queue is determined by using Stein's equation and Malliavin calculus.

math.PR↗

arXiv2020

A generalized Catoni's ${\rm M}$-estimator under finite {$α$-th moment assumption} with $α\in (1,2)$

We generalize the { ${\rm M}$-estimator} put forward by Catoni in his seminal paper [C12] to the case in which samples can have finite $α$-th moment with $α\in (1,2)$ rather than finite variance, our approach is by slightly modifying the influence function $φ$ therein. The choice of the new influence function is inspired by the Taylor-like expansion developed in [C-N-X]. We obtain a deviation bound of the estimator, as $α\rightarrow 2$, this bound is the same as that in [C12]. Experiment shows that our generalized ${\rm M}$-estimator performs better than the empirical mean estimator, the smaller the $α$ is, the better the performance will be. As an application, we study an $\ell_{1}$ regression considered by Zhang et al. [Z-Z] who assumed that samples have finite variance, and relax their assumption to be finite {$α$-th} moment with $α\in (1,2)$.

math.ST↗