arXiv subjects

Xinjie Dai

Publications and source records attributed to Xinjie Dai.

11 recordsLinked to original sources

arXiv2026

Density functions for the overdamped generalized Langevin equation and its Euler--Maruyama method: smoothness and convergence

This paper focuses on studying the convergence rate of the density function of the Euler--Maruyama (EM) method, when applied to the overdamped generalized Langevin equation with fractional noise which serves as an important model in many fields. Firstly, we give an improved upper bound estimate for the total variation distance between random variables by their Malliavin--Sobolev norms. Secondly, we establish the existence and smoothness of the density function for both the exact solution and the numerical one. Based on the above results, the convergence rate of the density function of the numerical solution is obtained, which relies on the regularity of the noise and kernel. This convergence result provides a powerful support for numerically capturing the statistical information of the exact solution through the EM method.

math.NA↗

arXiv2026

Fundamental weak convergence theorem for stochastic Volterra integral equations and its applications

We study weak convergence rates of numerical approximations for stochastic Volterra integral equations (SVIEs), a class of non-Markovian models that arises naturally in stochastic volatility modeling and other fields. The intrinsic non-Markovian nature prevents the direct application of classical weak error techniques developed for finite-dimensional Markov processes. To overcome this difficulty, we combine a Markovian lifting technique with a domino argument, Taylor expansions, and Fréchet differential calculus for path-dependent functionals, and establish a fundamental weak convergence theorem for nonsingular SVIEs, providing a unified approach to the weak error analysis for a broad class of numerical approximations. As applications, we derive the first-order weak convergence rate for the stochastic theta method and the Wong--Zakai approximation. Our results relax existing assumptions for Euler-type schemes by removing the boundedness requirement on the diffusion coefficient. Furthermore, to the best of our knowledge, this work provides the first weak convergence result for Wong--Zakai approximations of SVIEs. Numerical experiments for a stochastic volatility model corroborate the theoretical convergence rate.

math.NA↗

arXiv2026

Splitting AVF method for generalized Langevin equations: probability density function and geometric ergodicity

The generalized Langevin equation (GLE) constitutes a fundamental model for describing nonequilibrium dynamics with memory effects. To overcome the numerical challenges arising from superquadratically growing potentials and degenerate noise, we propose and analyze a structure-preserving splitting averaged vector field (AVF) method for a quasi-Markovian GLE. The core advantage of this method lies in its ability to simultaneously preserve the exponential integrability, Malliavin differentiability, and ergodicity of the underlying continuous system. Notably, by leveraging exponential integrability, Malliavin differentiability, and uniform non-degeneracy of the numerical solution, we obtain the existence and smoothness of its probability density function, which converges to that of the exact solution with first-order accuracy. Furthermore, by validating the Lyapunov condition and the minorization condition using a localized technique, we establish the geometric ergodicity of the numerical solution. Finally, numerical experiments are conducted to confirm the theoretical results.

math.NA↗

arXiv2026

Asymptotic error distribution of Mittag--Leffler Euler method for a fractional stochastic differential equation

In this paper, we investigate the asymptotic distribution of the normalized error for the Mittag--Leffler Euler (MLE) method applied to a class of multidimensional fractional stochastic differential equations. These equations are reformulated as stochastic Volterra equations (SVEs) featuring a non-diagonal, matrix-valued kernel $K(u)=u^{α-1}E_{α,α}(Au^α)$ with singular exponent $α\in (\frac{1}{2}, 1)$. To enhance computational efficiency, the singular kernel is discretized using the left-rectangle rule, posing technical challenges for the theoretical analysis. To address this, we introduce an auxiliary $K$-undiscretized scheme to bridge the gap between the exact solution and the MLE method, integrating Jacod's stable convergence theory for conditional Gaussian martingales with methodologies developed for SVEs. To the best of our knowledge, this is the first work to establish the asymptotic error distribution for numerical methods incorporating non-diagonal matrix-valued kernels.

math.NA↗

arXiv2026

Asymptotic error distribution for tamed Euler method with coupled monotonicity condition

This paper establishes the asymptotic error distribution of the tamed Euler method for stochastic differential equations (SDEs) with a coupled monotonicity condition, that is, the limit distribution of the corresponding normalized error process. Specifically, for SDEs driven by multiplicative noise, we first propose a tamed Euler method parameterized by $α\in (0, 1]$ and establish that its strong convergence rate is $α\wedge\frac{1}{2}$. Notably, $α$ can take arbitrary positive values by adjusting the regularization coefficient without altering the strong convergence rate. We then derive the asymptotic error distribution for this tamed Euler method. Further, we infer from the limit equation that among the tamed Euler method of strong order $\frac{1}{2}$, the one with $α= \frac{1}{2}$ yields the largest mean-square error after a long time, while those of $α>\frac{1}{2}$ share a unified asymptotic error distribution. In addition, our analysis is also extended to SDEs with additive noise and similar conclusions are obtained. Additional treatments are required to accommodate super-linearly growing coefficients, a feature that distinguishes our analysis on the asymptotic error distribution from established results.

math.NA↗

arXiv2025

Strong convergence rate of the positivity-preserving logarithmic truncated EM method for multi-dimensional stochastic differential equations with positive solutions

As a combination of the logarithmic transformation with the truncated Euler-Maruyama (TEM) scheme, the positivity-preserving logarithmic truncated Euler-Maruyama (LTEM) scheme has been generally developed for scalar stochastic differential equations (SDEs) with positive solutions. A subsequent question arises: can this method be extended to effectively solve general multidimensional SDEs with positive solutions? The answer to this question is affirmative. In this paper, we construct the positivity-preserving LTEM scheme to solve this type of system and demonstrate the suboptimal strong convergence rate of this scheme. On the other hand, when the underlying system degenerates into a scalar equation, the latest LTEM scheme analyzed by Tang & Mao (2024) is applicable to scalar SDEs with weak conditions, but its strong convergence rate is suboptimal. Based on this, we will theoretically demonstrate the optimal convergence rate of the LTEM method without infinitesimal factors in the scalar case. The proof strategy exactly improves its convergence rate from suboptimal to optimal. Finally, numerical examples are provided to validate the effectiveness and positivity-preserving of the LTEM method.

math.NA↗

arXiv2025

Strong convergence rate of positivity-preserving truncated Euler--Maruyama method for multi-dimensional stochastic differential equations with positive solutions

To construct positivity-preserving numerical methods, a vast majority of existing works employ transformation techniques such as the Lamperti transformation or logarithmic transformation. However, using these techniques often leads to the transformed stochastic differential equations (SDEs) not meeting the global monotonicity condition, particularly in multi-dimension case. This condition is essential for achieving strong convergence rates of numerical schemes. A pertinent question arises from this issue regarding the existence of an effective method with a convergence rate for solving multi-dimensional SDEs with positive solutions. This paper presents a positivity-preserving method that combines a novel truncated mapping with a truncated Euler--Maruyama discretization. We investigate both the strong convergence of the numerical method under some reasonable conditions. Furthermore, we demonstrate that this method achieves the optimal strong convergence order of 1/2 under certain additional assumptions. Numerical experiments are conducted to validate these theoretical results and demonstrate the positivity of the numerical solutions.

math.NA↗

arXiv2023

Error analysis of numerical methods on graded meshes for stochastic Volterra equations

This paper presents the error analysis of numerical methods on graded meshes for stochastic Volterra equations with weakly singular kernels. We first prove a novel regularity estimate for the exact solution via analyzing the associated convolution structure. This reveals that the exact solution exhibits an initial singularity in the sense that its Hölder continuous exponent on any neighborhood of $t=0$ is lower than that on every compact subset of $(0,T]$. Motivated by the initial singularity, we then construct the Euler--Maruyama method, fast Euler--Maruyama method, and Milstein method based on graded meshes. By establishing their pointwise-in-time error estimates, we give the grading exponents of meshes to attain the optimal uniform-in-time convergence orders, where the convergence orders improve those of the uniform mesh case. Numerical experiments are finally reported to confirm the sharpness of theoretical findings.

math.NA↗

arXiv2023

Mittag--Leffler Euler integrator and large deviations for stochastic space-time fractional diffusion equations

Stochastic space-time fractional diffusion equations often appear in the modeling of the heat propagation in non-homogeneous medium. In this paper, we firstly investigate the Mittag--Leffler Euler integrator of a class of stochastic space-time fractional diffusion equations, whose super-convergence order is obtained by developing a helpful decomposition way for the time-fractional integral. Here, the developed decomposition way is the key to dealing with the singularity of the solution operator. Moreover, we study the Freidlin--Wentzell type large deviation principles of the underlying equation and its Mittag--Leffler Euler integrator based on the weak convergence approach. In particular, we prove that the large deviation rate function of the Mittag--Leffler Euler integrator $Γ$-converges to that of the underlying equation.

math.NA↗

arXiv2022

Strong error analysis of Euler methods for overdamped generalized Langevin equations with fractional noise: Nonlinear case

This paper considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the fractional Brownian motion and the singular kernel, which is overcome by means of the Malliavin calculus and fine estimates of several multiple singular integrals. Consequently, these two methods are proved to be strongly convergent with order nearly $\min\{2(H+α-1), α\}$, where $H \in (1/2,1)$ and $α\in(1-H,1)$ respectively characterize the singularity levels of fractional noises and singular kernels in the underlying equation. This result improves the existing convergence order $H+α-1$ of Euler methods for the nonlinear case, and gives a positive answer to the open problem raised in [4]. As an application of the theoretical findings, we further investigate the complexity of the multilevel Monte Carlo simulation based on the fast Euler method, which turns out to behave better performance than the standard Monte Carlo simulation when computing the expectation of functionals of the considered equation.

math.NA↗

arXiv2021

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler--Maruyama approximation

This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler--Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler--Maruyama method can reach strong first-order superconvergence when $α= 1$. Finally, several numerical tests are reported for verification of the theoretical findings.

math.NA↗