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Diancong Jin

Publications and source records attributed to Diancong Jin.

At least 19 recordsLinked to original sources
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.

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arXiv2026

A uniform-in-time weakly convergent explicit numerical method for the underdamped Langevin equation with polynomial potentials

The underdamped Langevin equation is a fundamental model in statistical mechanics for sampling Gibbs measures and simulating molecular dynamics, for which numerical methods with uniform-in-time weak convergence are essential for accurately reproducing long-time statistical observables and invariant measures of the underlying dynamics. Currently, such uniform-in-time weak convergence is established for implicit schemes, but remains unknown for explicit ones under polynomially growing potentials. To improve efficiency in long-time simulations, we propose the first explicit numerical method for the underdamped Langevin equation with polynomially growing potentials that is proven to achieve uniform-in-time weak convergence. The explicit numerical method is constructed by introducing a dissipativity on the scalar auxiliary variable (SAV), which we call the DSAV method. The proposed DSAV method enables the approximation of the invariant measure for the underdamped Langevin equation with a precision of $\varepsilon$ at a significantly reduced computational cost of $\mathcal{O}(\varepsilon^{-1} \log(\varepsilon^{-1}))$. In addition, we establish the existence and positivity of the density function of the numerical solution without using the Malliavin calculus. Numerical experiments are performed to verify the theoretical findings and demonstrate the long-time stability of the proposed numerical method.

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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.

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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.

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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.

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arXiv2025

Asymptotic error distribution of numerical methods for parabolic SPDEs with multiplicative noise

This paper aims to investigate the asymptotic error distribution of several numerical methods for stochastic partial differential equations (SPDEs) with multiplicative noise. Firstly, we give the limit distribution of the normalized error process of the exponential Euler method in $\dot{H}^η$ for some $η>0$. A key finding is that the asymptotic error in distribution of the exponential Euler method is governed by a linear SPDE driven by infinitely many independent $Q$-Wiener processes. This characteristic represents a significant difference from numerical methods for both stochastic ordinary differential equations and SPDEs with additive noise. Secondly, as applications of the above result, we derive the asymptotic error distribution of a full discretization based on the temporal exponential Euler method and the spatial finite element method. As a concrete illustration, we provide the pointwise limit distribution of the normalized error process when the exponential Euler method is applied to a specific class of stochastic heat equations. Finally, by studying the asymptotic error of the spatial semi-discrete spectral Galerkin method, we demonstrate that the actual strong convergence speed of spatial semi-discrete numerical methods may be highly problem-dependent, rather than universally predictable.

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arXiv2025

Asymptotic error distribution for stochastic Runge--Kutta methods of strong order one

This work gives the asymptotic error distribution of the stochastic Runge--Kutta (SRK) method of strong order $1$ applied to Stratonovich-type stochastic differential equations. For dealing with the implicitness introduced in the diffusion term, we propose a framework to derive the asymptotic error distribution of diffusion-implicit or fully implicit numerical methods, which enables us to construct a fully explicit numerical method sharing the same asymptotic error distribution as the SRK method. Further, we show that the limit distribution $U(T)$ satisfies $\mathbf E|U(T)|^2\le e^{L_1T}(1+η_1)T^3$ for some $η_1\ge0$ only depending on the coefficients of the SRK method. Thus, we infer that $η_1$ is the key parameter reflecting the growth rate of the mean-square error of the SRK method. Especially, among the SRK methods of strong order $1$, those of weak order $2$ correspond to $η_1=0$, sharing the unified asymptotic error distribution, and have the smallest mean-square errors after a long time. This property is also found for the case of additive noise. It seems that we are the first to give the asymptotic error distribution of fully implicit numerical methods for stochastic differential equations.

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arXiv2024

Asymptotic error distribution of accelerated exponential Euler method for parabolic SPDEs

The asymptotic error distribution of numerical methods applied to stochastic ordinary differential equations has been well studied, which characterizes the evolution pattern of the error distribution in the small step-size regime. It is still open for stochastic partial differential equations whether the normalized error process of numerical methods admits a nontrivial limit distribution. We answer this question by presenting the asymptotic error distribution of the temporal accelerated exponential Euler (AEE) method when applied to parabolic stochastic partial differential equations. In order to overcome the difficulty caused by the infinite-dimensional setting, we establish a uniform approximation theorem for convergence in distribution. Based on it, we derive the limit distribution of the normalized error process of the AEE method by studying the limit distribution of its certain appropriate finite-dimensional approximation process. As applications of our main result, the asymptotic error distribution of a fully discrete AEE method for the original equation and that of the AEE method for a stochastic ordinary differential equation are also obtained.

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arXiv2024

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.

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arXiv2023

Convergence analysis for minimum action methods coupled with a finite difference method

The minimum action method (MAM) is an effective approach to numerically solving minimums and minimizers of Freidlin--Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are $1/2$ and $1$, respectively. Our main result also reveals the convergence of the stochastic $θ$-method for SDEs with small noise in terms of large deviations.

math.PR↗
arXiv2023

Central limit theorem for temporal average of backward Euler--Maruyama method

This work focuses on the temporal average of the backward Euler--Maruyama (BEM) method, which is used to approximate the ergodic limit of stochastic ordinary differential equations with super-linearly growing drift coefficients. We give the central limit theorem (CLT) of the temporal average, which characterizes the asymptotics in distribution of the temporal average. When the deviation order is smaller than the optimal strong order, we directly derive the CLT of the temporal average through that of original equations and the uniform strong order of the BEM method. For the case that the deviation order equals to the optimal strong order, the CLT is established via the Poisson equation associated with the generator of original equations. Numerical experiments are performed to illustrate the theoretical results.

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arXiv2023

Asymptotics of large deviations of finite difference method for stochastic Cahn--Hilliard equation

In this work, we establish the Freidlin--Wentzell large deviations principle (LDP) of the stochastic Cahn--Hilliard equation with small noise, which implies the one-point LDP. Further, we give the one-point LDP of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation. Our main result is the convergence of the one-point large deviations rate function (LDRF) of the spatial FDM, which is about the asymptotical limit of a parametric variational problem. The main idea for proving the convergence of the LDRF of the spatial FDM is via the $Γ$-convergence of objective functions, which relies on the qualitative analysis of skeleton equations of the original equation and the numerical method. In order to overcome the difficulty that the drift coefficient is not one-side Lipschitz, we use the equivalent characterization of the skeleton equation of the spatial FDM and the discrete interpolation inequality to obtain the uniform boundedness of the solution to the underlying skeleton equation. This plays an important role in deriving the $Γ$-convergence of objective functions.

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arXiv2022

Scheme-Induced Minimum Action Methods for SPDEs: A Variational Framework for Convergence Analysis

The minimum action method (MAM) is an important numerical tool for computing the most probable transition paths and the leading exponential rates of rare-event probabilities for stochastic systems with small noise. To the best of our knowledge, no rigorous convergence analysis is available for fully discrete MAMs for stochastic partial differential equations (SPDEs). This paper establishes the convergence of the constrained minima of fully discrete scheme-induced action functionals for terminal observations of scalar semilinear SPDEs. A main difficulty is that a consistent discrete approximation of an admissible control generally fails to satisfy the discrete terminal constraint exactly. To overcome this difficulty, we construct an asymptotically negligible correction that restores the constraint, and incorporate this argument into an abstract variational framework. We then propose a criterion that reduces the assumptions of this framework to verifiable conditions involving the continuous and discrete Green functions. Applying this criterion to fully discrete schemes for a stochastic wave equation and a fourth-order parabolic equation, we prove convergence of the constrained minima along arbitrary spatial and temporal refinement sequences, without a mesh-ratio condition.

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arXiv2022

Convergence analysis of a finite difference method for stochastic Cahn--Hilliard equation

This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green function, we prove that both the spatial semi-discrete numerical solution and its Malliavin derivative have strong convergence order $1$. Further, by showing the negative moment estimates of the exact solution, we obtain that the density of the spatial semi-discrete numerical solution converges in $L^1(\mathbb R)$ to the exact one. Finally, we apply an exponential Euler method to discretize the spatial semi-discrete numerical solution in time and show that the temporal strong convergence order is nearly $\frac38$, where a difficulty we overcome is to derive the optimal Hölder continuity of the spatial semi-discrete numerical solution.

math.NA↗
arXiv2022

Large deviations principle for stochastic delay differential equations with super-linearly growing coefficients

We utilize the weak convergence method to establish the Freidlin--Wentzell large deviations principle (LDP) for stochastic delay differential equations (SDDEs) with super-linearly growing coefficients, which covers a large class of cases with non-globally Lipschitz coefficients. The key ingredient in our proof is the uniform moment estimate of the controlled equation, where we handle the super-linear growth of the coefficients by an iterative argument. Our results allow both the drift and diffusion coefficients of the considered equations to super-linearly grow not only with respect to the delay variable but also to the state variable. This work extends the existing results which develop the LDPs for SDDEs with super-linearly growing coefficients only with respect to the delay variable.

math.PR↗
arXiv2021

Large deviations principles of sample paths and invariant measures of numerical methods for parabolic SPDEs

For parabolic stochastic partial differential equations (SPDEs), we show that the numerical methods, including the spatial spectral Galerkin method and further the full discretization via the temporal accelerated exponential Euler method, satisfy the uniform sample path large deviations. Combining the exponential tail estimate of invariant measures, we establish the large deviations principles (LDPs) of invariant measures of these numerical methods. Based on the error estimate between the rate function of the considered numerical methods and that of the original equation, we prove that these numerical methods can weakly asymptotically preserve the LDPs of sample paths and invariant measures of the original equation. This work provides an approach to proving the weakly asymptotical preservation for the above two LDPs for SPDEs with small noise via numerical methods, by means of the minimization sequences.

math.NA↗
arXiv2020

Numerically asymptotical preservation of the large deviations principles for invariant measures of Langevin equations

In this paper, we focus on two kinds of large deviations principles (LDPs) of the invariant measures of Langevin equations and their numerical methods, as the noise intensity $ε\to 0$ and the dissipation intensity $ν\to\infty$ respectively. First, by proving the weak LDP and the exponential tightness, we conclude that the invariant measure $\{μ_{ν,ε}\}$ of the exact solution satisfies the LDPs as $ε\to0$ and $ν\to\infty$ respectively. Then, we study whether there exist numerical methods asymptotically preserving these two LDPs of $\{μ_{ν,ε}\}$ in the sense that the rate functions of invariant measures of numerical methods converge pointwise to the rate function of $\{μ_{ν,ε}\}$ as the step-size tends to zero. The answer is positive for the linear Langevin equation. For the small noise case, we show that a large class of numerical methods can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ε>0}$ as $ε\to0$. For the strong dissipation case, we study the stochastic $θ$-method ($θ\in[1/2,1]$) and show that only the midpoint scheme ($θ=1/2$) can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ν>0}$ as $ν\to\infty$. These results indicate that in the linear case, the LDP as $ε\to0$ and the LDP as $ν\to\infty$ for the invariant measures of numerical methods have intrinsic differences: the common numerical methods can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ε>0}$ as $ε\to0$ while the asymptotical preservation of numerical methods for the LDP of $\{μ_{ν,ε}\}_{ν>0}$ as $ν\to\infty$ depends on the choice of numerical methods. To the best of our knowledge, this is the first result of investigating the relationship between the LDPs of invariant measures of stochastic differential equations and those of their numerical methods.

math.NA↗
arXiv2020

Large deviations principles for symplectic discretizations of stochastic linear Schrödinger Equation

In this paper, we consider the large deviations principles (LDPs) for the stochastic linear Schrödinger equation and its symplectic discretizations. These numerical discretizations are the spatial semi-discretization based on spectral Galerkin method, and the further full discretizations with symplectic schemes in temporal direction. First, by means of the abstract Gärtner--Ellis theorem, we prove that the observable $B_T=\frac{u(T)}{T}$, $T>0$ of the exact solution $u$ is exponentially tight and satisfies an LDP on $L^2(0, π; \mathbb C)$. Then, we present the LDPs for both $\{B^M_T\}_{T>0}$ of the spatial discretization $\{u^M\}_{M\in\mathbb N}$ and $\{B^M_N\}_{N\in \mathbb N}$ of the full discretization $\{u^M_N\}_{M,N\in\mathbb N}$, where $B^M_T=\frac{u^M(T)}{T}$ and $B^M_N=\frac{u^M_N}{Nτ}$ are the discrete approximations of $B_T$. Further, we show that both the semi-discretization $\{u^M\}_{M\in \mathbb N}$ and the full discretization $\{u^M_N\}_{M,N\in \mathbb N}$ based on temporal symplectic schemes can weakly asymptotically preserve the LDP of $\{B_T\}_{T>0}$. These results show the ability of symplectic discretizations to preserve the LDP of the stochastic linear \xde equation, and first provide an effective approach to approximating the LDP rate function in infinite dimensional space based on the numerical discretizations.

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