Diancong Jin

Jialin Hong, Diancong Jin, Derui Sheng

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.

Chuchu Chen, Jialin Hong, Diancong Jin

A novel class of conservative numerical methods for general conservative Stratonovich stochastic differential equations with multiple invariants is proposed and analyzed. These methods, which are called modified averaged vector field methods, are constructed by modifying the averaged vector field methods to preserve multiple invariants simultaneously. Based on the prior estimate for high order moments of the modification coefficient, the mean square convergence order $1$ of proposed methods is proved in the case of commutative noises. In addition, the effect of quadrature formula on the mean square convergence order and the preservation of invariants for the modified averaged vector field methods is considered. Numerical experiments are performed to verify the theoretical analyses and to show the superiority of the proposed methods in long time simulation.

Xinjie Dai, Baiping Zhang, Diancong Jin

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.

Xinjie Dai, Xingyu Liu, Diancong Jin et al.

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.