Longtime behaviors of $\theta$-Euler-Maruyama method for stochastic functional differential equations

• Main Authors: Chen, Chuchu, Dang, Tonghe, Hong, Jialin, Song, Guoting
• Format: Journal Article
• Language: English
• Published: 12-04-2024
• Subjects: Computer Science - Numerical Analysis; Mathematics - Numerical Analysis
• DOI: 10.48550/arxiv.2404.08891

This paper investigates longtime behaviors of the $\theta$-Euler-Maruyama
method for the stochastic functional differential equation with superlinearly
growing coefficients. We focus on the longtime convergence analysis in
mean-square sense and weak sense of the $\theta$-Euler-Maruyama method, the
convergence of the numerical invariant measure, the existence and convergence
of the numerical density function, and the Freidlin-Wentzell large deviation
principle of the method. The main contributions are outlined as follows. First,
we obtain the longtime mean-square convergence of the $\theta$-Euler-Maruyama
method and show that the mean-square convergence rate is $\frac12$. A key step
in the proof is to establish the time-independent boundedness of high-order
moments of the numerical functional solution. Second, based on the technique of
the Malliavin calculus, we present the longtime weak convergence of the
$\theta$-Euler-Maruyama method, which implies that the invariant measure of the
$\theta$-Euler-Maruyama functional solution converges to the exact one with
rate $1.$ Third, by the analysis of the test-functional-independent weak
convergence and negative moment estimates of the determinant of the
corresponding Malliavin covariance matrix, we derive the existence,
convergence, and the logarithmic estimate of the density function of the
$\theta$-Euler-Maruyama solution. At last, utilizing the weak convergence
method, we obtain the Freidlin-Wentzell large deviation principle for the
$\theta$-Euler-Maruyama solution on the infinite time horizon.