Normalized and self-normalized Cramér-type moderate deviations for the Euler-Maruyama scheme for the SDE (Q6595567)

Suppose we are given the stochastic differential equation \(d{X_t} = g({X_t})dt + \sigma d{B_t},\quad X(0) = {x_0},\) where \({X_t} \in {\mathbb{R}^d}\), \({B_t}\) is a \(d\)-dimensional standard Brownian motion, \(\sigma\) is an invertible \(d \times d\) matrix, \(g:{\mathbb{R}^d} \to {\mathbb{R}^d}\) satisfies the Lipschitz condition in the Euclidean norm and some additional requirements. Given a step size \(\eta \in (0,1)\), the Euler-Maruyama scheme for this equation is given by \({\theta _{k + 1}} = {\theta _k} + \eta g({\theta _k}) + \sqrt \eta \sigma {\xi _{k + 1}},\;k \geqslant 0\), where \({({\xi _k})_{k \geqslant 1}}\) are i.i.d. standard \(d\)-dimensional normal random vectors. The authors establish normalized and self-normalized Cramér-type moderate deviations for the Euler-Maruyama scheme, which refine those obtained earlier. Using these results, they obtain Berry-Esseen's bounds and moderate deviation principles.